%------------------------------------------------------------------------------
% File     : ITP217_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Time_Reasoning 00330_012302
% 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    : 0029_Time_Reasoning_00330_012302 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11936 (4807 unt;1916 typ;   0 def)
%            Number of atoms       : 14986 (8563 equ)
%            Maximal formula atoms :   38 (   1 avg)
%            Number of connectives : 16747 (1831   ~; 287   |;1290   &)
%                                         (2091 <=>;11248  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (   6 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       :  822 ( 581 fml; 241 var)
%            Number of X terms     :  511 (   0  []; 480 ite;  31 let)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  : 1665 (1364   >; 301   *;   0   +;   0  <<)
%            Number of predicates  :  218 ( 215 usr;   3 prp; 0-7 aty)
%            Number of functors    : 1708 (1708 usr;  77 con; 0-7 aty)
%            Number of variables   : 35552 (31862   !; 578   ?;35552   :)
%                                         (3112  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 16:31:57.365
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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_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_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_c,type,
    c: $tType ).

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

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

% Explicit typings (1893)
tff(sy_cl_Lattices_Obounded__lattice__top,type,
    bounded_lattice_top: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Lattices_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_Rings_Ocomm__semiring__1__cancel,type,
    comm_s4317794764714335236cancel: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
    condit1219197933456340205attice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__cy____,type,
    aTP_Lamp_cy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cz____,type,
    aTP_Lamp_cz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__da____,type,
    aTP_Lamp_da: 
      !>[A: $tType,B: $tType] : 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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__dq____,type,
    aTP_Lamp_dq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dr____,type,
    aTP_Lamp_dr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ds____,type,
    aTP_Lamp_ds: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dt____,type,
    aTP_Lamp_dt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__fi____,type,
    aTP_Lamp_fi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fj____,type,
    aTP_Lamp_fj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__fl____,type,
    aTP_Lamp_fl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fm____,type,
    aTP_Lamp_fm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fn____,type,
    aTP_Lamp_fn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ld____,type,
    aTP_Lamp_ld: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lw____,type,
    aTP_Lamp_lw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lx____,type,
    aTP_Lamp_lx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

tff(sy_c_ATP_058Lamp__mc____,type,
    aTP_Lamp_mc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__md____,type,
    aTP_Lamp_md: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__me____,type,
    aTP_Lamp_me: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mf____,type,
    aTP_Lamp_mf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mg____,type,
    aTP_Lamp_mg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__wo____,type,
    aTP_Lamp_wo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wp____,type,
    aTP_Lamp_wp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wq____,type,
    aTP_Lamp_wq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wr____,type,
    aTP_Lamp_wr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ws____,type,
    aTP_Lamp_ws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wt____,type,
    aTP_Lamp_wt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xz____,type,
    aTP_Lamp_xz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ya____,type,
    aTP_Lamp_ya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yb____,type,
    aTP_Lamp_yb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yc____,type,
    aTP_Lamp_yc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yd____,type,
    aTP_Lamp_yd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ye____,type,
    aTP_Lamp_ye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yf____,type,
    aTP_Lamp_yf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Array__Time_Oalloc,type,
    array_alloc: 
      !>[A: $tType] : ( ( list(A) * heap_ext(product_unit) ) > product_prod(array(A),heap_ext(product_unit)) ) ).

tff(sy_c_Array__Time_Ofreeze,type,
    array_freeze: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_Array__Time_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_Omake_H,type,
    array_make2: 
      !>[B: $tType,A: $tType] : ( ( code_integer * fun(B,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_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_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

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: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,$o)) ) ).

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(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C)))) ).

tff(sy_c_BNF__Def_Oeq__onp,type,
    bNF_eq_onp: 
      !>[A: $tType] : ( fun(A,$o) > fun(A,fun(A,$o)) ) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(fun(A,B),fun(fun(C,D),$o)) ) ).

tff(sy_c_BNF__Def_Orel__set,type,
    bNF_rel_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(set(A),fun(set(B),$o)) ) ).

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(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_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: $tType] : ( ( fun(C,A) * fun(D,B) * fun(C,fun(D,$o)) ) > fun(A,fun(B,$o)) ) ).

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_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_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_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: $tType] : ( fun(A,$o) > fun(fun(B,$o),fun(fun(A,B),$o)) ) ).

tff(sy_c_Basic__BNFs_Opred__prod,type,
    basic_pred_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(B,$o) ) > fun(product_prod(A,B),$o) ) ).

tff(sy_c_Basic__BNFs_Orel__prod,type,
    basic_rel_prod: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,$o)) * fun(C,fun(D,$o)) ) > fun(product_prod(A,C),fun(product_prod(B,D),$o)) ) ).

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),$o)) ).

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),$o)) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : fun(A,fun(nat,$o)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__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),$o)) ).

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_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_Code__Numeral_ONat,type,
    code_Nat: fun(code_integer,code_natural) ).

tff(sy_c_Code__Numeral_ONeg,type,
    code_Neg: num > code_integer ).

tff(sy_c_Code__Numeral_OSuc,type,
    code_Suc: fun(code_natural,code_natural) ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: 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__nat,type,
    code_integer_of_nat: fun(nat,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__natural,type,
    code_i5400310926305786745atural: fun(code_natural,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: fun(code_integer,nat) ).

tff(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
    code_case_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,T) ) > fun(code_natural,T) ) ).

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_Onatural_Orec__natural,type,
    code_rec_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) ) > fun(code_natural,T) ) ).

tff(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
    code_rec_set_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) * code_natural ) > fun(T,$o) ) ).

tff(sy_c_Code__Numeral_Onatural__of__integer,type,
    code_n4118661773612635043nteger: fun(code_integer,code_natural) ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: fun(code_integer,num) ).

tff(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: fun(int,fun(code_integer,$o)) ).

tff(sy_c_Code__Numeral_Opositive,type,
    code_positive: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Osize__nat,type,
    code_size_nat: fun(code_natural,nat) ).

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: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,$o) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
    comple7512665784863727008ratesp: 
      !>[A: $tType] : ( fun(A,A) > fun(A,$o) ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
    condit8047198070973881523_above: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
    condit8119078960628432327_below: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

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_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_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] : fun(A,nat) ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
    euclid7384307370059645450egment: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilter_OAbs__filter,type,
    abs_filter: 
      !>[A: $tType] : ( fun(fun(A,$o),$o) > 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: $tType] : ( ( fun(A,fun(B,$o)) * filter(A) * filter(B) ) > $o ) ).

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_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),$o) ).

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

tff(sy_c_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(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_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

tff(sy_c_Fun__Def_Ois__measure,type,
    fun_is_measure: 
      !>[A: $tType] : ( fun(A,nat) > $o ) ).

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_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_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
    bounde6485984586167503788ce_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
    bounde2362111253966948842tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__class_Olcm,type,
    gcd_lcm: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : fun(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(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(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_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_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: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_HOL_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,$o) > $o ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : 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_Oassert,type,
    heap_Time_assert: 
      !>[A: $tType] : ( ( fun(A,$o) * A ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Obind,type,
    heap_Time_bind: 
      !>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).

tff(sy_c_Heap__Time__Monad_Oeffect,type,
    heap_Time_effect: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Oexecute,type,
    heap_Time_execute: 
      !>[A: $tType] : ( heap_Time_Heap(A) > fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

tff(sy_c_Heap__Time__Monad_Oguard,type,
    heap_Time_guard: 
      !>[A: $tType] : ( ( fun(heap_ext(product_unit),$o) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oheap,type,
    heap_Time_heap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Olift,type,
    heap_Time_lift: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Time_Heap(B)) ) ).

tff(sy_c_Heap__Time__Monad_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: $tType] : fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o)) ).

tff(sy_c_Heap__Time__Monad_Oureturn,type,
    heap_Time_ureturn: 
      !>[A: $tType] : fun(A,heap_Time_Heap(A)) ).

tff(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_Time_Heap(product_unit) ).

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : fun($o,fun(A,fun(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_Ocr__int,type,
    cr_int: fun(product_prod(nat,nat),fun(int,$o)) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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,$o)) ).

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: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_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: $tType] : ( ( fun(B,A) * fun(B,$o) ) > 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: $tType] : ( ( fun(B,A) * fun(B,$o) ) > 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: $tType] : ( ( fun(B,A) * fun(B,$o) ) > fun(B,$o) ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) ) > fun(B,$o) ) ).

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: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $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: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,B) * fun(B,A) * fun(A,fun(B,$o)) ) > $o ) ).

tff(sy_c_Lifting_Orel__pred__comp,type,
    rel_pred_comp: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,$o) * A ) > $o ) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : 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: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),$o)) ).

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] : fun(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: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * 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: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * 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 > fun(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: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * 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: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

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: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

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_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oord_Olexordp,type,
    lexordp2: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),fun(list(A),$o)) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

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_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),$o)) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_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: $tType] : ( fun(A,$o) > 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: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod($o,A),product_prod($o,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: $tType] : fun(fun(A,$o),fun(list(A),list(A))) ).

tff(sy_c_Misc_Ofilter__rev__aux,type,
    filter_rev_aux: 
      !>[A: $tType] : ( list(A) > fun(fun(A,$o),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: $tType] : fun(product_prod(list(list(A)),list(list(A))),fun(product_prod(list(list(A)),list(list(A))),$o)) ).

tff(sy_c_Misc_Omerge__rel,type,
    merge_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort,type,
    mergesort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Omergesort__by__rel,type,
    mergesort_by_rel: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
    merges2244889521215249637ge_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__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: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),$o)) ).

tff(sy_c_Misc_Opartition__rev,type,
    partition_rev: 
      !>[A: $tType] : ( ( fun(A,$o) * 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: $tType] : fun(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o)) ).

tff(sy_c_Misc_Oquicksort__by__rel,type,
    quicksort_by_rel: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Oquicksort__by__rel__rel,type,
    quicksort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_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: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

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: $tType] : fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),$o)) ).

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_Ocomm__monoid__mult__class_Oprod__mset,type,
    comm_m9189036328036947845d_mset: 
      !>[A: $tType] : ( multiset(A) > A ) ).

tff(sy_c_Multiset_Ofilter__mset,type,
    filter_mset: 
      !>[A: $tType] : fun(fun(A,$o),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_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: $tType] : ( ( fun(A,fun(A,$o)) * multiset(A) * multiset(A) ) > $o ) ).

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: $tType] : ( fun(A,fun(A,$o)) > fun(multiset(A),fun(multiset(A),$o)) ) ).

tff(sy_c_Multiset_Omultp__code,type,
    multp_code: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * multiset(A) * multiset(A) ) > $o ) ).

tff(sy_c_Multiset_Opcr__multiset,type,
    pcr_multiset: 
      !>[C: $tType,B: $tType] : ( fun(C,fun(B,$o)) > fun(fun(C,nat),fun(multiset(B),$o)) ) ).

tff(sy_c_Multiset_Orel__mset,type,
    rel_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(multiset(A),fun(multiset(B),$o)) ) ).

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: $tType] : ( ( list(A) * list(A) ) > option($o) ) ).

tff(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
    subset751672762298770561pl_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_Multiset_Osubset__mset,type,
    subset_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).

tff(sy_c_Multiset_Osubseteq__mset,type,
    subseteq_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).

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] : fun(T,fun(fun(nat,fun(T,T)),fun(nat,T))) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

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,$o)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list(nat) > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: 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,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_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,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))) ).

tff(sy_c_Num_Onum_Orec__num,type,
    rec_num: 
      !>[A: $tType] : fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(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: fun(nat,num) ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Num_Osqr,type,
    sqr: num > num ).

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_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),A) ).

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

tff(sy_c_Order__Relation_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder_OGreatest,type,
    greatest: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).

tff(sy_c_Orderings_Oorder_Omono,type,
    mono: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,B),$o) ) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > 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: $tType] : fun(fun(A,B),$o) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * 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_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : fun(A,fun(nat,A)) ).

tff(sy_c_Predicate_Oiterate__upto__rel,type,
    iterate_upto_rel: 
      !>[A: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o)) ).

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_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_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_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_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_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: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o)) ).

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),$o)) ).

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_OFrct,type,
    frct: product_prod(int,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_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,$o) ).

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),$o)) ).

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_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_Refine__Imp__Hol_Oassert_H,type,
    refine_Imp_assert: $o > heap_Time_Heap(product_unit) ).

tff(sy_c_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $o ) ).

tff(sy_c_Relation_ODomainp,type,
    domainp: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,$o)),fun(A,$o)) ).

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_Oasymp,type,
    asymp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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: $tType] : ( fun(A,fun(B,$o)) > fun(B,fun(A,$o)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oirreflp,type,
    irreflp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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: $tType] : fun(fun(A,fun(B,$o)),fun(fun(B,fun(C,$o)),fun(A,fun(C,$o)))) ).

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: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Relation_Otrans,type,
    trans: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $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: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun($o,A) ).

tff(sy_c_Set_OBall,type,
    ball: 
      !>[A: $tType] : fun(set(A),fun(fun(A,$o),$o)) ).

tff(sy_c_Set_OBex,type,
    bex: 
      !>[A: $tType] : fun(set(A),fun(fun(A,$o),$o)) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : ( A > fun(set(A),set(A)) ) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set_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: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

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: ( $o * $o * $o * $o * $o * $o * $o ) > fun($o,char) ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Time__Reasoning_OEQ,type,
    time_EQ: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Time__Reasoning_OTBOUND,type,
    time_TBOUND: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Ofails,type,
    time_fails: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

tff(sy_c_Time__Reasoning_Othe__heap,type,
    time_the_heap: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

tff(sy_c_Time__Reasoning_Othe__res,type,
    time_the_res: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > A ) ).

tff(sy_c_Time__Reasoning_Otime,type,
    time_time: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > nat ) ).

tff(sy_c_Transfer_Obi__total,type,
    bi_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Obi__unique,type,
    bi_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Oleft__total,type,
    left_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Oleft__unique,type,
    left_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Oright__total,type,
    right_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Otransfer__bforall,type,
    transfer_bforall: 
      !>[A: $tType] : ( ( fun(A,$o) * fun(A,$o) ) > $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: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

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: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),fun(set(A),$o)) ) ).

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: $tType] : ( ( fun(A,$o) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : fun(fun(A,$o),$o) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fNot,type,
    fNot: fun($o,$o) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > fun(set(A),$o) ) ).

tff(sy_v_P,type,
    p: $o ).

tff(sy_v_h,type,
    h: heap_ext(product_unit) ).

tff(sy_v_m,type,
    m: heap_Time_Heap(a) ).

tff(sy_v_t,type,
    t: nat ).

% Relevant facts (9331)
tff(fact_0__092_060open_062_092_060And_062h_O_AP_A_092_060Longrightarrow_062_Atime_Am_Ah_A_092_060le_062_At_092_060close_062,axiom,
    ! [H: heap_ext(product_unit)] :
      ( p
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(a,m,H)),t) ) ).

% \<open>\<And>h. P \<Longrightarrow> time m h \<le> t\<close>
tff(fact_1_assms,axiom,
    ( p
   => time_TBOUND(a,m,t) ) ).

% assms
tff(fact_2_Heap__Time__Monad_Obind__bind,axiom,
    ! [D: $tType,B: $tType,C: $tType,F: heap_Time_Heap(D),G: fun(D,heap_Time_Heap(C)),K: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,heap_Time_bind(D,C,F,G),K) = heap_Time_bind(D,B,F,aa(fun(C,heap_Time_Heap(B)),fun(D,heap_Time_Heap(B)),aTP_Lamp_aa(fun(D,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(D,heap_Time_Heap(B))),G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_3_order__refl,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),X) ) ).

% order_refl
tff(fact_4_dual__order_Orefl,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),A2) ) ).

% dual_order.refl
tff(fact_5_distrib__if__bind,axiom,
    ! [B: $tType,C: $tType,B2: $o,C2: heap_Time_Heap(C),D2: heap_Time_Heap(C),F: fun(C,heap_Time_Heap(B))] :
      heap_Time_bind(C,B,
        $ite((B2),C2,D2),
        F) = $ite((B2),heap_Time_bind(C,B,C2,F),heap_Time_bind(C,B,D2,F)) ).

% distrib_if_bind
tff(fact_6_TBOUNDD,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),Ta: nat,H: heap_ext(product_unit)] :
      ( time_TBOUND(B,Ma,Ta)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(B,Ma,H)),Ta) ) ).

% TBOUNDD
tff(fact_7_TBOUNDI,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),Ta: nat] :
      ( ! [H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(B,Ma,H2)),Ta)
     => time_TBOUND(B,Ma,Ta) ) ).

% TBOUNDI
tff(fact_8_TBOUND__def,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),Ta: nat] :
      ( time_TBOUND(B,Ma,Ta)
    <=> ! [H3: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(B,Ma,H3)),Ta) ) ).

% TBOUND_def
tff(fact_9_le__refl,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N) ).

% le_refl
tff(fact_10_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_11_eq__imp__le,axiom,
    ! [Ma: nat,N: nat] :
      ( ( Ma = N )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% eq_imp_le
tff(fact_12_le__antisym,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
       => ( Ma = N ) ) ) ).

% le_antisym
tff(fact_13_nat__le__linear,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma) ) ).

% nat_le_linear
tff(fact_14_TBOUND__mono,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),Ta: nat,T2: nat] :
      ( time_TBOUND(B,C2,Ta)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ta),T2)
       => time_TBOUND(B,C2,T2) ) ) ).

% TBOUND_mono
tff(fact_15_TBOUND__eqI,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),Ta: nat] :
      ( ! [H2: heap_ext(product_unit)] : time_time(B,Ma,H2) = Ta
     => time_TBOUND(B,Ma,Ta) ) ).

% TBOUND_eqI
tff(fact_16_order__antisym__conv,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_17_linorder__le__cases,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% linorder_le_cases
tff(fact_18_ord__le__eq__subst,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ord(C)
        & ord(B) )
     => ! [A2: B,B2: B,F: fun(B,C),C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( ( aa(B,C,F,B2) = C2 )
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_19_ord__eq__le__subst,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ord(C)
        & ord(B) )
     => ! [A2: B,F: fun(C,B),B2: C,C2: C] :
          ( ( A2 = aa(C,B,F,B2) )
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),B2),C2)
           => ( ! [X2: C,Y2: C] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X2)),aa(C,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(C,B,F,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_20_linorder__linear,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
          | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% linorder_linear
tff(fact_21_order__eq__refl,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( ( X = Y )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% order_eq_refl
tff(fact_22_order__subst2,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,B2: B,F: fun(B,C),C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,B2)),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_23_order__subst1,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,F: fun(C,B),B2: C,C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(C,B,F,B2))
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),B2),C2)
           => ( ! [X2: C,Y2: C] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X2)),aa(C,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(C,B,F,C2)) ) ) ) ) ).

% order_subst1
tff(fact_24_Orderings_Oorder__eq__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = B2 )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_25_le__fun__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),G: fun(B,C)] :
          ( aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less_eq(fun(B,C)),F),G)
        <=> ! [X3: B] : aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X3)),aa(B,C,G,X3)) ) ) ).

% le_fun_def
tff(fact_26_le__funI,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),G: fun(B,C)] :
          ( ! [X2: B] : aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2))
         => aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less_eq(fun(B,C)),F),G) ) ) ).

% le_funI
tff(fact_27_le__funE,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),G: fun(B,C),X: B] :
          ( aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less_eq(fun(B,C)),F),G)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,G,X)) ) ) ).

% le_funE
tff(fact_28_le__funD,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),G: fun(B,C),X: B] :
          ( aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less_eq(fun(B,C)),F),G)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,G,X)) ) ) ).

% le_funD
tff(fact_29_antisym,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_30_dual__order_Otrans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2) ) ) ) ).

% dual_order.trans
tff(fact_31_dual__order_Oantisym,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_32_dual__order_Oeq__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = B2 )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_33_linorder__wlog,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,fun(B,$o)),A2: B,B2: B] :
          ( ! [A3: B,B3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B3)
             => aa(B,$o,aa(B,fun(B,$o),Pa,A3),B3) )
         => ( ! [A3: B,B3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),Pa,B3),A3)
               => aa(B,$o,aa(B,fun(B,$o),Pa,A3),B3) )
           => aa(B,$o,aa(B,fun(B,$o),Pa,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_34_order__trans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z) ) ) ) ).

% order_trans
tff(fact_35_order_Otrans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% order.trans
tff(fact_36_order__antisym,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_37_ord__le__eq__trans,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( ( B2 = C2 )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_38_ord__eq__le__trans,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 = B2 )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_39_order__class_Oorder__eq__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( ( X = Y )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ) ).

% order_class.order_eq_iff
tff(fact_40_le__cases3,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) )
         => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z) )
           => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z)
               => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),Y) )
             => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),Y)
                 => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) )
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z)
                   => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X) )
                 => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X)
                     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_41_nle__le,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_42_Nat_Oex__has__greatest__nat,axiom,
    ! [Pa: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,Pa,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),B2) )
       => ? [X2: nat] :
            ( aa(nat,$o,Pa,X2)
            & ! [Y3: nat] :
                ( aa(nat,$o,Pa,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_43_raise__bind,axiom,
    ! [C: $tType,B: $tType,E2: list(char),F: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,heap_Time_raise(C,E2),F) = heap_Time_raise(B,E2) ).

% raise_bind
tff(fact_44_Greatest__equality,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o),X: B] :
          ( aa(B,$o,Pa,X)
         => ( ! [Y2: B] :
                ( aa(B,$o,Pa,Y2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X) )
           => ( order_Greatest(B,Pa) = X ) ) ) ) ).

% Greatest_equality
tff(fact_45_mem__Collect__eq,axiom,
    ! [B: $tType,A2: B,Pa: fun(B,$o)] :
      ( aa(set(B),$o,member(B,A2),aa(fun(B,$o),set(B),collect(B),Pa))
    <=> aa(B,$o,Pa,A2) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [B: $tType,A4: set(B)] : aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4)) = A4 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] :
      ( ! [X2: B] :
          ( aa(B,$o,Pa,X2)
        <=> aa(B,$o,Q,X2) )
     => ( aa(fun(B,$o),set(B),collect(B),Pa) = aa(fun(B,$o),set(B),collect(B),Q) ) ) ).

% Collect_cong
tff(fact_48_ext,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),G: fun(B,C)] :
      ( ! [X2: B] : aa(B,C,F,X2) = aa(B,C,G,X2)
     => ( F = G ) ) ).

% ext
tff(fact_49_GreatestI2__order,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o),X: B,Q: fun(B,$o)] :
          ( aa(B,$o,Pa,X)
         => ( ! [Y2: B] :
                ( aa(B,$o,Pa,Y2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X) )
           => ( ! [X2: B] :
                  ( aa(B,$o,Pa,X2)
                 => ( ! [Y3: B] :
                        ( aa(B,$o,Pa,Y3)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),X2) )
                   => aa(B,$o,Q,X2) ) )
             => aa(B,$o,Q,order_Greatest(B,Pa)) ) ) ) ) ).

% GreatestI2_order
tff(fact_50_TBOUND__assert_H__bind,axiom,
    ! [B: $tType,Pa: $o,P: $o,Ma: heap_Time_Heap(B),Ta: nat] :
      ( time_EQ($o,(Pa),(P))
     => ( ( (Pa)
         => time_TBOUND(B,Ma,Ta) )
       => time_TBOUND(B,heap_Time_bind(product_unit,B,refine_Imp_assert((Pa)),aTP_Lamp_ac(heap_Time_Heap(B),fun(product_unit,heap_Time_Heap(B)),Ma)),
            $ite((P),Ta,zero_zero(nat))) ) ) ).

% TBOUND_assert'_bind
tff(fact_51_ex__has__least__nat,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,Ma: fun(B,nat)] :
      ( aa(B,$o,Pa,K)
     => ? [X2: B] :
          ( aa(B,$o,Pa,X2)
          & ! [Y3: B] :
              ( aa(B,$o,Pa,Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,Ma,X2)),aa(B,nat,Ma,Y3)) ) ) ) ).

% ex_has_least_nat
tff(fact_52_bounded__Max__nat,axiom,
    ! [Pa: fun(nat,$o),X: nat,M: nat] :
      ( aa(nat,$o,Pa,X)
     => ( ! [X2: nat] :
            ( aa(nat,$o,Pa,X2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),M) )
       => ~ ! [M2: nat] :
              ( aa(nat,$o,Pa,M2)
             => ~ ! [X4: nat] :
                    ( aa(nat,$o,Pa,X4)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_53_le__rel__bool__arg__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X5: fun($o,B),Y4: fun($o,B)] :
          ( aa(fun($o,B),$o,aa(fun($o,B),fun(fun($o,B),$o),ord_less_eq(fun($o,B)),X5),Y4)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa($o,B,X5,$false)),aa($o,B,Y4,$false))
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa($o,B,X5,$true)),aa($o,B,Y4,$true)) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_54_verit__la__disequality,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = B2 )
          | ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
          | ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_55_ord__eq__le__eq__trans,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( A2 = B2 )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ( C2 = D2 )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),D2) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_56_verit__comp__simplify1_I2_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_57_refines__assert_H__bind,axiom,
    ! [B: $tType,P2: heap_Time_Heap(B),Q2: heap_Time_Heap(B),Phi: $o] :
      ( refine_Imp_refines(B,P2,Q2)
     => refine_Imp_refines(B,P2,heap_Time_bind(product_unit,B,refine_Imp_assert((Phi)),aTP_Lamp_ac(heap_Time_Heap(B),fun(product_unit,heap_Time_Heap(B)),Q2))) ) ).

% refines_assert'_bind
tff(fact_58_ureturn__bind,axiom,
    ! [B: $tType,C: $tType,X: C,F: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,aa(C,heap_Time_Heap(C),heap_Time_ureturn(C),X),F) = aa(C,heap_Time_Heap(B),F,X) ).

% ureturn_bind
tff(fact_59_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_60_le0,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),N) ).

% le0
tff(fact_61_refines__refl,axiom,
    ! [B: $tType,P2: heap_Time_Heap(B)] : refine_Imp_refines(B,P2,P2) ).

% refines_refl
tff(fact_62_refines__If,axiom,
    ! [B: $tType,B2: $o,Ta: heap_Time_Heap(B),T2: heap_Time_Heap(B),E2: heap_Time_Heap(B),E3: heap_Time_Heap(B)] :
      ( ( (B2)
       => refine_Imp_refines(B,Ta,T2) )
     => ( ( ~ (B2)
         => refine_Imp_refines(B,E2,E3) )
       => refine_Imp_refines(B,
            $ite((B2),Ta,E2),
            $ite((B2),T2,E3)) ) ) ).

% refines_If
tff(fact_63_EQD,axiom,
    ! [B: $tType,A2: B,B2: B] :
      ( time_EQ(B,A2,B2)
     => ( A2 = B2 ) ) ).

% EQD
tff(fact_64_EQI,axiom,
    ! [B: $tType,A2: B,B2: B] :
      ( ( A2 = B2 )
     => time_EQ(B,A2,B2) ) ).

% EQI
tff(fact_65_EQ__def,axiom,
    ! [B: $tType,A2: B,B2: B] :
      ( time_EQ(B,A2,B2)
    <=> ( A2 = B2 ) ) ).

% EQ_def
tff(fact_66_EQ__refl,axiom,
    ! [B: $tType,A2: B] : time_EQ(B,A2,A2) ).

% EQ_refl
tff(fact_67_time__raise,axiom,
    ! [B: $tType,Msg: list(char),H: heap_ext(product_unit)] : time_time(B,heap_Time_raise(B,Msg),H) = zero_zero(nat) ).

% time_raise
tff(fact_68_refines__bind,axiom,
    ! [C: $tType,B: $tType,Ma: heap_Time_Heap(B),M3: heap_Time_Heap(B),F: fun(B,heap_Time_Heap(C)),F2: fun(B,heap_Time_Heap(C))] :
      ( refine_Imp_refines(B,Ma,M3)
     => ( ! [X2: B] : refine_Imp_refines(C,aa(B,heap_Time_Heap(C),F,X2),aa(B,heap_Time_Heap(C),F2,X2))
       => refine_Imp_refines(C,heap_Time_bind(B,C,Ma,F),heap_Time_bind(B,C,M3,F2)) ) ) ).

% refines_bind
tff(fact_69_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),N) ).

% less_eq_nat.simps(1)
tff(fact_70_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_71_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_72_le__0__eq,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),zero_zero(nat))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_73_time__assert_H,axiom,
    ! [Pa: $o,H: heap_ext(product_unit)] : time_time(product_unit,refine_Imp_assert((Pa)),H) = zero_zero(nat) ).

% time_assert'
tff(fact_74_TBOUND__refines,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),Ta: nat,D2: heap_Time_Heap(B)] :
      ( time_TBOUND(B,C2,Ta)
     => ( refine_Imp_refines(B,C2,D2)
       => time_TBOUND(B,D2,Ta) ) ) ).

% TBOUND_refines
tff(fact_75_GreatestI__nat,axiom,
    ! [Pa: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,Pa,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),B2) )
       => aa(nat,$o,Pa,order_Greatest(nat,Pa)) ) ) ).

% GreatestI_nat
tff(fact_76_Greatest__le__nat,axiom,
    ! [Pa: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,Pa,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,Pa)) ) ) ).

% Greatest_le_nat
tff(fact_77_GreatestI__ex__nat,axiom,
    ! [Pa: fun(nat,$o),B2: nat] :
      ( ? [X_1: nat] : aa(nat,$o,Pa,X_1)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),B2) )
       => aa(nat,$o,Pa,order_Greatest(nat,Pa)) ) ) ).

% GreatestI_ex_nat
tff(fact_78_bind__ureturn,axiom,
    ! [B: $tType,F: heap_Time_Heap(B)] : heap_Time_bind(B,B,F,heap_Time_ureturn(B)) = F ).

% bind_ureturn
tff(fact_79_le__zero__eq,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [N: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N),zero_zero(B))
        <=> ( N = zero_zero(B) ) ) ) ).

% le_zero_eq
tff(fact_80_le__numeral__extra_I3_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),zero_zero(B)) ) ).

% le_numeral_extra(3)
tff(fact_81_zero__le,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X) ) ).

% zero_le
tff(fact_82_time__refines,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),C3: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( refine_Imp_refines(B,C2,C3)
     => ( ~ time_fails(B,C3,H)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(B,C2,H)),time_time(B,C3,H)) ) ) ).

% time_refines
tff(fact_83_fun__cong__unused__0,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( zero(C)
     => ! [F: fun(fun(B,C),D),G: D] :
          ( ! [X2: fun(B,C)] : aa(fun(B,C),D,F,X2) = G
         => ( aa(fun(B,C),D,F,aTP_Lamp_ad(B,C)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_84_is__arg__max__linorder,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),X: B] :
          ( aa(B,$o,lattic501386751176901750rg_max(B,C,F,Pa),X)
        <=> ( aa(B,$o,Pa,X)
            & ! [Y5: B] :
                ( aa(B,$o,Pa,Y5)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y5)),aa(B,C,F,X)) ) ) ) ) ).

% is_arg_max_linorder
tff(fact_85_of__nat__le__0__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),Ma)),zero_zero(B))
        <=> ( Ma = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_86_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_87_zero__reorient,axiom,
    ! [B: $tType] :
      ( zero(B)
     => ! [X: B] :
          ( ( zero_zero(B) = X )
        <=> ( X = zero_zero(B) ) ) ) ).

% zero_reorient
tff(fact_88_of__nat__aux_Osimps_I1_J,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Inc: fun(B,B),I: B] : semiri8178284476397505188at_aux(B,Inc,zero_zero(nat),I) = I ) ).

% of_nat_aux.simps(1)
tff(fact_89_arg__max__equality,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [Pa: fun(B,$o),K: B,F: fun(B,C)] :
          ( aa(B,$o,Pa,K)
         => ( ! [X2: B] :
                ( aa(B,$o,Pa,X2)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,K)) )
           => ( aa(B,C,F,lattices_ord_arg_max(B,C,F,Pa)) = aa(B,C,F,K) ) ) ) ) ).

% arg_max_equality
tff(fact_90_of__nat__eq__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [Ma: nat,N: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),Ma) = aa(nat,B,semiring_1_of_nat(B),N) )
        <=> ( Ma = N ) ) ) ).

% of_nat_eq_iff
tff(fact_91_of__nat__eq__0__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [Ma: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),Ma) = zero_zero(B) )
        <=> ( Ma = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_92_of__nat__0__eq__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] :
          ( ( zero_zero(B) = aa(nat,B,semiring_1_of_nat(B),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_93_of__nat__0,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( aa(nat,B,semiring_1_of_nat(B),zero_zero(nat)) = zero_zero(B) ) ) ).

% of_nat_0
tff(fact_94_of__nat__le__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% of_nat_le_iff
tff(fact_95_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_96_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(3)
tff(fact_97_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_98_of__nat__0__le__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_0_le_iff
tff(fact_99_of__nat__mono,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),I)),aa(nat,B,semiring_1_of_nat(B),J)) ) ) ).

% of_nat_mono
tff(fact_100_fails__refines,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),C3: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( refine_Imp_refines(B,C2,C3)
     => ( time_fails(B,C2,H)
       => time_fails(B,C3,H) ) ) ).

% fails_refines
tff(fact_101_fails__assert_H,axiom,
    ! [Pa: $o,H: heap_ext(product_unit)] :
      ( time_fails(product_unit,refine_Imp_assert((Pa)),H)
    <=> ~ (Pa) ) ).

% fails_assert'
tff(fact_102_arg__max__on__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),S: set(B)] : lattic1883929316492267755max_on(B,C,F,S) = lattices_ord_arg_max(B,C,F,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S)) ) ).

% arg_max_on_def
tff(fact_103_real__arch__simple,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(nat,B,semiring_1_of_nat(B),N2)) ) ).

% real_arch_simple
tff(fact_104_time__wait,axiom,
    ! [N: nat,H: heap_ext(product_unit)] : time_time(product_unit,heap_Time_wait(N),H) = N ).

% time_wait
tff(fact_105_arg__max__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),Pa: fun(B,$o)] : lattices_ord_arg_max(B,C,F,Pa) = fChoice(B,lattic501386751176901750rg_max(B,C,F,Pa)) ) ).

% arg_max_def
tff(fact_106_zle__int,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% zle_int
tff(fact_107_dbl__simps_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl(B,zero_zero(B)) = zero_zero(B) ) ) ).

% dbl_simps(2)
tff(fact_108_fails__bind,axiom,
    ! [B: $tType,C: $tType,Ma: heap_Time_Heap(C),F: fun(C,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      ( time_fails(B,heap_Time_bind(C,B,Ma,F),H)
    <=> ( ~ time_fails(C,Ma,H)
       => time_fails(B,aa(C,heap_Time_Heap(B),F,time_the_res(C,Ma,H)),time_the_heap(C,Ma,H)) ) ) ).

% fails_bind
tff(fact_109_Heap_Osize__neq,axiom,
    ! [B: $tType,X: heap_Time_Heap(B)] : aa(heap_Time_Heap(B),nat,size_size(heap_Time_Heap(B)),X) != zero_zero(nat) ).

% Heap.size_neq
tff(fact_110_TBOUND__bind,axiom,
    ! [B: $tType,C: $tType,Ma: heap_Time_Heap(B),T_1: nat,F: fun(B,heap_Time_Heap(C)),T_2: nat] :
      ( time_TBOUND(B,Ma,T_1)
     => ( ! [X2: B,H2: heap_ext(product_unit)] :
            ( ( X2 = time_the_res(B,Ma,H2) )
           => ( ~ time_fails(B,Ma,H2)
             => time_TBOUND(C,aa(B,heap_Time_Heap(C),F,X2),T_2) ) )
       => time_TBOUND(C,heap_Time_bind(B,C,Ma,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),T_1),T_2)) ) ) ).

% TBOUND_bind
tff(fact_111_timeFrame__zero,axiom,
    ! [B: $tType,H: option(product_prod(B,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(B,zero_zero(nat),H) = H ).

% timeFrame_zero
tff(fact_112_arg__min__equality,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [Pa: fun(B,$o),K: B,F: fun(B,C)] :
          ( aa(B,$o,Pa,K)
         => ( ! [X2: B] :
                ( aa(B,$o,Pa,X2)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,K)),aa(B,C,F,X2)) )
           => ( aa(B,C,F,lattices_ord_arg_min(B,C,F,Pa)) = aa(B,C,F,K) ) ) ) ) ).

% arg_min_equality
tff(fact_113_old_Onat_Osimps_I6_J,axiom,
    ! [B: $tType,F1: B,F22: fun(nat,fun(B,B))] : aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),F1),F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_114_add__left__cancel,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_115_add__right__cancel,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_116_add__le__cancel__left,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_117_add__le__cancel__right,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_118_double__eq__0__iff,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% double_eq_0_iff
tff(fact_119_add__0,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),zero_zero(B)),A2) = A2 ) ).

% add_0
tff(fact_120_zero__eq__add__iff__both__eq__0,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [X: B,Y: B] :
          ( ( zero_zero(B) = aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y) )
        <=> ( ( X = zero_zero(B) )
            & ( Y = zero_zero(B) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_121_add__eq__0__iff__both__eq__0,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y) = zero_zero(B) )
        <=> ( ( X = zero_zero(B) )
            & ( Y = zero_zero(B) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_122_add__cancel__right__right,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) )
        <=> ( B2 = zero_zero(B) ) ) ) ).

% add_cancel_right_right
tff(fact_123_add__cancel__right__left,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) )
        <=> ( B2 = zero_zero(B) ) ) ) ).

% add_cancel_right_left
tff(fact_124_add__cancel__left__right,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = A2 )
        <=> ( B2 = zero_zero(B) ) ) ) ).

% add_cancel_left_right
tff(fact_125_add__cancel__left__left,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [B2: B,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) = A2 )
        <=> ( B2 = zero_zero(B) ) ) ) ).

% add_cancel_left_left
tff(fact_126_double__zero__sym,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( ( zero_zero(B) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% double_zero_sym
tff(fact_127_add_Oright__neutral,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),zero_zero(B)) = A2 ) ).

% add.right_neutral
tff(fact_128_add__is__0,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N) = zero_zero(nat) )
    <=> ( ( Ma = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_129_Nat_Oadd__0__right,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),zero_zero(nat)) = Ma ).

% Nat.add_0_right
tff(fact_130_nat__add__left__cancel__le,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% nat_add_left_cancel_le
tff(fact_131_timeFrame__assoc,axiom,
    ! [B: $tType,N: nat,N3: nat,F: option(product_prod(B,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(B,N,heap_Time_timeFrame(B,N3,F)) = heap_Time_timeFrame(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N3),F) ).

% timeFrame_assoc
tff(fact_132_add__le__same__cancel1,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% add_le_same_cancel1
tff(fact_133_add__le__same__cancel2,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% add_le_same_cancel2
tff(fact_134_le__add__same__cancel1,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_135_le__add__same__cancel2,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_136_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_137_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_138_of__nat__add,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_add
tff(fact_139_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_140_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ~ ! [N2: nat] : K != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_141_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ? [N2: nat] : K = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_142_int__if,axiom,
    ! [Pa: $o,A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),
        $ite((Pa),A2,B2)) = $ite((Pa),aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_if
tff(fact_143_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_144_nat__int__comparison_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A2) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_145_is__num__normalize_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_146_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K) = aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_147_group__cancel_Oadd1,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: B,K: B,A2: B,B2: B] :
          ( ( A4 = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),A2) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A4),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_148_group__cancel_Oadd2,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [B4: B,K: B,B2: B,A2: B] :
          ( ( B4 = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),B2) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_149_dbl__def,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [X: B] : neg_numeral_dbl(B,X) = aa(B,B,aa(B,fun(B,B),plus_plus(B),X),X) ) ).

% dbl_def
tff(fact_150_add_Oassoc,axiom,
    ! [B: $tType] :
      ( semigroup_add(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ).

% add.assoc
tff(fact_151_add_Oleft__cancel,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_152_add_Oright__cancel,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_153_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [B: $tType] :
      ( ab_semigroup_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_154_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: $tType] :
      ( ab_semigroup_add(B)
     => ! [B2: B,A2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_155_add__left__imp__eq,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_156_add__right__imp__eq,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_157_int__int__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Ma) = aa(nat,int,semiring_1_of_nat(int),N) )
    <=> ( Ma = N ) ) ).

% int_int_eq
tff(fact_158_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),X)
      | ( A2 = X )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),A2) ) ).

% verit_la_generic
tff(fact_159_fails__wait,axiom,
    ! [X: nat,H: heap_ext(product_unit)] : ~ time_fails(product_unit,heap_Time_wait(X),H) ).

% fails_wait
tff(fact_160_verit__sko__forall__indirect2,axiom,
    ! [B: $tType,X: B,Pa: fun(B,$o),P: fun(B,$o)] :
      ( ( X = fChoice(B,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)) )
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
          <=> aa(B,$o,P,X2) )
       => ( ! [X_12: B] : aa(B,$o,P,X_12)
        <=> aa(B,$o,Pa,X) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_161_verit__sko__forall__indirect,axiom,
    ! [B: $tType,X: B,Pa: fun(B,$o)] :
      ( ( X = fChoice(B,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)) )
     => ( ! [X_12: B] : aa(B,$o,Pa,X_12)
      <=> aa(B,$o,Pa,X) ) ) ).

% verit_sko_forall_indirect
tff(fact_162_verit__sko__ex__indirect2,axiom,
    ! [B: $tType,X: B,Pa: fun(B,$o),P: fun(B,$o)] :
      ( ( X = fChoice(B,Pa) )
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
          <=> aa(B,$o,P,X2) )
       => ( ? [X_12: B] : aa(B,$o,P,X_12)
        <=> aa(B,$o,Pa,X) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_163_verit__sko__ex__indirect,axiom,
    ! [B: $tType,X: B,Pa: fun(B,$o)] :
      ( ( X = fChoice(B,Pa) )
     => ( ? [X_12: B] : aa(B,$o,Pa,X_12)
      <=> aa(B,$o,Pa,X) ) ) ).

% verit_sko_ex_indirect
tff(fact_164_verit__sko__forall_H_H,axiom,
    ! [B: $tType,B4: B,A4: B,Pa: fun(B,$o)] :
      ( ( B4 = A4 )
     => ( ( fChoice(B,Pa) = A4 )
      <=> ( fChoice(B,Pa) = B4 ) ) ) ).

% verit_sko_forall''
tff(fact_165_verit__sko__forall_H,axiom,
    ! [B: $tType,Pa: fun(B,$o),A4: $o] :
      ( ( aa(B,$o,Pa,fChoice(B,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)))
      <=> (A4) )
     => ( ! [X_12: B] : aa(B,$o,Pa,X_12)
      <=> (A4) ) ) ).

% verit_sko_forall'
tff(fact_166_verit__sko__forall,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ! [X_12: B] : aa(B,$o,Pa,X_12)
    <=> aa(B,$o,Pa,fChoice(B,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa))) ) ).

% verit_sko_forall
tff(fact_167_verit__sko__ex_H,axiom,
    ! [B: $tType,Pa: fun(B,$o),A4: $o] :
      ( ( aa(B,$o,Pa,fChoice(B,Pa))
      <=> (A4) )
     => ( ? [X_12: B] : aa(B,$o,Pa,X_12)
      <=> (A4) ) ) ).

% verit_sko_ex'
tff(fact_168_size__neq__size__imp__neq,axiom,
    ! [B: $tType] :
      ( size(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,nat,size_size(B),X) != aa(B,nat,size_size(B),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_169_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),J)
            & ( K = L ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_170_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( ( I = J )
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K),L) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_171_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),J)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K),L) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_172_add__mono,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),D2)) ) ) ) ).

% add_mono
tff(fact_173_add__left__mono,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)) ) ) ).

% add_left_mono
tff(fact_174_less__eqE,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ~ ! [C4: B] : B2 != aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C4) ) ) ).

% less_eqE
tff(fact_175_add__right__mono,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ) ).

% add_right_mono
tff(fact_176_le__iff__add,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ? [C5: B] : B2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C5) ) ) ).

% le_iff_add
tff(fact_177_add__le__imp__le__left,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_178_add__le__imp__le__right,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_179_verit__sum__simplify,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),zero_zero(B)) = A2 ) ).

% verit_sum_simplify
tff(fact_180_add_Ogroup__left__neutral,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),zero_zero(B)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_181_add_Ocomm__neutral,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),zero_zero(B)) = A2 ) ).

% add.comm_neutral
tff(fact_182_comm__monoid__add__class_Oadd__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),zero_zero(B)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_183_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_184_add__eq__self__zero,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N) = Ma )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_185_add__leE,axiom,
    ! [Ma: nat,K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),N)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) ) ) ).

% add_leE
tff(fact_186_le__add1,axiom,
    ! [N: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma)) ).

% le_add1
tff(fact_187_le__add2,axiom,
    ! [N: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) ).

% le_add2
tff(fact_188_add__leD1,axiom,
    ! [Ma: nat,K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% add_leD1
tff(fact_189_add__leD2,axiom,
    ! [Ma: nat,K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) ) ).

% add_leD2
tff(fact_190_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_191_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_192_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_193_trans__le__add1,axiom,
    ! [I: nat,J: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Ma)) ) ).

% trans_le_add1
tff(fact_194_trans__le__add2,axiom,
    ! [I: nat,J: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),J)) ) ).

% trans_le_add2
tff(fact_195_nat__le__iff__add,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
    <=> ? [K2: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K2) ) ).

% nat_le_iff_add
tff(fact_196_add__decreasing,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_197_add__increasing,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_198_add__decreasing2,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_199_add__increasing2,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)) ) ) ) ).

% add_increasing2
tff(fact_200_add__nonneg__nonneg,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_201_add__nonpos__nonpos,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),zero_zero(B)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_202_add__nonneg__eq__0__iff,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y) = zero_zero(B) )
            <=> ( ( X = zero_zero(B) )
                & ( Y = zero_zero(B) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_203_add__nonpos__eq__0__iff,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),zero_zero(B))
           => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y) = zero_zero(B) )
            <=> ( ( X = zero_zero(B) )
                & ( Y = zero_zero(B) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_204_arg__min__nat__le,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,Ma: fun(B,nat)] :
      ( aa(B,$o,Pa,X)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,Ma,lattices_ord_arg_min(B,nat,Ma,Pa))),aa(B,nat,Ma,X)) ) ).

% arg_min_nat_le
tff(fact_205_arg__min__nat__lemma,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,Ma: fun(B,nat)] :
      ( aa(B,$o,Pa,K)
     => ( aa(B,$o,Pa,lattices_ord_arg_min(B,nat,Ma,Pa))
        & ! [Y3: B] :
            ( aa(B,$o,Pa,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,Ma,lattices_ord_arg_min(B,nat,Ma,Pa))),aa(B,nat,Ma,Y3)) ) ) ) ).

% arg_min_nat_lemma
tff(fact_206_time__bind,axiom,
    ! [C: $tType,B: $tType,Ma: heap_Time_Heap(C),F: fun(C,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      time_time(B,heap_Time_bind(C,B,Ma,F),H) = $ite(
        ( ~ time_fails(C,Ma,H)
        & ~ time_fails(B,aa(C,heap_Time_Heap(B),F,time_the_res(C,Ma,H)),time_the_heap(C,Ma,H)) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),time_time(C,Ma,H)),time_time(B,aa(C,heap_Time_Heap(B),F,time_the_res(C,Ma,H)),time_the_heap(C,Ma,H))),
        zero_zero(nat) ) ).

% time_bind
tff(fact_207_some__sym__eq__trivial,axiom,
    ! [B: $tType,X: B] : fChoice(B,aa(B,fun(B,$o),fequal(B),X)) = X ).

% some_sym_eq_trivial
tff(fact_208_some__eq__trivial,axiom,
    ! [B: $tType,X: B] : fChoice(B,aTP_Lamp_af(B,fun(B,$o),X)) = X ).

% some_eq_trivial
tff(fact_209_some__equality,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B] :
      ( aa(B,$o,Pa,A2)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => ( X2 = A2 ) )
       => ( fChoice(B,Pa) = A2 ) ) ) ).

% some_equality
tff(fact_210_rec__nat__0__imp,axiom,
    ! [B: $tType,F: fun(nat,B),F1: B,F22: fun(nat,fun(B,B))] :
      ( ( F = aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),F1),F22) )
     => ( aa(nat,B,F,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_211_imp__le__cong,axiom,
    ! [X: int,X6: int,Pa: $o,P: $o] :
      ( ( X = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (Pa)
          <=> (P) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
           => (Pa) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
           => (P) ) ) ) ) ).

% imp_le_cong
tff(fact_212_conj__le__cong,axiom,
    ! [X: int,X6: int,Pa: $o,P: $o] :
      ( ( X = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (Pa)
          <=> (P) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
            & (Pa) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
            & (P) ) ) ) ) ).

% conj_le_cong
tff(fact_213_Euclid__induct,axiom,
    ! [Pa: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,A3),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),Pa,B3),A3) )
     => ( ! [A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,A3),zero_zero(nat))
       => ( ! [A3: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,A3),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),Pa,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),Pa,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_214_add__0__iff,axiom,
    ! [B: $tType] :
      ( semiri1453513574482234551roduct(B)
     => ! [B2: B,A2: B] :
          ( ( B2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% add_0_iff
tff(fact_215_arg__min__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),Pa: fun(B,$o)] : lattices_ord_arg_min(B,C,F,Pa) = fChoice(B,lattic501386751177426532rg_min(B,C,F,Pa)) ) ).

% arg_min_def
tff(fact_216_of__nat__code,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] : aa(nat,B,semiring_1_of_nat(B),N) = semiri8178284476397505188at_aux(B,aTP_Lamp_ag(B,B),N,zero_zero(B)) ) ).

% of_nat_code
tff(fact_217_some1__equality,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B] :
      ( ? [X4: B] :
          ( aa(B,$o,Pa,X4)
          & ! [Y2: B] :
              ( aa(B,$o,Pa,Y2)
             => ( Y2 = X4 ) ) )
     => ( aa(B,$o,Pa,A2)
       => ( fChoice(B,Pa) = A2 ) ) ) ).

% some1_equality
tff(fact_218_some__eq__ex,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( aa(B,$o,Pa,fChoice(B,Pa))
    <=> ? [X_12: B] : aa(B,$o,Pa,X_12) ) ).

% some_eq_ex
tff(fact_219_of__nat__eq__1__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),N) = one_one(B) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_220_of__nat__1__eq__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] :
          ( ( one_one(B) = aa(nat,B,semiring_1_of_nat(B),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_221_of__nat__1,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( aa(nat,B,semiring_1_of_nat(B),one_one(nat)) = one_one(B) ) ) ).

% of_nat_1
tff(fact_222_arg__min__natI,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,Ma: fun(B,nat)] :
      ( aa(B,$o,Pa,K)
     => aa(B,$o,Pa,lattices_ord_arg_min(B,nat,Ma,Pa)) ) ).

% arg_min_natI
tff(fact_223_one__reorient,axiom,
    ! [B: $tType] :
      ( one(B)
     => ! [X: B] :
          ( ( one_one(B) = X )
        <=> ( X = one_one(B) ) ) ) ).

% one_reorient
tff(fact_224_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_225_int__ge__induct,axiom,
    ! [K: int,I: int,Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,Pa,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,Pa,I2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,Pa,I) ) ) ) ).

% int_ge_induct
tff(fact_226_is__arg__min__arg__min__nat,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,Ma: fun(B,nat)] :
      ( aa(B,$o,Pa,X)
     => aa(B,$o,lattic501386751177426532rg_min(B,nat,Ma,Pa),lattices_ord_arg_min(B,nat,Ma,Pa)) ) ).

% is_arg_min_arg_min_nat
tff(fact_227_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_228_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_229_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_230_le__numeral__extra_I4_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),one_one(B)) ) ).

% le_numeral_extra(4)
tff(fact_231_zadd__int__left,axiom,
    ! [Ma: nat,N: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N))),Z) ).

% zadd_int_left
tff(fact_232_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_233_int__plus,axiom,
    ! [N: nat,Ma: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma)) = 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),Ma)) ).

% int_plus
tff(fact_234_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z)
    <=> ? [N4: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N4)) ) ).

% zle_iff_zadd
tff(fact_235_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),X)
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_236_is__arg__min__linorder,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),X: B] :
          ( aa(B,$o,lattic501386751177426532rg_min(B,C,F,Pa),X)
        <=> ( aa(B,$o,Pa,X)
            & ! [Y5: B] :
                ( aa(B,$o,Pa,Y5)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,Y5)) ) ) ) ) ).

% is_arg_min_linorder
tff(fact_237_is__arg__min__antimono,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),X: B,Y: B] :
          ( aa(B,$o,lattic501386751177426532rg_min(B,C,F,Pa),X)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y)),aa(B,C,F,X))
           => ( aa(B,$o,Pa,Y)
             => aa(B,$o,lattic501386751177426532rg_min(B,C,F,Pa),Y) ) ) ) ) ).

% is_arg_min_antimono
tff(fact_238_someI,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B] :
      ( aa(B,$o,Pa,X)
     => aa(B,$o,Pa,fChoice(B,Pa)) ) ).

% someI
tff(fact_239_Eps__cong,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] :
      ( ! [X2: B] :
          ( aa(B,$o,Pa,X2)
        <=> aa(B,$o,Q,X2) )
     => ( fChoice(B,Pa) = fChoice(B,Q) ) ) ).

% Eps_cong
tff(fact_240_tfl__some,axiom,
    ! [B: $tType,P3: fun(B,$o),X4: B] :
      ( aa(B,$o,P3,X4)
     => aa(B,$o,P3,fChoice(B,P3)) ) ).

% tfl_some
tff(fact_241_some__eq__imp,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B,B2: B] :
      ( ( fChoice(B,Pa) = A2 )
     => ( aa(B,$o,Pa,B2)
       => aa(B,$o,Pa,A2) ) ) ).

% some_eq_imp
tff(fact_242_someI2,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B,Q: fun(B,$o)] :
      ( aa(B,$o,Pa,A2)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => aa(B,$o,Q,X2) )
       => aa(B,$o,Q,fChoice(B,Pa)) ) ) ).

% someI2
tff(fact_243_someI__ex,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ? [X_1: B] : aa(B,$o,Pa,X_1)
     => aa(B,$o,Pa,fChoice(B,Pa)) ) ).

% someI_ex
tff(fact_244_someI2__ex,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] :
      ( ? [X_1: B] : aa(B,$o,Pa,X_1)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => aa(B,$o,Q,X2) )
       => aa(B,$o,Q,fChoice(B,Pa)) ) ) ).

% someI2_ex
tff(fact_245_someI2__bex,axiom,
    ! [B: $tType,A4: set(B),Pa: fun(B,$o),Q: fun(B,$o)] :
      ( ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),A4)
          & aa(B,$o,Pa,X4) )
     => ( ! [X2: B] :
            ( ( aa(set(B),$o,member(B,X2),A4)
              & aa(B,$o,Pa,X2) )
           => aa(B,$o,Q,X2) )
       => aa(B,$o,Q,fChoice(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa))) ) ) ).

% someI2_bex
tff(fact_246_not__one__le__zero,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),zero_zero(B)) ) ).

% not_one_le_zero
tff(fact_247_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),one_one(B)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_248_zero__less__one__class_Ozero__le__one,axiom,
    ! [B: $tType] :
      ( zero_less_one(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),one_one(B)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_249_size__union,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(B),nat,size_size(multiset(B)),M)),aa(multiset(B),nat,size_size(multiset(B)),N5)) ).

% size_union
tff(fact_250_size__eq__0__iff__empty,axiom,
    ! [B: $tType,M: multiset(B)] :
      ( ( aa(multiset(B),nat,size_size(multiset(B)),M) = zero_zero(nat) )
    <=> ( M = zero_zero(multiset(B)) ) ) ).

% size_eq_0_iff_empty
tff(fact_251_size__empty,axiom,
    ! [B: $tType] : aa(multiset(B),nat,size_size(multiset(B)),zero_zero(multiset(B))) = zero_zero(nat) ).

% size_empty
tff(fact_252_zero__neq__one,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ( zero_zero(B) != one_one(B) ) ) ).

% zero_neq_one
tff(fact_253_return__bind,axiom,
    ! [B: $tType,C: $tType,X: C,F: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,aa(C,heap_Time_Heap(C),heap_Time_return(C),X),F) = heap_Time_bind(product_unit,B,heap_Time_wait(one_one(nat)),aa(fun(C,heap_Time_Heap(B)),fun(product_unit,heap_Time_Heap(B)),aTP_Lamp_ai(C,fun(fun(C,heap_Time_Heap(B)),fun(product_unit,heap_Time_Heap(B))),X),F)) ).

% return_bind
tff(fact_254_bind__return,axiom,
    ! [B: $tType,F: heap_Time_Heap(B)] : heap_Time_bind(B,B,F,heap_Time_return(B)) = heap_Time_bind(product_unit,B,heap_Time_wait(one_one(nat)),aTP_Lamp_ac(heap_Time_Heap(B),fun(product_unit,heap_Time_Heap(B)),F)) ).

% bind_return
tff(fact_255_time__array__nth,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [P2: array(B),I: nat,H: heap_ext(product_unit)] :
          time_time(B,array_nth(B,P2,I),H) = $ite(time_fails(B,array_nth(B,P2,I),H),zero_zero(nat),one_one(nat)) ) ).

% time_array_nth
tff(fact_256_dbl__inc__simps_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl_inc(B,zero_zero(B)) = one_one(B) ) ) ).

% dbl_inc_simps(2)
tff(fact_257_union__eq__empty,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5) = zero_zero(multiset(B)) )
    <=> ( ( M = zero_zero(multiset(B)) )
        & ( N5 = zero_zero(multiset(B)) ) ) ) ).

% union_eq_empty
tff(fact_258_empty__eq__union,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] :
      ( ( zero_zero(multiset(B)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5) )
    <=> ( ( M = zero_zero(multiset(B)) )
        & ( N5 = zero_zero(multiset(B)) ) ) ) ).

% empty_eq_union
tff(fact_259_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B)] :
      ( ( zero_zero(multiset(B)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Y) )
    <=> ( ( X = zero_zero(multiset(B)) )
        & ( Y = zero_zero(multiset(B)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_260_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Y) = zero_zero(multiset(B)) )
    <=> ( ( X = zero_zero(multiset(B)) )
        & ( Y = zero_zero(multiset(B)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_261_empty__neutral_I1_J,axiom,
    ! [B: $tType,X: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),zero_zero(multiset(B))),X) = X ).

% empty_neutral(1)
tff(fact_262_empty__neutral_I2_J,axiom,
    ! [B: $tType,X: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),zero_zero(multiset(B))) = X ).

% empty_neutral(2)
tff(fact_263_multi__union__self__other__eq,axiom,
    ! [B: $tType,A4: multiset(B),X5: multiset(B),Y4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),X5) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),Y4) )
     => ( X5 = Y4 ) ) ).

% multi_union_self_other_eq
tff(fact_264_union__right__cancel,axiom,
    ! [B: $tType,M: multiset(B),K3: multiset(B),N5: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),K3) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),K3) )
    <=> ( M = N5 ) ) ).

% union_right_cancel
tff(fact_265_union__left__cancel,axiom,
    ! [B: $tType,K3: multiset(B),M: multiset(B),N5: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),K3),M) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),K3),N5) )
    <=> ( M = N5 ) ) ).

% union_left_cancel
tff(fact_266_union__commute,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),M) ).

% union_commute
tff(fact_267_union__lcomm,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B),K3: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),K3)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),K3)) ).

% union_lcomm
tff(fact_268_union__assoc,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B),K3: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)),K3) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),K3)) ).

% union_assoc
tff(fact_269_mset__distrib,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),M: multiset(B),N5: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5) )
     => ~ ! [Am: multiset(B),An: multiset(B)] :
            ( ( A4 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Am),An) )
           => ! [Bm: multiset(B),Bn: multiset(B)] :
                ( ( B4 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Bm),Bn) )
               => ( ( M = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Am),Bm) )
                 => ( N5 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_270_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_271_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_272_fold__if__return,axiom,
    ! [B: $tType,B2: $o,C2: B,D2: B] :
      $ite((B2),aa(B,heap_Time_Heap(B),heap_Time_return(B),C2),aa(B,heap_Time_Heap(B),heap_Time_return(B),D2)) = aa(B,heap_Time_Heap(B),heap_Time_return(B),
        $ite((B2),C2,D2)) ).

% fold_if_return
tff(fact_273_time__return,axiom,
    ! [B: $tType,X: B,H: heap_ext(product_unit)] : time_time(B,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),H) = one_one(nat) ).

% time_return
tff(fact_274_fails__return,axiom,
    ! [B: $tType,X: B,H: heap_ext(product_unit)] : ~ time_fails(B,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),H) ).

% fails_return
tff(fact_275_dbl__inc__def,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [X: B] : neg_numeral_dbl_inc(B,X) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),X)),one_one(B)) ) ).

% dbl_inc_def
tff(fact_276_bind__lift,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(C),G: fun(C,B)] : heap_Time_bind(C,B,F,heap_Time_lift(C,B,G)) = heap_Time_bind(C,B,F,aTP_Lamp_aj(fun(C,B),fun(C,heap_Time_Heap(B)),G)) ).

% bind_lift
tff(fact_277_lift__collapse,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),X: C] : aa(C,heap_Time_Heap(B),heap_Time_lift(C,B,F),X) = aa(B,heap_Time_Heap(B),heap_Time_return(B),aa(C,B,F,X)) ).

% lift_collapse
tff(fact_278_lookup__chain,axiom,
    ! [C: $tType,B: $tType] :
      ( heap(C)
     => ! [R: ref(C),F: heap_Time_Heap(B)] : heap_Time_bind(C,B,ref_lookup(C,R),aTP_Lamp_ak(heap_Time_Heap(B),fun(C,heap_Time_Heap(B)),F)) = heap_Time_bind(product_unit,B,heap_Time_wait(one_one(nat)),aTP_Lamp_ac(heap_Time_Heap(B),fun(product_unit,heap_Time_Heap(B)),F)) ) ).

% lookup_chain
tff(fact_279_univ__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),X5: set(C)] : bNF_Greatest_univ(C,B,F,X5) = aa(C,B,F,fChoice(C,aTP_Lamp_al(set(C),fun(C,$o),X5))) ).

% univ_def
tff(fact_280_ceiling__add__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),one_one(B))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(B,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_281_ceiling__le__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,X)),one_one(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),one_one(B)) ) ) ).

% ceiling_le_one
tff(fact_282_ceiling__le__zero,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,X)),zero_zero(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B)) ) ) ).

% ceiling_le_zero
tff(fact_283_of__int__le__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_284_of__int__1__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_285_size__multiset__union,axiom,
    ! [B: $tType,F: fun(B,nat),M: multiset(B),N5: multiset(B)] : aa(multiset(B),nat,size_multiset(B,F),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(B),nat,size_multiset(B,F),M)),aa(multiset(B),nat,size_multiset(B,F),N5)) ).

% size_multiset_union
tff(fact_286_of__int__eq__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [W: int,Z: int] :
          ( ( aa(int,B,ring_1_of_int(B),W) = aa(int,B,ring_1_of_int(B),Z) )
        <=> ( W = Z ) ) ) ).

% of_int_eq_iff
tff(fact_287_ceiling__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int] : archimedean_ceiling(B,aa(int,B,ring_1_of_int(B),Z)) = Z ) ).

% ceiling_of_int
tff(fact_288_of__int__0,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ( aa(int,B,ring_1_of_int(B),zero_zero(int)) = zero_zero(B) ) ) ).

% of_int_0
tff(fact_289_of__int__0__eq__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Z: int] :
          ( ( zero_zero(B) = aa(int,B,ring_1_of_int(B),Z) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_290_of__int__eq__0__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Z: int] :
          ( ( aa(int,B,ring_1_of_int(B),Z) = zero_zero(B) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_291_of__int__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [W: int,Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),W)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% of_int_le_iff
tff(fact_292_of__int__eq__1__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Z: int] :
          ( ( aa(int,B,ring_1_of_int(B),Z) = one_one(B) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_293_of__int__1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ( aa(int,B,ring_1_of_int(B),one_one(int)) = one_one(B) ) ) ).

% of_int_1
tff(fact_294_of__int__add,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [W: int,Z: int] : aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),W)),aa(int,B,ring_1_of_int(B),Z)) ) ).

% of_int_add
tff(fact_295_ceiling__zero,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ( archimedean_ceiling(B,zero_zero(B)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_296_of__int__of__nat__eq,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] : aa(int,B,ring_1_of_int(B),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,B,semiring_1_of_nat(B),N) ) ).

% of_int_of_nat_eq
tff(fact_297_ceiling__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ( archimedean_ceiling(B,one_one(B)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_298_ceiling__of__nat,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [N: nat] : archimedean_ceiling(B,aa(nat,B,semiring_1_of_nat(B),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% ceiling_of_nat
tff(fact_299_size__multiset__eq__0__iff__empty,axiom,
    ! [B: $tType,F: fun(B,nat),M: multiset(B)] :
      ( ( aa(multiset(B),nat,size_multiset(B,F),M) = zero_zero(nat) )
    <=> ( M = zero_zero(multiset(B)) ) ) ).

% size_multiset_eq_0_iff_empty
tff(fact_300_size__multiset__empty,axiom,
    ! [B: $tType,F: fun(B,nat)] : aa(multiset(B),nat,size_multiset(B,F),zero_zero(multiset(B))) = zero_zero(nat) ).

% size_multiset_empty
tff(fact_301_ceiling__add__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] : archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(int,B,ring_1_of_int(B),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(B,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_302_of__int__le__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Z)),zero_zero(B))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_303_of__int__0__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_304_le__of__int__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,X))) ) ).

% le_of_int_ceiling
tff(fact_305_ceiling__le__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,X)),Z)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(int,B,ring_1_of_int(B),Z)) ) ) ).

% ceiling_le_iff
tff(fact_306_ex__le__of__int,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [Z2: int] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(int,B,ring_1_of_int(B),Z2)) ) ).

% ex_le_of_int
tff(fact_307_ceiling__mono,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,Y)),archimedean_ceiling(B,X)) ) ) ).

% ceiling_mono
tff(fact_308_ceiling__add__le,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(B,X)),archimedean_ceiling(B,Y))) ) ).

% ceiling_add_le
tff(fact_309_of__int__nonneg,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(int,B,ring_1_of_int(B),Z)) ) ) ).

% of_int_nonneg
tff(fact_310_size__multiset__overloaded__def,axiom,
    ! [B: $tType] : size_size(multiset(B)) = size_multiset(B,aTP_Lamp_am(B,nat)) ).

% size_multiset_overloaded_def
tff(fact_311_change__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [F: fun(B,B),R: ref(B)] : ref_change(B,F,R) = heap_Time_bind(B,B,ref_lookup(B,R),aa(ref(B),fun(B,heap_Time_Heap(B)),aTP_Lamp_ao(fun(B,B),fun(ref(B),fun(B,heap_Time_Heap(B))),F),R)) ) ).

% change_def
tff(fact_312_one__le__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X) ) ) ).

% one_le_ceiling
tff(fact_313_ceiling__less__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(B,X)),one_one(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B)) ) ) ).

% ceiling_less_one
tff(fact_314_of__nat__nat,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => ( aa(nat,B,semiring_1_of_nat(B),aa(int,nat,nat2,Z)) = aa(int,B,ring_1_of_int(B),Z) ) ) ) ).

% of_nat_nat
tff(fact_315_one__le__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),X) ) ) ).

% one_le_floor
tff(fact_316_arg__min__on__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),S: set(B)] : lattic7623131987881927897min_on(B,C,F,S) = lattices_ord_arg_min(B,C,F,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S)) ) ).

% arg_min_on_def
tff(fact_317_power__int__decreasing,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [N: int,N5: int,A2: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B))
             => ( ( ( A2 != zero_zero(B) )
                  | ( N5 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,A2,N5)),power_int(B,A2,N)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_318_power__int__le__one,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,X,N)),one_one(B)) ) ) ) ) ).

% power_int_le_one
tff(fact_319_zero__le__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X) ) ) ).

% zero_le_floor
tff(fact_320_rotate1__length01,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat))
     => ( rotate1(B,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_321_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_322_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_323_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% neq0_conv
tff(fact_324_less__nat__zero__code,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_325_nat__add__left__cancel__less,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% nat_add_left_cancel_less
tff(fact_326_not__gr__zero,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [N: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),N)
        <=> ( N = zero_zero(B) ) ) ) ).

% not_gr_zero
tff(fact_327_add__less__cancel__left,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_328_add__less__cancel__right,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_329_of__nat__less__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% of_nat_less_iff
tff(fact_330_add__gr__0,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% add_gr_0
tff(fact_331_less__one,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),one_one(nat))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_332_power__int__1__left,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [N: int] : power_int(B,one_one(B),N) = one_one(B) ) ).

% power_int_1_left
tff(fact_333_nat__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),N)) = N ).

% nat_int
tff(fact_334_power__int__1__right,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & monoid_mult(B) )
     => ! [Y: B] : power_int(B,Y,one_one(int)) = Y ) ).

% power_int_1_right
tff(fact_335_floor__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int] : archim6421214686448440834_floor(B,aa(int,B,ring_1_of_int(B),Z)) = Z ) ).

% floor_of_int
tff(fact_336_length__rotate1,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),rotate1(B,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_rotate1
tff(fact_337_add__less__same__cancel1,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% add_less_same_cancel1
tff(fact_338_add__less__same__cancel2,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% add_less_same_cancel2
tff(fact_339_less__add__same__cancel1,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_340_less__add__same__cancel2,axiom,
    ! [B: $tType] :
      ( ordere1937475149494474687imp_le(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_341_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_342_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_343_of__int__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [W: int,Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),W)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% of_int_less_iff
tff(fact_344_power__int__eq__0__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,N: int] :
          ( ( power_int(B,X,N) = zero_zero(B) )
        <=> ( ( X = zero_zero(B) )
            & ( N != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_345_power__int__0__left,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Ma: int] :
          ( ( Ma != zero_zero(int) )
         => ( power_int(B,zero_zero(B),Ma) = zero_zero(B) ) ) ) ).

% power_int_0_left
tff(fact_346_floor__zero,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ( archim6421214686448440834_floor(B,zero_zero(B)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_347_power__int__0__right,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & power(B) )
     => ! [X: B] : power_int(B,X,zero_zero(int)) = one_one(B) ) ).

% power_int_0_right
tff(fact_348_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% zless_nat_conj
tff(fact_349_floor__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ( archim6421214686448440834_floor(B,one_one(B)) = one_one(int) ) ) ).

% floor_one
tff(fact_350_floor__of__nat,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [N: nat] : archim6421214686448440834_floor(B,aa(nat,B,semiring_1_of_nat(B),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% floor_of_nat
tff(fact_351_of__nat__0__less__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,semiring_1_of_nat(B),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% of_nat_0_less_iff
tff(fact_352_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_353_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_354_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_355_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ).

% zle_add1_eq_le
tff(fact_356_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_357_of__int__less__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),Z)),zero_zero(B))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_358_of__int__0__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_359_power__int__mono__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,A2,N)),power_int(B,B2,N))
              <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_360_of__int__less__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_361_of__int__1__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_362_floor__less__zero,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(B,X)),zero_zero(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),zero_zero(B)) ) ) ).

% floor_less_zero
tff(fact_363_zero__less__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),X) ) ) ).

% zero_less_floor
tff(fact_364_floor__le__zero,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),zero_zero(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),one_one(B)) ) ) ).

% floor_le_zero
tff(fact_365_zero__less__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X) ) ) ).

% zero_less_ceiling
tff(fact_366_floor__less__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(B,X)),one_one(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),one_one(B)) ) ) ).

% floor_less_one
tff(fact_367_one__less__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),X) ) ) ).

% one_less_ceiling
tff(fact_368_neq__if__length__neq,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) != aa(list(B),nat,size_size(list(B)),Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_369_Ex__list__of__length,axiom,
    ! [B: $tType,N: nat] :
    ? [Xs2: list(B)] : aa(list(B),nat,size_size(list(B)),Xs2) = N ).

% Ex_list_of_length
tff(fact_370_length__induct,axiom,
    ! [B: $tType,Pa: fun(list(B),$o),Xs: list(B)] :
      ( ! [Xs2: list(B)] :
          ( ! [Ys2: list(B)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Ys2)),aa(list(B),nat,size_size(list(B)),Xs2))
             => aa(list(B),$o,Pa,Ys2) )
         => aa(list(B),$o,Pa,Xs2) )
     => aa(list(B),$o,Pa,Xs) ) ).

% length_induct
tff(fact_371_floor__less__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(B,X)),Z)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(int,B,ring_1_of_int(B),Z)) ) ) ).

% floor_less_iff
tff(fact_372_minf_I11_J,axiom,
    ! [B: $tType,C: $tType] :
      ( ord(B)
     => ! [F3: C] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ( F3 = F3 ) ) ) ).

% minf(11)
tff(fact_373_minf_I7_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ta),X4) ) ) ).

% minf(7)
tff(fact_374_minf_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Ta) ) ) ).

% minf(5)
tff(fact_375_minf_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ( X4 != Ta ) ) ) ).

% minf(4)
tff(fact_376_minf_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ( X4 != Ta ) ) ) ).

% minf(3)
tff(fact_377_minf_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o),P: fun(B,$o),Q: fun(B,$o),Q3: fun(B,$o)] :
          ( ? [Z3: B] :
            ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Z3)
             => ( aa(B,$o,Pa,X2)
              <=> aa(B,$o,P,X2) ) )
         => ( ? [Z3: B] :
              ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Z3)
               => ( aa(B,$o,Q,X2)
                <=> aa(B,$o,Q3,X2) ) )
           => ? [Z2: B] :
              ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
               => ( ( aa(B,$o,Pa,X4)
                    | aa(B,$o,Q,X4) )
                <=> ( aa(B,$o,P,X4)
                    | aa(B,$o,Q3,X4) ) ) ) ) ) ) ).

% minf(2)
tff(fact_378_minf_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o),P: fun(B,$o),Q: fun(B,$o),Q3: fun(B,$o)] :
          ( ? [Z3: B] :
            ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Z3)
             => ( aa(B,$o,Pa,X2)
              <=> aa(B,$o,P,X2) ) )
         => ( ? [Z3: B] :
              ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Z3)
               => ( aa(B,$o,Q,X2)
                <=> aa(B,$o,Q3,X2) ) )
           => ? [Z2: B] :
              ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
               => ( ( aa(B,$o,Pa,X4)
                    & aa(B,$o,Q,X4) )
                <=> ( aa(B,$o,P,X4)
                    & aa(B,$o,Q3,X4) ) ) ) ) ) ) ).

% minf(1)
tff(fact_379_pinf_I11_J,axiom,
    ! [B: $tType,C: $tType] :
      ( ord(B)
     => ! [F3: C] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ( F3 = F3 ) ) ) ).

% pinf(11)
tff(fact_380_pinf_I7_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ta),X4) ) ) ).

% pinf(7)
tff(fact_381_pinf_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Ta) ) ) ).

% pinf(5)
tff(fact_382_pinf_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ( X4 != Ta ) ) ) ).

% pinf(4)
tff(fact_383_pinf_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ( X4 != Ta ) ) ) ).

% pinf(3)
tff(fact_384_pinf_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o),P: fun(B,$o),Q: fun(B,$o),Q3: fun(B,$o)] :
          ( ? [Z3: B] :
            ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z3),X2)
             => ( aa(B,$o,Pa,X2)
              <=> aa(B,$o,P,X2) ) )
         => ( ? [Z3: B] :
              ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z3),X2)
               => ( aa(B,$o,Q,X2)
                <=> aa(B,$o,Q3,X2) ) )
           => ? [Z2: B] :
              ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
               => ( ( aa(B,$o,Pa,X4)
                    | aa(B,$o,Q,X4) )
                <=> ( aa(B,$o,P,X4)
                    | aa(B,$o,Q3,X4) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_385_pinf_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o),P: fun(B,$o),Q: fun(B,$o),Q3: fun(B,$o)] :
          ( ? [Z3: B] :
            ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z3),X2)
             => ( aa(B,$o,Pa,X2)
              <=> aa(B,$o,P,X2) ) )
         => ( ? [Z3: B] :
              ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z3),X2)
               => ( aa(B,$o,Q,X2)
                <=> aa(B,$o,Q3,X2) ) )
           => ? [Z2: B] :
              ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
               => ( ( aa(B,$o,Pa,X4)
                    & aa(B,$o,Q,X4) )
                <=> ( aa(B,$o,P,X4)
                    & aa(B,$o,Q3,X4) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_386_lt__ex,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [X: B] :
        ? [Y2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y2),X) ) ).

% lt_ex
tff(fact_387_gt__ex,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [X: B] :
        ? [X_13: B] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),X_13) ) ).

% gt_ex
tff(fact_388_dense,axiom,
    ! [B: $tType] :
      ( dense_order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ? [Z2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Z2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),Y) ) ) ) ).

% dense
tff(fact_389_less__imp__neq,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_390_order_Oasym,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ).

% order.asym
tff(fact_391_ord__eq__less__trans,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 = B2 )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_392_ord__less__eq__trans,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( ( B2 = C2 )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_393_less__induct,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),A2: B] :
          ( ! [X2: B] :
              ( ! [Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y3),X2)
                 => aa(B,$o,Pa,Y3) )
             => aa(B,$o,Pa,X2) )
         => aa(B,$o,Pa,A2) ) ) ).

% less_induct
tff(fact_394_antisym__conv3,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Y: B,X: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X)
         => ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_395_linorder__cases,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( ( X != Y )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ) ).

% linorder_cases
tff(fact_396_dual__order_Oasym,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% dual_order.asym
tff(fact_397_dual__order_Oirrefl,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),A2) ) ).

% dual_order.irrefl
tff(fact_398_exists__least__iff,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o)] :
          ( ? [X_12: B] : aa(B,$o,Pa,X_12)
        <=> ? [N4: B] :
              ( aa(B,$o,Pa,N4)
              & ! [M4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),M4),N4)
                 => ~ aa(B,$o,Pa,M4) ) ) ) ) ).

% exists_least_iff
tff(fact_399_linorder__less__wlog,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,fun(B,$o)),A2: B,B2: B] :
          ( ! [A3: B,B3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),B3)
             => aa(B,$o,aa(B,fun(B,$o),Pa,A3),B3) )
         => ( ! [A3: B] : aa(B,$o,aa(B,fun(B,$o),Pa,A3),A3)
           => ( ! [A3: B,B3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),Pa,B3),A3)
                 => aa(B,$o,aa(B,fun(B,$o),Pa,A3),B3) )
             => aa(B,$o,aa(B,fun(B,$o),Pa,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_400_order_Ostrict__trans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),C2) ) ) ) ).

% order.strict_trans
tff(fact_401_not__less__iff__gr__or__eq,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X)
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_402_dual__order_Ostrict__trans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_403_order_Ostrict__implies__not__eq,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_404_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_405_measure__induct,axiom,
    ! [C: $tType,B: $tType] :
      ( wellorder(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),A2: B] :
          ( ! [X2: B] :
              ( ! [Y3: B] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,Y3)),aa(B,C,F,X2))
                 => aa(B,$o,Pa,Y3) )
             => aa(B,$o,Pa,X2) )
         => aa(B,$o,Pa,A2) ) ) ).

% measure_induct
tff(fact_406_measure__induct__rule,axiom,
    ! [C: $tType,B: $tType] :
      ( wellorder(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),A2: B] :
          ( ! [X2: B] :
              ( ! [Y3: B] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,Y3)),aa(B,C,F,X2))
                 => aa(B,$o,Pa,Y3) )
             => aa(B,$o,Pa,X2) )
         => aa(B,$o,Pa,A2) ) ) ).

% measure_induct_rule
tff(fact_407_linorder__neqE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ( X != Y )
         => ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ) ).

% linorder_neqE
tff(fact_408_order__less__asym,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ).

% order_less_asym
tff(fact_409_linorder__neq__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ( X != Y )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
            | aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ) ).

% linorder_neq_iff
tff(fact_410_order__less__asym_H,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ).

% order_less_asym'
tff(fact_411_order__less__trans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),Z)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Z) ) ) ) ).

% order_less_trans
tff(fact_412_ord__eq__less__subst,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ord(C)
        & ord(B) )
     => ! [A2: B,F: fun(C,B),B2: C,C2: C] :
          ( ( A2 = aa(C,B,F,B2) )
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),B2),C2)
           => ( ! [X2: C,Y2: C] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,X2)),aa(C,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_413_ord__less__eq__subst,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ord(C)
        & ord(B) )
     => ! [A2: B,B2: B,F: fun(B,C),C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( ( aa(B,C,F,B2) = C2 )
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_414_order__less__irrefl,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),X) ) ).

% order_less_irrefl
tff(fact_415_order__less__subst1,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,F: fun(C,B),B2: C,C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,B2))
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),B2),C2)
           => ( ! [X2: C,Y2: C] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,X2)),aa(C,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_416_order__less__subst2,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,B2: B,F: fun(B,C),C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,B2)),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_417_order__less__not__sym,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ).

% order_less_not_sym
tff(fact_418_order__less__imp__triv,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B,Pa: $o] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X)
           => (Pa) ) ) ) ).

% order_less_imp_triv
tff(fact_419_linorder__less__linear,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
          | ( X = Y )
          | aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ).

% linorder_less_linear
tff(fact_420_order__less__imp__not__eq,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_421_order__less__imp__not__eq2,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_422_order__less__imp__not__less,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ).

% order_less_imp_not_less
tff(fact_423_verit__comp__simplify1_I1_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_424_floor__less__cancel,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(B,X)),archim6421214686448440834_floor(B,Y))
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ).

% floor_less_cancel
tff(fact_425_zless__nat__eq__int__zless,axiom,
    ! [Ma: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Ma)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_426_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_mono_iff
tff(fact_427_power__int__strict__increasing,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [N: int,N5: int,A2: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),power_int(B,A2,N)),power_int(B,A2,N5)) ) ) ) ).

% power_int_strict_increasing
tff(fact_428_zero__less__power__int,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),power_int(B,X,N)) ) ) ).

% zero_less_power_int
tff(fact_429_less__imp__of__nat__less,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ) ).

% less_imp_of_nat_less
tff(fact_430_of__nat__less__imp__less,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% of_nat_less_imp_less
tff(fact_431_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(2)
tff(fact_432_power__int__strict__decreasing,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [N: int,N5: int,A2: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),power_int(B,A2,N5)),power_int(B,A2,N)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_433_power__int__strict__antimono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),zero_zero(int))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),power_int(B,B2,N)),power_int(B,A2,N)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_434_ceiling__less__cancel,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(B,X)),archimedean_ceiling(B,Y))
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ).

% ceiling_less_cancel
tff(fact_435_one__less__power__int,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),power_int(B,A2,N)) ) ) ) ).

% one_less_power_int
tff(fact_436_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_437_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_438_power__int__antimono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),zero_zero(int))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,B2,N)),power_int(B,A2,N)) ) ) ) ) ).

% power_int_antimono
tff(fact_439_power__int__strict__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),power_int(B,A2,N)),power_int(B,B2,N)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_440_power__int__le__imp__less__exp,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Ma: int,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),power_int(B,X,Ma)),power_int(B,X,N))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ma),N) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_441_nat__less__iff,axiom,
    ! [W: int,Ma: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),Ma)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),Ma)) ) ) ).

% nat_less_iff
tff(fact_442_less__ceiling__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),Z)),X) ) ) ).

% less_ceiling_iff
tff(fact_443_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B5: B,A5: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B5),A5)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A5),B5) ) ) ).

% verit_comp_simplify1(3)
tff(fact_444_leD,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ).

% leD
tff(fact_445_leI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% leI
tff(fact_446_nless__le,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
        <=> ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_447_antisym__conv1,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_448_antisym__conv2,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_449_dense__ge,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [Z: B,Y: B] :
          ( ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z),X2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) ) ) ).

% dense_ge
tff(fact_450_dense__le,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [Y: B,Z: B] :
          ( ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Z) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) ) ) ).

% dense_le
tff(fact_451_less__le__not__le,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
            & ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ) ).

% less_le_not_le
tff(fact_452_not__le__imp__less,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Y: B,X: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ).

% not_le_imp_less
tff(fact_453_order_Oorder__iff__strict,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_454_order_Ostrict__iff__order,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_455_order_Ostrict__trans1,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_456_order_Ostrict__trans2,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_457_order_Ostrict__iff__not,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            & ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_458_dense__ge__bounded,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [Z: B,X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z),X)
         => ( ! [W2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z),W2)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),W2),X)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),W2) ) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_459_dense__le__bounded,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( ! [W2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),W2)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),W2),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),W2),Z) ) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_460_dual__order_Oorder__iff__strict,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_461_dual__order_Ostrict__iff__order,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_462_dual__order_Ostrict__trans1,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_463_dual__order_Ostrict__trans2,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_464_dual__order_Ostrict__iff__not,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            & ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_465_order_Ostrict__implies__order,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_466_dual__order_Ostrict__implies__order,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_467_order__le__less,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_468_order__less__le,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_469_linorder__not__le,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ).

% linorder_not_le
tff(fact_470_linorder__not__less,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% linorder_not_less
tff(fact_471_order__less__imp__le,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% order_less_imp_le
tff(fact_472_order__le__neq__trans,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( ( A2 != B2 )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_473_order__neq__le__trans,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != B2 )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_474_order__le__less__trans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),Z)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Z) ) ) ) ).

% order_le_less_trans
tff(fact_475_order__less__le__trans,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Z) ) ) ) ).

% order_less_le_trans
tff(fact_476_order__le__less__subst1,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,F: fun(C,B),B2: C,C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(C,B,F,B2))
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),B2),C2)
           => ( ! [X2: C,Y2: C] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,X2)),aa(C,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_477_order__le__less__subst2,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,B2: B,F: fun(B,C),C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,B2)),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_478_order__less__le__subst1,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,F: fun(C,B),B2: C,C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,B2))
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),B2),C2)
           => ( ! [X2: C,Y2: C] :
                  ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X2)),aa(C,B,F,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_479_order__less__le__subst2,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [A2: B,B2: B,F: fun(B,C),C2: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,B2)),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_480_linorder__le__less__linear,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
          | aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),X) ) ) ).

% linorder_le_less_linear
tff(fact_481_order__le__imp__less__or__eq,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_482_minf_I8_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ta),X4) ) ) ).

% minf(8)
tff(fact_483_minf_I6_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Ta) ) ) ).

% minf(6)
tff(fact_484_pinf_I8_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ta),X4) ) ) ).

% pinf(8)
tff(fact_485_pinf_I6_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ta: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Ta) ) ) ).

% pinf(6)
tff(fact_486_zero__less__iff__neq__zero,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [N: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),N)
        <=> ( N != zero_zero(B) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_487_gr__implies__not__zero,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [Ma: B,N: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),N)
         => ( N != zero_zero(B) ) ) ) ).

% gr_implies_not_zero
tff(fact_488_not__less__zero,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [N: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),N),zero_zero(B)) ) ).

% not_less_zero
tff(fact_489_gr__zeroI,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [N: B] :
          ( ( N != zero_zero(B) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),N) ) ) ).

% gr_zeroI
tff(fact_490_less__numeral__extra_I3_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),zero_zero(B)) ) ).

% less_numeral_extra(3)
tff(fact_491_add__mono__thms__linordered__field_I5_J,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),I),J)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),L) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_492_add__mono__thms__linordered__field_I2_J,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( ( I = J )
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),L) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_493_add__mono__thms__linordered__field_I1_J,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),I),J)
            & ( K = L ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_494_add__strict__mono,axiom,
    ! [B: $tType] :
      ( strict9044650504122735259up_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),D2)) ) ) ) ).

% add_strict_mono
tff(fact_495_add__strict__left__mono,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)) ) ) ).

% add_strict_left_mono
tff(fact_496_add__strict__right__mono,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ) ).

% add_strict_right_mono
tff(fact_497_add__less__imp__less__left,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_498_add__less__imp__less__right,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_499_less__numeral__extra_I4_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),one_one(B)) ) ).

% less_numeral_extra(4)
tff(fact_500_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_501_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% gr0I
tff(fact_502_not__gr0,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_503_not__less0,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),zero_zero(nat)) ).

% not_less0
tff(fact_504_less__zeroE,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),zero_zero(nat)) ).

% less_zeroE
tff(fact_505_gr__implies__not0,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_506_infinite__descent0,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( ~ aa(nat,$o,Pa,N2)
             => ? [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N2)
                  & ~ aa(nat,$o,Pa,M5) ) ) )
       => aa(nat,$o,Pa,N) ) ) ).

% infinite_descent0
tff(fact_507_infinite__descent0__measure,axiom,
    ! [B: $tType,V: fun(B,nat),Pa: fun(B,$o),X: B] :
      ( ! [X2: B] :
          ( ( aa(B,nat,V,X2) = zero_zero(nat) )
         => aa(B,$o,Pa,X2) )
     => ( ! [X2: B] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,V,X2))
           => ( ~ aa(B,$o,Pa,X2)
             => ? [Y3: B] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,V,Y3)),aa(B,nat,V,X2))
                  & ~ aa(B,$o,Pa,Y3) ) ) )
       => aa(B,$o,Pa,X) ) ) ).

% infinite_descent0_measure
tff(fact_508_less__fun__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),G: fun(B,C)] :
          ( aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less(fun(B,C)),F),G)
        <=> ( aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less_eq(fun(B,C)),F),G)
            & ~ aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),ord_less_eq(fun(B,C)),G),F) ) ) ) ).

% less_fun_def
tff(fact_509_reals__Archimedean2,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(nat,B,semiring_1_of_nat(B),N2)) ) ).

% reals_Archimedean2
tff(fact_510_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_511_exists__leI,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ( ! [N6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N6),N)
           => ~ aa(nat,$o,Pa,N6) )
       => aa(nat,$o,Pa,N) )
     => ? [N7: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
          & aa(nat,$o,Pa,N7) ) ) ).

% exists_leI
tff(fact_512_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,F: fun(B,nat),B2: nat] :
      ( aa(B,$o,Pa,K)
     => ( ! [Y2: B] :
            ( aa(B,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,Y2)),B2) )
       => ? [X2: B] :
            ( aa(B,$o,Pa,X2)
            & ! [Y3: B] :
                ( aa(B,$o,Pa,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,Y3)),aa(B,nat,F,X2)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_513_nat__less__le,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
        & ( Ma != N ) ) ) ).

% nat_less_le
tff(fact_514_less__imp__le__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% less_imp_le_nat
tff(fact_515_le__eq__less__or__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | ( Ma = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_516_less__or__eq__imp__le,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | ( Ma = N ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% less_or_eq_imp_le
tff(fact_517_le__neq__implies__less,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( ( Ma != N )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% le_neq_implies_less
tff(fact_518_less__mono__imp__le__mono,axiom,
    ! [F: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F,I2)),aa(nat,nat,F,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F,I)),aa(nat,nat,F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_519_arg__max__nat__le,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,F: fun(B,nat),B2: nat] :
      ( aa(B,$o,Pa,X)
     => ( ! [Y2: B] :
            ( aa(B,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,Y2)),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,X)),aa(B,nat,F,lattices_ord_arg_max(B,nat,F,Pa))) ) ) ).

% arg_max_nat_le
tff(fact_520_arg__max__nat__lemma,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,F: fun(B,nat),B2: nat] :
      ( aa(B,$o,Pa,K)
     => ( ! [Y2: B] :
            ( aa(B,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,Y2)),B2) )
       => ( aa(B,$o,Pa,lattices_ord_arg_max(B,nat,F,Pa))
          & ! [Y3: B] :
              ( aa(B,$o,Pa,Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,Y3)),aa(B,nat,F,lattices_ord_arg_max(B,nat,F,Pa))) ) ) ) ) ).

% arg_max_nat_lemma
tff(fact_521_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_522_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_523_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_524_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_525_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_526_trans__less__add1,axiom,
    ! [I: nat,J: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Ma)) ) ).

% trans_less_add1
tff(fact_527_trans__less__add2,axiom,
    ! [I: nat,J: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),J)) ) ).

% trans_less_add2
tff(fact_528_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% less_add_eq_less
tff(fact_529_union__less__mono,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: multiset(B),C6: multiset(B),B4: multiset(B),D3: multiset(B)] :
          ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),A4),C6)
         => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),B4),D3)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),D3)) ) ) ) ).

% union_less_mono
tff(fact_530_union__le__mono2,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B4: multiset(B),D3: multiset(B),C6: multiset(B)] :
          ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),B4),D3)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),B4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),D3)) ) ) ).

% union_le_mono2
tff(fact_531_union__le__mono1,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [B4: multiset(B),D3: multiset(B),C6: multiset(B)] :
          ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),B4),D3)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),ord_less(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),D3),C6)) ) ) ).

% union_le_mono1
tff(fact_532_ex__less__of__int,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [Z2: int] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(int,B,ring_1_of_int(B),Z2)) ) ).

% ex_less_of_int
tff(fact_533_ex__of__int__less,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [Z2: int] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),Z2)),X) ) ).

% ex_of_int_less
tff(fact_534_of__nat__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [R: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),R)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),aa(int,nat,nat2,archim6421214686448440834_floor(B,R)))),R) ) ) ).

% of_nat_floor
tff(fact_535_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_536_split__nat,axiom,
    ! [Pa: fun(nat,$o),I: int] :
      ( aa(nat,$o,Pa,aa(int,nat,nat2,I))
    <=> ( ! [N4: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N4) )
           => aa(nat,$o,Pa,N4) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,Pa,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_537_of__int__pos,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(int,B,ring_1_of_int(B),Z)) ) ) ).

% of_int_pos
tff(fact_538_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_539_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).

% pos_int_cases
tff(fact_540_of__nat__less__of__int__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat,X: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),N)),aa(int,B,ring_1_of_int(B),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),X) ) ) ).

% of_nat_less_of_int_iff
tff(fact_541_arg__minI,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [Pa: fun(B,$o),X: B,F: fun(B,C),Q: fun(B,$o)] :
          ( aa(B,$o,Pa,X)
         => ( ! [Y2: B] :
                ( aa(B,$o,Pa,Y2)
               => ~ aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,Y2)),aa(B,C,F,X)) )
           => ( ! [X2: B] :
                  ( aa(B,$o,Pa,X2)
                 => ( ! [Y3: B] :
                        ( aa(B,$o,Pa,Y3)
                       => ~ aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,Y3)),aa(B,C,F,X2)) )
                   => aa(B,$o,Q,X2) ) )
             => aa(B,$o,Q,lattices_ord_arg_min(B,C,F,Pa)) ) ) ) ) ).

% arg_minI
tff(fact_542_arg__maxI,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [Pa: fun(B,$o),X: B,F: fun(B,C),Q: fun(B,$o)] :
          ( aa(B,$o,Pa,X)
         => ( ! [Y2: B] :
                ( aa(B,$o,Pa,Y2)
               => ~ aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X)),aa(B,C,F,Y2)) )
           => ( ! [X2: B] :
                  ( aa(B,$o,Pa,X2)
                 => ( ! [Y3: B] :
                        ( aa(B,$o,Pa,Y3)
                       => ~ aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X2)),aa(B,C,F,Y3)) )
                   => aa(B,$o,Q,X2) ) )
             => aa(B,$o,Q,lattices_ord_arg_max(B,C,F,Pa)) ) ) ) ) ).

% arg_maxI
tff(fact_543_is__arg__min__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),X: B] :
          ( aa(B,$o,lattic501386751177426532rg_min(B,C,F,Pa),X)
        <=> ( aa(B,$o,Pa,X)
            & ~ ? [Y5: B] :
                  ( aa(B,$o,Pa,Y5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,Y5)),aa(B,C,F,X)) ) ) ) ) ).

% is_arg_min_def
tff(fact_544_zero__le__power__int,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),power_int(B,X,N)) ) ) ).

% zero_le_power_int
tff(fact_545_power__int__le__imp__le__exp,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Ma: int,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,X,Ma)),power_int(B,X,N))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ma),N) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_546_floor__split,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Pa: fun(int,$o),Ta: B] :
          ( aa(int,$o,Pa,archim6421214686448440834_floor(B,Ta))
        <=> ! [I3: int] :
              ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),I3)),Ta)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ta),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),I3)),one_one(B))) )
             => aa(int,$o,Pa,I3) ) ) ) ).

% floor_split
tff(fact_547_floor__eq__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,A2: int] :
          ( ( archim6421214686448440834_floor(B,X) = A2 )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),A2)),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),A2)),one_one(B))) ) ) ) ).

% floor_eq_iff
tff(fact_548_floor__unique,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Z)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B)))
           => ( archim6421214686448440834_floor(B,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_549_less__floor__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))),X) ) ) ).

% less_floor_iff
tff(fact_550_power__int__not__zero,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,N: int] :
          ( ( ( X != zero_zero(B) )
            | ( N = zero_zero(int) ) )
         => ( power_int(B,X,N) != zero_zero(B) ) ) ) ).

% power_int_not_zero
tff(fact_551_floor__le__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),Z)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))) ) ) ).

% floor_le_iff
tff(fact_552_floor__correct,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,X))),X)
          & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_553_floor__mono,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),archim6421214686448440834_floor(B,Y)) ) ) ).

% floor_mono
tff(fact_554_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_555_of__int__floor__le,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,X))),X) ) ).

% of_int_floor_le
tff(fact_556_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_557_ex__nat,axiom,
    ! [Pa: fun(nat,$o)] :
      ( ? [X_12: nat] : aa(nat,$o,Pa,X_12)
    <=> ? [X3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X3)
          & aa(nat,$o,Pa,aa(int,nat,nat2,X3)) ) ) ).

% ex_nat
tff(fact_558_all__nat,axiom,
    ! [Pa: fun(nat,$o)] :
      ( ! [X_12: nat] : aa(nat,$o,Pa,X_12)
    <=> ! [X3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X3)
         => aa(nat,$o,Pa,aa(int,nat,nat2,X3)) ) ) ).

% all_nat
tff(fact_559_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_560_is__arg__max__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),X: B] :
          ( aa(B,$o,lattic501386751176901750rg_max(B,C,F,Pa),X)
        <=> ( aa(B,$o,Pa,X)
            & ~ ? [Y5: B] :
                  ( aa(B,$o,Pa,Y5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X)),aa(B,C,F,Y5)) ) ) ) ) ).

% is_arg_max_def
tff(fact_561_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_562_add__less__le__mono,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_563_add__le__less__mono,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_564_add__mono__thms__linordered__field_I3_J,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),I),J)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K),L) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_565_add__mono__thms__linordered__field_I4_J,axiom,
    ! [B: $tType] :
      ( ordere580206878836729694up_add(B)
     => ! [I: B,J: B,K: B,L: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),J)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),L) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_566_add__less__zeroD,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),zero_zero(B))
            | aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),zero_zero(B)) ) ) ) ).

% add_less_zeroD
tff(fact_567_add__neg__neg,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),zero_zero(B)) ) ) ) ).

% add_neg_neg
tff(fact_568_add__pos__pos,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% add_pos_pos
tff(fact_569_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ ! [C4: B] :
                ( ( B2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C4) )
               => ( C4 = zero_zero(B) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_570_pos__add__strict,axiom,
    ! [B: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_571_zero__less__one,axiom,
    ! [B: $tType] :
      ( zero_less_one(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),one_one(B)) ) ).

% zero_less_one
tff(fact_572_not__one__less__zero,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),zero_zero(B)) ) ).

% not_one_less_zero
tff(fact_573_less__numeral__extra_I1_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),one_one(B)) ) ).

% less_numeral_extra(1)
tff(fact_574_less__add__one,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))) ) ).

% less_add_one
tff(fact_575_add__mono1,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),one_one(B))) ) ) ).

% add_mono1
tff(fact_576_floor__le__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),archimedean_ceiling(B,X)) ) ).

% floor_le_ceiling
tff(fact_577_of__nat__less__0__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: nat] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),Ma)),zero_zero(B)) ) ).

% of_nat_less_0_iff
tff(fact_578_ex__least__nat__le,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,N)
     => ( ~ aa(nat,$o,Pa,zero_zero(nat))
       => ? [K4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K4)
               => ~ aa(nat,$o,Pa,I4) )
            & aa(nat,$o,Pa,K4) ) ) ) ).

% ex_least_nat_le
tff(fact_579_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K4)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K4) = J ) ) ) ).

% less_imp_add_positive
tff(fact_580_ex__has__greatest__nat__lemma,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,F: fun(B,nat),N: nat] :
      ( aa(B,$o,Pa,K)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => ? [Y3: B] :
                ( aa(B,$o,Pa,Y3)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,Y3)),aa(B,nat,F,X2)) ) )
       => ? [Y2: B] :
            ( aa(B,$o,Pa,Y2)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,Y2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,F,K)),N)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_581_mono__nat__linear__lb,axiom,
    ! [F: fun(nat,nat),Ma: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F,M2)),aa(nat,nat,F,N2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F,Ma)),K)),aa(nat,nat,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K))) ) ).

% mono_nat_linear_lb
tff(fact_582_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_583_int__gr__induct,axiom,
    ! [K: int,I: int,Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,Pa,I2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,Pa,I) ) ) ) ).

% int_gr_induct
tff(fact_584_power__int__0__left__If,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Ma: int] :
          power_int(B,zero_zero(B),Ma) = $ite(Ma = zero_zero(int),one_one(B),zero_zero(B)) ) ).

% power_int_0_left_If
tff(fact_585_power__int__increasing,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [N: int,N5: int,A2: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,A2,N)),power_int(B,A2,N5)) ) ) ) ).

% power_int_increasing
tff(fact_586_le__floor__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Z)),X) ) ) ).

% le_floor_iff
tff(fact_587_le__floor__add,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),archim6421214686448440834_floor(B,Y))),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y))) ) ).

% le_floor_add
tff(fact_588_int__add__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(B,X)) = archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),Z)),X)) ) ).

% int_add_floor
tff(fact_589_floor__add__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),Z) = archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(int,B,ring_1_of_int(B),Z))) ) ).

% floor_add_int
tff(fact_590_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,X)),N)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nat_le_iff
tff(fact_591_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_592_int__eq__iff,axiom,
    ! [Ma: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Ma) = Z )
    <=> ( ( Ma = aa(int,nat,nat2,Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_593_of__nat__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [R: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),R),aa(nat,B,semiring_1_of_nat(B),aa(int,nat,nat2,archimedean_ceiling(B,R)))) ) ).

% of_nat_ceiling
tff(fact_594_nat__int__add,axiom,
    ! [A2: 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),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2) ).

% nat_int_add
tff(fact_595_add__neg__nonpos,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),zero_zero(B)) ) ) ) ).

% add_neg_nonpos
tff(fact_596_add__nonneg__pos,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_597_add__nonpos__neg,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),zero_zero(B)) ) ) ) ).

% add_nonpos_neg
tff(fact_598_add__pos__nonneg,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_599_add__strict__increasing,axiom,
    ! [B: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_600_add__strict__increasing2,axiom,
    ! [B: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_601_zero__less__two,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),one_one(B))) ) ).

% zero_less_two
tff(fact_602_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_603_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_604_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% add1_zle_eq
tff(fact_605_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_606_nonempty__has__size,axiom,
    ! [B: $tType,S: multiset(B)] :
      ( ( S != zero_zero(multiset(B)) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(multiset(B),nat,size_size(multiset(B)),S)) ) ).

% nonempty_has_size
tff(fact_607_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_608_power__int__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),power_int(B,X,N)),power_int(B,Y,N)) ) ) ) ) ).

% power_int_mono
tff(fact_609_one__le__power__int,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),power_int(B,X,N)) ) ) ) ).

% one_le_power_int
tff(fact_610_one__add__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),one_one(int)) = archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),one_one(B))) ) ).

% one_add_floor
tff(fact_611_nat__eq__iff2,axiom,
    ! [Ma: nat,W: int] :
      ( ( Ma = aa(int,nat,nat2,W) )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),Ma),Ma = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_612_nat__eq__iff,axiom,
    ! [W: int,Ma: nat] :
      ( ( aa(int,nat,nat2,W) = Ma )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),Ma),Ma = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_613_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K) ) ) ).

% le_nat_iff
tff(fact_614_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_615_ceiling__altdef,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          archimedean_ceiling(B,X) = $ite(X = aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,X)),archim6421214686448440834_floor(B,X),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_616_floor__exists,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [Z2: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Z2)),X)
          & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))) ) ) ).

% floor_exists
tff(fact_617_floor__exists1,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
        ? [X2: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),X2)),X)
          & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int))))
          & ! [Y3: int] :
              ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Y3)),X)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int)))) )
             => ( Y3 = X2 ) ) ) ) ).

% floor_exists1
tff(fact_618_discrete,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),B2) ) ) ).

% discrete
tff(fact_619_field__le__epsilon,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( ! [E4: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),E4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),Y),E4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% field_le_epsilon
tff(fact_620_floor__add,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),archimedean_frac(B,X)),archimedean_frac(B,Y))),one_one(B)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),archim6421214686448440834_floor(B,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(B,X)),archim6421214686448440834_floor(B,Y))),one_one(int))) ) ).

% floor_add
tff(fact_621_complete__interval,axiom,
    ! [B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A2: B,B2: B,Pa: fun(B,$o)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,Pa,A2)
           => ( ~ aa(B,$o,Pa,B2)
             => ? [C4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C4),B2)
                  & ! [X4: B] :
                      ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                        & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),C4) )
                     => aa(B,$o,Pa,X4) )
                  & ! [D4: B] :
                      ( ! [X2: B] :
                          ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),D4) )
                         => aa(B,$o,Pa,X2) )
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D4),C4) ) ) ) ) ) ) ).

% complete_interval
tff(fact_622_zero__le__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),one_one(B))),X) ) ) ).

% zero_le_ceiling
tff(fact_623_ceiling__less__zero,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(B,X)),zero_zero(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,uminus_uminus(B),one_one(B))) ) ) ).

% ceiling_less_zero
tff(fact_624_time__array__of__list,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B),H: heap_ext(product_unit)] : time_time(array(B),array_of_list(B,Xs),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)) ) ).

% time_array_of_list
tff(fact_625_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_626_gen__length__def,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),nat,gen_length(B,N),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(B),nat,size_size(list(B)),Xs)) ).

% gen_length_def
tff(fact_627_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_628_neg__equal__iff__equal,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,uminus_uminus(B),A2) = aa(B,B,uminus_uminus(B),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_629_add_Oinverse__inverse,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,uminus_uminus(B),aa(B,B,uminus_uminus(B),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_630_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_631_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_632_frac__frac,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_frac(B,archimedean_frac(B,X)) = archimedean_frac(B,X) ) ).

% frac_frac
tff(fact_633_neg__le__iff__le,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,uminus_uminus(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_634_add_Oinverse__neutral,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ( aa(B,B,uminus_uminus(B),zero_zero(B)) = zero_zero(B) ) ) ).

% add.inverse_neutral
tff(fact_635_neg__0__equal__iff__equal,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] :
          ( ( zero_zero(B) = aa(B,B,uminus_uminus(B),A2) )
        <=> ( zero_zero(B) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_636_neg__equal__0__iff__equal,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] :
          ( ( aa(B,B,uminus_uminus(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_637_equal__neg__zero,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( ( A2 = aa(B,B,uminus_uminus(B),A2) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% equal_neg_zero
tff(fact_638_neg__equal__zero,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( ( aa(B,B,uminus_uminus(B),A2) = A2 )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% neg_equal_zero
tff(fact_639_neg__less__iff__less,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,uminus_uminus(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_640_minus__add__distrib,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),aa(B,B,uminus_uminus(B),B2)) ) ).

% minus_add_distrib
tff(fact_641_minus__add__cancel,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_642_add__minus__cancel,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_643_lessI,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,N)) ).

% lessI
tff(fact_644_Suc__mono,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,N)) ) ).

% Suc_mono
tff(fact_645_Suc__less__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% Suc_less_eq
tff(fact_646_Suc__le__mono,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,Ma))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma) ) ).

% Suc_le_mono
tff(fact_647_add__Suc__right,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) ).

% add_Suc_right
tff(fact_648_negative__eq__positive,axiom,
    ! [N: nat,Ma: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),Ma) )
    <=> ( ( N = zero_zero(nat) )
        & ( Ma = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_649_of__int__minus,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: int] : aa(int,B,ring_1_of_int(B),aa(int,int,uminus_uminus(int),Z)) = aa(B,B,uminus_uminus(B),aa(int,B,ring_1_of_int(B),Z)) ) ).

% of_int_minus
tff(fact_650_negative__zle,axiom,
    ! [N: nat,Ma: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),Ma)) ).

% negative_zle
tff(fact_651_old_Onat_Osimps_I7_J,axiom,
    ! [B: $tType,F1: B,F22: fun(nat,fun(B,B)),Nat: nat] : aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),F1),F22),aa(nat,nat,suc,Nat)) = aa(B,B,aa(nat,fun(B,B),F22,Nat),aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),F1),F22),Nat)) ).

% old.nat.simps(7)
tff(fact_652_neg__less__eq__nonneg,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_653_less__eq__neg__nonpos,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,uminus_uminus(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% less_eq_neg_nonpos
tff(fact_654_neg__le__0__iff__le,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_655_neg__0__le__iff__le,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,uminus_uminus(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% neg_0_le_iff_le
tff(fact_656_neg__less__0__iff__less,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_657_neg__0__less__iff__less,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,uminus_uminus(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% neg_0_less_iff_less
tff(fact_658_neg__less__pos,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),A2)),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% neg_less_pos
tff(fact_659_less__neg__neg,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,uminus_uminus(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% less_neg_neg
tff(fact_660_ab__left__minus,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),A2) = zero_zero(B) ) ).

% ab_left_minus
tff(fact_661_add_Oright__inverse,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,uminus_uminus(B),A2)) = zero_zero(B) ) ).

% add.right_inverse
tff(fact_662_zero__less__Suc,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N)) ).

% zero_less_Suc
tff(fact_663_less__Suc0,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_664_negative__zless,axiom,
    ! [N: nat,Ma: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),Ma)) ).

% negative_zless
tff(fact_665_nat__zminus__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_666_floor__uminus__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int] : archim6421214686448440834_floor(B,aa(B,B,uminus_uminus(B),aa(int,B,ring_1_of_int(B),Z))) = aa(int,int,uminus_uminus(int),Z) ) ).

% floor_uminus_of_int
tff(fact_667_frac__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int] : archimedean_frac(B,aa(int,B,ring_1_of_int(B),Z)) = zero_zero(B) ) ).

% frac_of_int
tff(fact_668_dbl__inc__simps_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl_inc(B,aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% dbl_inc_simps(4)
tff(fact_669_add__neg__numeral__special_I8_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),one_one(B))),one_one(B)) = zero_zero(B) ) ) ).

% add_neg_numeral_special(8)
tff(fact_670_add__neg__numeral__special_I7_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,uminus_uminus(B),one_one(B))) = zero_zero(B) ) ) ).

% add_neg_numeral_special(7)
tff(fact_671_of__nat__Suc,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Ma: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,Ma)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(nat,B,semiring_1_of_nat(B),Ma)) ) ).

% of_nat_Suc
tff(fact_672_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_673_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_674_arg__max__natI,axiom,
    ! [B: $tType,Pa: fun(B,$o),K: B,F: fun(B,nat),B2: nat] :
      ( aa(B,$o,Pa,K)
     => ( ! [Y2: B] :
            ( aa(B,$o,Pa,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,Y2)),B2) )
       => aa(B,$o,Pa,lattices_ord_arg_max(B,nat,F,Pa)) ) ) ).

% arg_max_natI
tff(fact_675_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_676_Suc__lessD,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% Suc_lessD
tff(fact_677_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_678_Suc__lessI,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ( ( aa(nat,nat,suc,Ma) != N )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),N) ) ) ).

% Suc_lessI
tff(fact_679_less__SucE,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,N))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
       => ( Ma = N ) ) ) ).

% less_SucE
tff(fact_680_less__SucI,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,N)) ) ).

% less_SucI
tff(fact_681_Ex__less__Suc,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,N))
          & aa(nat,$o,Pa,I3) )
    <=> ( aa(nat,$o,Pa,N)
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
            & aa(nat,$o,Pa,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_682_less__Suc__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | ( Ma = N ) ) ) ).

% less_Suc_eq
tff(fact_683_nat__neq__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( ( Ma != N )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma) ) ) ).

% nat_neq_iff
tff(fact_684_not__less__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,Ma)) ) ).

% not_less_eq
tff(fact_685_All__less__Suc,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,N))
         => aa(nat,$o,Pa,I3) )
    <=> ( aa(nat,$o,Pa,N)
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
           => aa(nat,$o,Pa,I3) ) ) ) ).

% All_less_Suc
tff(fact_686_Suc__less__eq2,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),Ma)
    <=> ? [M6: nat] :
          ( ( Ma = aa(nat,nat,suc,M6) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M6) ) ) ).

% Suc_less_eq2
tff(fact_687_less__antisym,axiom,
    ! [N: nat,Ma: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,Ma))
       => ( Ma = N ) ) ) ).

% less_antisym
tff(fact_688_Suc__less__SucD,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,N))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% Suc_less_SucD
tff(fact_689_less__not__refl,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N) ).

% less_not_refl
tff(fact_690_less__not__refl2,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
     => ( Ma != N ) ) ).

% less_not_refl2
tff(fact_691_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_692_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_693_less__Suc__induct,axiom,
    ! [I: nat,J: nat,Pa: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K4)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,J2),K4)
                   => aa(nat,$o,aa(nat,fun(nat,$o),Pa,I2),K4) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),Pa,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_694_less__irrefl__nat,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N) ).

% less_irrefl_nat
tff(fact_695_nat__less__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N2)
             => aa(nat,$o,Pa,M5) )
         => aa(nat,$o,Pa,N2) )
     => aa(nat,$o,Pa,N) ) ).

% nat_less_induct
tff(fact_696_infinite__descent,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( ! [N2: nat] :
          ( ~ aa(nat,$o,Pa,N2)
         => ? [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N2)
              & ~ aa(nat,$o,Pa,M5) ) )
     => aa(nat,$o,Pa,N) ) ).

% infinite_descent
tff(fact_697_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_698_strict__inc__induct,axiom,
    ! [I: nat,J: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,Pa,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,Pa,aa(nat,nat,suc,I2))
               => aa(nat,$o,Pa,I2) ) )
         => aa(nat,$o,Pa,I) ) ) ) ).

% strict_inc_induct
tff(fact_699_not__less__less__Suc__eq,axiom,
    ! [N: nat,Ma: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,Ma))
      <=> ( N = Ma ) ) ) ).

% not_less_less_Suc_eq
tff(fact_700_infinite__descent__measure,axiom,
    ! [B: $tType,Pa: fun(B,$o),V: fun(B,nat),X: B] :
      ( ! [X2: B] :
          ( ~ aa(B,$o,Pa,X2)
         => ? [Y3: B] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,V,Y3)),aa(B,nat,V,X2))
              & ~ aa(B,$o,Pa,Y3) ) )
     => aa(B,$o,Pa,X) ) ).

% infinite_descent_measure
tff(fact_701_verit__negate__coefficient_I3_J,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = B2 )
         => ( aa(B,B,uminus_uminus(B),A2) = aa(B,B,uminus_uminus(B),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_702_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_703_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_704_minus__equation__iff,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,uminus_uminus(B),A2) = B2 )
        <=> ( aa(B,B,uminus_uminus(B),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_705_equation__minus__iff,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,uminus_uminus(B),B2) )
        <=> ( B2 = aa(B,B,uminus_uminus(B),A2) ) ) ) ).

% equation_minus_iff
tff(fact_706_fails__array__of__list,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B),H: heap_ext(product_unit)] : ~ time_fails(array(B),array_of_list(B,Xs),H) ) ).

% fails_array_of_list
tff(fact_707_int__of__nat__induct,axiom,
    ! [Pa: fun(int,$o),Z: int] :
      ( ! [N2: nat] : aa(int,$o,Pa,aa(nat,int,semiring_1_of_nat(int),N2))
     => ( ! [N2: nat] : aa(int,$o,Pa,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))))
       => aa(int,$o,Pa,Z) ) ) ).

% int_of_nat_induct
tff(fact_708_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_709_ceiling__def,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_ceiling(B,X) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(B,aa(B,B,uminus_uminus(B),X))) ) ).

% ceiling_def
tff(fact_710_floor__minus,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archim6421214686448440834_floor(B,aa(B,B,uminus_uminus(B),X)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(B,X)) ) ).

% floor_minus
tff(fact_711_ceiling__minus,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_ceiling(B,aa(B,B,uminus_uminus(B),X)) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(B,X)) ) ).

% ceiling_minus
tff(fact_712_le__imp__neg__le,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,uminus_uminus(B),A2)) ) ) ).

% le_imp_neg_le
tff(fact_713_minus__le__iff,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_714_le__minus__iff,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,uminus_uminus(B),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,uminus_uminus(B),A2)) ) ) ).

% le_minus_iff
tff(fact_715_verit__negate__coefficient_I2_J,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,uminus_uminus(B),A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_716_minus__less__iff,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),A2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),A2) ) ) ).

% minus_less_iff
tff(fact_717_less__minus__iff,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,uminus_uminus(B),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,uminus_uminus(B),A2)) ) ) ).

% less_minus_iff
tff(fact_718_add_Oinverse__distrib__swap,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,uminus_uminus(B),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_719_group__cancel_Oneg1,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A4: B,K: B,A2: B] :
          ( ( A4 = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),A2) )
         => ( aa(B,B,uminus_uminus(B),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),K)),aa(B,B,uminus_uminus(B),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_720_is__num__normalize_I8_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [A2: B,B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,uminus_uminus(B),A2)) ) ).

% is_num_normalize(8)
tff(fact_721_one__neq__neg__one,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ( one_one(B) != aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% one_neq_neg_one
tff(fact_722_lift__Suc__mono__less__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B),N: nat,Ma: nat] :
          ( ! [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,F,N2)),aa(nat,B,F,aa(nat,nat,suc,N2)))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,F,N)),aa(nat,B,F,Ma))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_723_lift__Suc__mono__less,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B),N: nat,N3: nat] :
          ( ! [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,F,N2)),aa(nat,B,F,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,F,N)),aa(nat,B,F,N3)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_724_dependent__nat__choice,axiom,
    ! [B: $tType,Pa: fun(nat,fun(B,$o)),Q: fun(nat,fun(B,fun(B,$o)))] :
      ( ? [X_1: B] : aa(B,$o,aa(nat,fun(B,$o),Pa,zero_zero(nat)),X_1)
     => ( ! [X2: B,N2: nat] :
            ( aa(B,$o,aa(nat,fun(B,$o),Pa,N2),X2)
           => ? [Y3: B] :
                ( aa(B,$o,aa(nat,fun(B,$o),Pa,aa(nat,nat,suc,N2)),Y3)
                & aa(B,$o,aa(B,fun(B,$o),aa(nat,fun(B,fun(B,$o)),Q,N2),X2),Y3) ) )
       => ? [F4: fun(nat,B)] :
          ! [N8: nat] :
            ( aa(B,$o,aa(nat,fun(B,$o),Pa,N8),aa(nat,B,F4,N8))
            & aa(B,$o,aa(B,fun(B,$o),aa(nat,fun(B,fun(B,$o)),Q,N8),aa(nat,B,F4,N8)),aa(nat,B,F4,aa(nat,nat,suc,N8))) ) ) ) ).

% dependent_nat_choice
tff(fact_725_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M2: nat] : N = aa(nat,nat,suc,M2) ) ).

% not0_implies_Suc
tff(fact_726_Zero__not__Suc,axiom,
    ! [Ma: nat] : zero_zero(nat) != aa(nat,nat,suc,Ma) ).

% Zero_not_Suc
tff(fact_727_Zero__neq__Suc,axiom,
    ! [Ma: nat] : zero_zero(nat) != aa(nat,nat,suc,Ma) ).

% Zero_neq_Suc
tff(fact_728_Suc__neq__Zero,axiom,
    ! [Ma: nat] : aa(nat,nat,suc,Ma) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_729_zero__induct,axiom,
    ! [Pa: fun(nat,$o),K: nat] :
      ( aa(nat,$o,Pa,K)
     => ( ! [N2: nat] :
            ( aa(nat,$o,Pa,aa(nat,nat,suc,N2))
           => aa(nat,$o,Pa,N2) )
       => aa(nat,$o,Pa,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_730_diff__induct,axiom,
    ! [Pa: fun(nat,fun(nat,$o)),Ma: nat,N: nat] :
      ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,X2),zero_zero(nat))
     => ( ! [Y2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,zero_zero(nat)),aa(nat,nat,suc,Y2))
       => ( ! [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,X2),Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),Pa,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y2)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),Pa,Ma),N) ) ) ) ).

% diff_induct
tff(fact_731_nat__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,Pa,N2)
           => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,Pa,N) ) ) ).

% nat_induct
tff(fact_732_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_733_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_734_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_735_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_736_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_737_exists__least__lemma,axiom,
    ! [Pa: fun(nat,$o)] :
      ( ~ aa(nat,$o,Pa,zero_zero(nat))
     => ( ? [X_1: nat] : aa(nat,$o,Pa,X_1)
       => ? [N2: nat] :
            ( ~ aa(nat,$o,Pa,N2)
            & aa(nat,$o,Pa,aa(nat,nat,suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_738_less__Suc__eq__0__disj,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,N))
    <=> ( ( Ma = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( Ma = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),N) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_739_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ? [M2: nat] : N = aa(nat,nat,suc,M2) ) ).

% gr0_implies_Suc
tff(fact_740_All__less__Suc2,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,N))
         => aa(nat,$o,Pa,I3) )
    <=> ( aa(nat,$o,Pa,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
           => aa(nat,$o,Pa,aa(nat,nat,suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_741_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
    <=> ? [M4: nat] : N = aa(nat,nat,suc,M4) ) ).

% gr0_conv_Suc
tff(fact_742_Ex__less__Suc2,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,N))
          & aa(nat,$o,Pa,I3) )
    <=> ( aa(nat,$o,Pa,zero_zero(nat))
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
            & aa(nat,$o,Pa,aa(nat,nat,suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_743_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_744_transitive__stepwise__le,axiom,
    ! [Ma: nat,N: nat,R2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R2,X2),Y2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R2,Y2),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R2,X2),Z2) ) )
         => ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,N2),aa(nat,nat,suc,N2))
           => aa(nat,$o,aa(nat,fun(nat,$o),R2,Ma),N) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_745_nat__induct__at__least,axiom,
    ! [Ma: nat,N: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,$o,Pa,Ma)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N2)
             => ( aa(nat,$o,Pa,N2)
               => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,Pa,N) ) ) ) ).

% nat_induct_at_least
tff(fact_746_full__nat__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( ! [N2: nat] :
          ( ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M5)),N2)
             => aa(nat,$o,Pa,M5) )
         => aa(nat,$o,Pa,N2) )
     => aa(nat,$o,Pa,N) ) ).

% full_nat_induct
tff(fact_747_not__less__eq__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),Ma) ) ).

% not_less_eq_eq
tff(fact_748_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),N) ).

% Suc_n_not_le_n
tff(fact_749_le__Suc__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
        | ( Ma = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_750_Suc__le__D,axiom,
    ! [N: nat,M3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),M3)
     => ? [M2: nat] : M3 = aa(nat,nat,suc,M2) ) ).

% Suc_le_D
tff(fact_751_le__SucI,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N)) ) ).

% le_SucI
tff(fact_752_le__SucE,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
       => ( Ma = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_753_Suc__leD,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% Suc_leD
tff(fact_754_Suc__leI,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),N) ) ).

% Suc_leI
tff(fact_755_Suc__le__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% Suc_le_eq
tff(fact_756_dec__induct,axiom,
    ! [I: nat,J: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,Pa,I)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),J)
               => ( aa(nat,$o,Pa,N2)
                 => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) ) ) )
         => aa(nat,$o,Pa,J) ) ) ) ).

% dec_induct
tff(fact_757_inc__induct,axiom,
    ! [I: nat,J: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,Pa,J)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),J)
               => ( aa(nat,$o,Pa,aa(nat,nat,suc,N2))
                 => aa(nat,$o,Pa,N2) ) ) )
         => aa(nat,$o,Pa,I) ) ) ) ).

% inc_induct
tff(fact_758_Suc__le__lessD,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% Suc_le_lessD
tff(fact_759_le__less__Suc__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,suc,Ma))
      <=> ( N = Ma ) ) ) ).

% le_less_Suc_eq
tff(fact_760_less__Suc__eq__le,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% less_Suc_eq_le
tff(fact_761_less__eq__Suc__le,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),Ma) ) ).

% less_eq_Suc_le
tff(fact_762_le__imp__less__Suc,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,suc,N)) ) ).

% le_imp_less_Suc
tff(fact_763_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_764_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_765_add__Suc__shift,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Ma)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_766_add__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Ma)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) ).

% add_Suc
tff(fact_767_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A2: nat] :
      ( ( A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_768_less__imp__Suc__add,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ? [K4: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K4)) ) ).

% less_imp_Suc_add
tff(fact_769_less__iff__Suc__add,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
    <=> ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K2)) ) ).

% less_iff_Suc_add
tff(fact_770_less__add__Suc2,axiom,
    ! [I: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),I))) ).

% less_add_Suc2
tff(fact_771_less__add__Suc1,axiom,
    ! [I: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma))) ).

% less_add_Suc1
tff(fact_772_less__natE,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ~ ! [Q4: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Q4)) ) ).

% less_natE
tff(fact_773_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% int_cases2
tff(fact_774_not__zle__0__negative,axiom,
    ! [N: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))) ).

% not_zle_0_negative
tff(fact_775_negD,axiom,
    ! [X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),zero_zero(int))
     => ? [N2: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_776_negative__zless__0,axiom,
    ! [N: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_777_of__nat__aux_Osimps_I2_J,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Inc: fun(B,B),N: nat,I: B] : semiri8178284476397505188at_aux(B,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(B,Inc,N,aa(B,B,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_778_rec__nat__Suc__imp,axiom,
    ! [B: $tType,F: fun(nat,B),F1: B,F22: fun(nat,fun(B,B)),N: nat] :
      ( ( F = aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),F1),F22) )
     => ( aa(nat,B,F,aa(nat,nat,suc,N)) = aa(B,B,aa(nat,fun(B,B),F22,N),aa(nat,B,F,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_779_neg__eq__iff__add__eq__0,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,uminus_uminus(B),A2) = B2 )
        <=> ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = zero_zero(B) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_780_eq__neg__iff__add__eq__0,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,uminus_uminus(B),B2) )
        <=> ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = zero_zero(B) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_781_add_Oinverse__unique,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = zero_zero(B) )
         => ( aa(B,B,uminus_uminus(B),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_782_ab__group__add__class_Oab__left__minus,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),A2) = zero_zero(B) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_783_add__eq__0__iff,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = zero_zero(B) )
        <=> ( B2 = aa(B,B,uminus_uminus(B),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_784_le__minus__one__simps_I4_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% le_minus_one_simps(4)
tff(fact_785_le__minus__one__simps_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),one_one(B))),one_one(B)) ) ).

% le_minus_one_simps(2)
tff(fact_786_zero__neq__neg__one,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ( zero_zero(B) != aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% zero_neq_neg_one
tff(fact_787_less__minus__one__simps_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),one_one(B))),one_one(B)) ) ).

% less_minus_one_simps(2)
tff(fact_788_less__minus__one__simps_I4_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% less_minus_one_simps(4)
tff(fact_789_lift__Suc__antimono__le,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B),N: nat,N3: nat] :
          ( ! [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,aa(nat,nat,suc,N2))),aa(nat,B,F,N2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,N3)),aa(nat,B,F,N)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_790_lift__Suc__mono__le,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B),N: nat,N3: nat] :
          ( ! [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,N2)),aa(nat,B,F,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,N)),aa(nat,B,F,N3)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_791_of__nat__neq__0,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,N)) != zero_zero(B) ) ).

% of_nat_neq_0
tff(fact_792_nat__compl__induct_H,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N2)
               => aa(nat,$o,Pa,Nn) )
           => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,Pa,N) ) ) ).

% nat_compl_induct'
tff(fact_793_nat__compl__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N2)
               => aa(nat,$o,Pa,Nn) )
           => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,Pa,N) ) ) ).

% nat_compl_induct
tff(fact_794_ex__least__nat__less,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,N)
     => ( ~ aa(nat,$o,Pa,zero_zero(nat))
       => ? [K4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K4),N)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),K4)
               => ~ aa(nat,$o,Pa,I4) )
            & aa(nat,$o,Pa,aa(nat,nat,suc,K4)) ) ) ) ).

% ex_least_nat_less
tff(fact_795_one__is__add,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N) )
    <=> ( ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( Ma = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_796_add__is__1,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( Ma = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_797_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_798_nat__induct__non__zero,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,Pa,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(nat,$o,Pa,N2)
               => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,Pa,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_799_int__cases4,axiom,
    ! [Ma: int] :
      ( ! [N2: nat] : Ma != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( Ma != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_800_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_801_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_802_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_803_not__int__zless__negative,axiom,
    ! [N: nat,Ma: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Ma))) ).

% not_int_zless_negative
tff(fact_804_of__int__of__nat,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [K: int] :
          aa(int,B,ring_1_of_int(B),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,B,semiring_1_of_nat(B),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_805_frac__ge__0,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),archimedean_frac(B,X)) ) ).

% frac_ge_0
tff(fact_806_frac__lt__1,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),archimedean_frac(B,X)),one_one(B)) ) ).

% frac_lt_1
tff(fact_807_frac__1__eq,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_frac(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),one_one(B))) = archimedean_frac(B,X) ) ).

% frac_1_eq
tff(fact_808_le__minus__one__simps_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),one_one(B))),zero_zero(B)) ) ).

% le_minus_one_simps(1)
tff(fact_809_le__minus__one__simps_I3_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% le_minus_one_simps(3)
tff(fact_810_less__minus__one__simps_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),one_one(B))),zero_zero(B)) ) ).

% less_minus_one_simps(1)
tff(fact_811_less__minus__one__simps_I3_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% less_minus_one_simps(3)
tff(fact_812_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ) ).

% int_cases3
tff(fact_813_int__zle__neg,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Ma)))
    <=> ( ( N = zero_zero(nat) )
        & ( Ma = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_814_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N2: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_815_negative__zle__0,axiom,
    ! [N: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int)) ).

% negative_zle_0
tff(fact_816_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
    <=> ? [N4: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_817_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_818_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_819_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).

% neg_int_cases
tff(fact_820_Suc__as__int,axiom,
    ! [X4: nat] : aa(nat,nat,suc,X4) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),one_one(int))) ).

% Suc_as_int
tff(fact_821_frac__eq,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( ( archimedean_frac(B,X) = X )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),one_one(B)) ) ) ) ).

% frac_eq
tff(fact_822_length__code,axiom,
    ! [B: $tType] : size_size(list(B)) = gen_length(B,zero_zero(nat)) ).

% length_code
tff(fact_823_compl__le__compl__iff,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),X)),aa(B,B,uminus_uminus(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% compl_le_compl_iff
tff(fact_824_power__int__minus__one__minus,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [N: int] : power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),aa(int,int,uminus_uminus(int),N)) = power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),N) ) ).

% power_int_minus_one_minus
tff(fact_825_dbl__dec__simps_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl_dec(B,zero_zero(B)) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% dbl_dec_simps(2)
tff(fact_826_nat__descend__induct,axiom,
    ! [N: nat,Pa: fun(nat,$o),Ma: nat] :
      ( ! [K4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K4)
         => aa(nat,$o,Pa,K4) )
     => ( ! [K4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
           => ( ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K4),I4)
                 => aa(nat,$o,Pa,I4) )
             => aa(nat,$o,Pa,K4) ) )
       => aa(nat,$o,Pa,Ma) ) ) ).

% nat_descend_induct
tff(fact_827_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_828_time__array__new,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,X: B,H: heap_ext(product_unit)] : time_time(array(B),array_new(B,N,X),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ) ).

% time_array_new
tff(fact_829_compl__le__swap2,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),Y)),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),X)),Y) ) ) ).

% compl_le_swap2
tff(fact_830_compl__le__swap1,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),aa(B,B,uminus_uminus(B),X))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,uminus_uminus(B),Y)) ) ) ).

% compl_le_swap1
tff(fact_831_compl__mono,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),Y)),aa(B,B,uminus_uminus(B),X)) ) ) ).

% compl_mono
tff(fact_832_dbl__dec__simps_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl_dec(B,one_one(B)) = one_one(B) ) ) ).

% dbl_dec_simps(3)
tff(fact_833_fails__array__new,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,X: B,H: heap_ext(product_unit)] : ~ time_fails(array(B),array_new(B,N,X),H) ) ).

% fails_array_new
tff(fact_834_Nitpick_OEps__psimp,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,Y: B] :
      ( aa(B,$o,Pa,X)
     => ( ~ aa(B,$o,Pa,Y)
       => ( ( fChoice(B,Pa) = Y )
         => ( fChoice(B,Pa) = X ) ) ) ) ).

% Nitpick.Eps_psimp
tff(fact_835_time__array__upd,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,P2: array(B),H: heap_ext(product_unit)] :
          time_time(array(B),array_upd(B,I,X,P2),H) = $ite(time_fails(array(B),array_upd(B,I,X,P2),H),zero_zero(nat),one_one(nat)) ) ).

% time_array_upd
tff(fact_836_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_837_time__array__make,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,F: fun(nat,B),H: heap_ext(product_unit)] : time_time(array(B),array_make(B,N,F),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ) ).

% time_array_make
tff(fact_838_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ap(nat,fun(nat,$o)),aTP_Lamp_aq(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_839_power__int__add__1_H,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: int] :
          ( ( ( X != zero_zero(B) )
            | ( Ma != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(B,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),one_one(int))) = aa(B,B,aa(B,fun(B,B),times_times(B),X),power_int(B,X,Ma)) ) ) ) ).

% power_int_add_1'
tff(fact_840_power__int__add__1,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: int] :
          ( ( ( X != zero_zero(B) )
            | ( Ma != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(B,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),one_one(int))) = aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,Ma)),X) ) ) ) ).

% power_int_add_1
tff(fact_841_length__ge__1__conv,axiom,
    ! [B: $tType,L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(B),nat,size_size(list(B)),L))
    <=> ( L != nil(B) ) ) ).

% length_ge_1_conv
tff(fact_842_foldr__length,axiom,
    ! [B: $tType,L: list(B)] : aa(nat,nat,foldr(B,nat,aTP_Lamp_ar(B,fun(nat,nat)),L),zero_zero(nat)) = aa(list(B),nat,size_size(list(B)),L) ).

% foldr_length
tff(fact_843_le__ceiling__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))),X) ) ) ).

% le_ceiling_iff
tff(fact_844_diff__self__eq__0,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),Ma) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_845_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_846_mult__cancel2,axiom,
    ! [Ma: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( Ma = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_847_mult__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( Ma = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_848_mult__0__right,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_849_mult__is__0,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) = zero_zero(nat) )
    <=> ( ( Ma = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_850_Suc__diff__diff,axiom,
    ! [Ma: nat,N: nat,K: 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,Ma)),N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),K) ).

% Suc_diff_diff
tff(fact_851_diff__Suc__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) ).

% diff_Suc_Suc
tff(fact_852_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_853_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_854_diff__diff__add__mset,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B),Pa: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),N5)),Pa) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),Pa)) ).

% diff_diff_add_mset
tff(fact_855_nat__1__eq__mult__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) )
    <=> ( ( Ma = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_856_nat__mult__eq__1__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) = one_one(nat) )
    <=> ( ( Ma = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_857_mult__zero__left,axiom,
    ! [B: $tType] :
      ( mult_zero(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% mult_zero_left
tff(fact_858_mult__zero__right,axiom,
    ! [B: $tType] :
      ( mult_zero(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),zero_zero(B)) = zero_zero(B) ) ).

% mult_zero_right
tff(fact_859_mult__eq__0__iff,axiom,
    ! [B: $tType] :
      ( semiri3467727345109120633visors(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) = zero_zero(B) )
        <=> ( ( A2 = zero_zero(B) )
            | ( B2 = zero_zero(B) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_860_mult__cancel__left,axiom,
    ! [B: $tType] :
      ( semiri6575147826004484403cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2) )
        <=> ( ( C2 = zero_zero(B) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_861_mult__cancel__right,axiom,
    ! [B: $tType] :
      ( semiri6575147826004484403cancel(B)
     => ! [A2: B,C2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2) )
        <=> ( ( C2 = zero_zero(B) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_862_diff__self,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),A2) = zero_zero(B) ) ).

% diff_self
tff(fact_863_diff__0__right,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),zero_zero(B)) = A2 ) ).

% diff_0_right
tff(fact_864_zero__diff,axiom,
    ! [B: $tType] :
      ( comm_monoid_diff(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% zero_diff
tff(fact_865_diff__zero,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),zero_zero(B)) = A2 ) ).

% diff_zero
tff(fact_866_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),A2) = zero_zero(B) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_867_mult__1,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),one_one(B)),A2) = A2 ) ).

% mult_1
tff(fact_868_mult_Oright__neutral,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),one_one(B)) = A2 ) ).

% mult.right_neutral
tff(fact_869_add__diff__cancel__right_H,axiom,
    ! [B: $tType] :
      ( cancel2418104881723323429up_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_870_add__diff__cancel__right,axiom,
    ! [B: $tType] :
      ( cancel2418104881723323429up_add(B)
     => ! [A2: B,C2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_871_add__diff__cancel__left_H,axiom,
    ! [B: $tType] :
      ( cancel2418104881723323429up_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_872_add__diff__cancel__left,axiom,
    ! [B: $tType] :
      ( cancel2418104881723323429up_add(B)
     => ! [C2: B,A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_873_diff__add__cancel,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_874_add__diff__cancel,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_875_minus__diff__eq,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2) ) ).

% minus_diff_eq
tff(fact_876_one__eq__mult__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) )
    <=> ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_877_mult__eq__1__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Ma = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_878_zero__less__diff,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% zero_less_diff
tff(fact_879_nat__0__less__mult__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% nat_0_less_mult_iff
tff(fact_880_mult__less__cancel2,axiom,
    ! [Ma: nat,K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% mult_less_cancel2
tff(fact_881_of__nat__mult,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_mult
tff(fact_882_diff__is__0__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% diff_is_0_eq
tff(fact_883_diff__is__0__eq_H,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_884_mult__Suc__right,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)) ).

% mult_Suc_right
tff(fact_885_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_886_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_887_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_888_of__int__mult,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [W: int,Z: int] : aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(int,B,ring_1_of_int(B),W)),aa(int,B,ring_1_of_int(B),Z)) ) ).

% of_int_mult
tff(fact_889_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_890_of__int__diff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [W: int,Z: int] : aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),minus_minus(int),W),Z)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),W)),aa(int,B,ring_1_of_int(B),Z)) ) ).

% of_int_diff
tff(fact_891_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_892_diff__ge__0__iff__ge,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_893_zero__comp__diff__simps_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_894_diff__gt__0__iff__gt,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_895_zero__comp__diff__simps_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_896_le__add__diff__inverse,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_897_le__add__diff__inverse2,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_898_mult__cancel__right2,axiom,
    ! [B: $tType] :
      ( ring_15535105094025558882visors(B)
     => ! [A2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(B) )
            | ( A2 = one_one(B) ) ) ) ) ).

% mult_cancel_right2
tff(fact_899_mult__cancel__right1,axiom,
    ! [B: $tType] :
      ( ring_15535105094025558882visors(B)
     => ! [C2: B,B2: B] :
          ( ( C2 = aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2) )
        <=> ( ( C2 = zero_zero(B) )
            | ( B2 = one_one(B) ) ) ) ) ).

% mult_cancel_right1
tff(fact_900_mult__cancel__left2,axiom,
    ! [B: $tType] :
      ( ring_15535105094025558882visors(B)
     => ! [C2: B,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(B) )
            | ( A2 = one_one(B) ) ) ) ) ).

% mult_cancel_left2
tff(fact_901_mult__cancel__left1,axiom,
    ! [B: $tType] :
      ( ring_15535105094025558882visors(B)
     => ! [C2: B,B2: B] :
          ( ( C2 = aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2) )
        <=> ( ( C2 = zero_zero(B) )
            | ( B2 = one_one(B) ) ) ) ) ).

% mult_cancel_left1
tff(fact_902_diff__add__zero,axiom,
    ! [B: $tType] :
      ( comm_monoid_diff(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = zero_zero(B) ) ).

% diff_add_zero
tff(fact_903_diff__numeral__special_I9_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),one_one(B)) = zero_zero(B) ) ) ).

% diff_numeral_special(9)
tff(fact_904_diff__0,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),A2) = aa(B,B,uminus_uminus(B),A2) ) ).

% diff_0
tff(fact_905_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_906_mult__minus1__right,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: B] : aa(B,B,aa(B,fun(B,B),times_times(B),Z),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),Z) ) ).

% mult_minus1_right
tff(fact_907_mult__minus1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),one_one(B))),Z) = aa(B,B,uminus_uminus(B),Z) ) ).

% mult_minus1
tff(fact_908_uminus__add__conv__diff,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),B2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_909_diff__minus__eq__add,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,uminus_uminus(B),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_910_Suc__pred,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_911_length__0__conv,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = zero_zero(nat) )
    <=> ( Xs = nil(B) ) ) ).

% length_0_conv
tff(fact_912_one__le__mult__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N) ) ) ).

% one_le_mult_iff
tff(fact_913_mult__le__cancel2,axiom,
    ! [Ma: nat,K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% mult_le_cancel2
tff(fact_914_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_915_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_916_Suc__diff,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Ma)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_917_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% zle_diff1_eq
tff(fact_918_floor__diff__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] : archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(int,B,ring_1_of_int(B),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(B,X)),Z) ) ).

% floor_diff_of_int
tff(fact_919_ceiling__diff__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] : archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(int,B,ring_1_of_int(B),Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(B,X)),Z) ) ).

% ceiling_diff_of_int
tff(fact_920_diff__numeral__special_I12_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(B,B,uminus_uminus(B),one_one(B))) = zero_zero(B) ) ) ).

% diff_numeral_special(12)
tff(fact_921_length__greater__0__conv,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(B),nat,size_size(list(B)),Xs))
    <=> ( Xs != nil(B) ) ) ).

% length_greater_0_conv
tff(fact_922_Suc__diff__1,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_923_power__int__minus__one__mult__self,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Ma: int] : aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),Ma)),power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),Ma)) = one_one(B) ) ).

% power_int_minus_one_mult_self
tff(fact_924_power__int__minus__one__mult__self_H,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Ma: int,B2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),Ma)),aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),Ma)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_925_floor__diff__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),one_one(B))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(B,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_926_ceiling__diff__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),one_one(B))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(B,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_927_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_928_ordering__top_Oextremum__uniqueI,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),Top: B,A2: B] :
      ( ordering_top(B,Less_eq,Less,Top)
     => ( aa(B,$o,aa(B,fun(B,$o),Less_eq,Top),A2)
       => ( A2 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_929_ordering__top_Onot__eq__extremum,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),Top: B,A2: B] :
      ( ordering_top(B,Less_eq,Less,Top)
     => ( ( A2 != Top )
      <=> aa(B,$o,aa(B,fun(B,$o),Less,A2),Top) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_930_ordering__top_Oextremum__unique,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),Top: B,A2: B] :
      ( ordering_top(B,Less_eq,Less,Top)
     => ( aa(B,$o,aa(B,fun(B,$o),Less_eq,Top),A2)
      <=> ( A2 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_931_ordering__top_Oextremum__strict,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),Top: B,A2: B] :
      ( ordering_top(B,Less_eq,Less,Top)
     => ~ aa(B,$o,aa(B,fun(B,$o),Less,Top),A2) ) ).

% ordering_top.extremum_strict
tff(fact_932_ordering__top_Oextremum,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),Top: B,A2: B] :
      ( ordering_top(B,Less_eq,Less,Top)
     => aa(B,$o,aa(B,fun(B,$o),Less_eq,A2),Top) ) ).

% ordering_top.extremum
tff(fact_933_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [B: $tType] :
      ( cancel2418104881723323429up_add(B)
     => ! [A2: B,C2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2)),B2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_934_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: $tType] :
      ( ab_semigroup_mult(B)
     => ! [B2: B,A2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_935_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [B: $tType] :
      ( ab_semigroup_mult(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) = aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_936_mult_Oassoc,axiom,
    ! [B: $tType] :
      ( semigroup_mult(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ).

% mult.assoc
tff(fact_937_diff__eq__diff__eq,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_938_inf__period_I1_J,axiom,
    ! [B: $tType] :
      ( ( comm_ring(B)
        & dvd(B) )
     => ! [Pa: fun(B,$o),D3: B,Q: fun(B,$o)] :
          ( ! [X2: B,K4: B] :
              ( aa(B,$o,Pa,X2)
            <=> aa(B,$o,Pa,aa(B,B,aa(B,fun(B,B),minus_minus(B),X2),aa(B,B,aa(B,fun(B,B),times_times(B),K4),D3))) )
         => ( ! [X2: B,K4: B] :
                ( aa(B,$o,Q,X2)
              <=> aa(B,$o,Q,aa(B,B,aa(B,fun(B,B),minus_minus(B),X2),aa(B,B,aa(B,fun(B,B),times_times(B),K4),D3))) )
           => ! [X4: B,K5: B] :
                ( ( aa(B,$o,Pa,X4)
                  & aa(B,$o,Q,X4) )
              <=> ( aa(B,$o,Pa,aa(B,B,aa(B,fun(B,B),minus_minus(B),X4),aa(B,B,aa(B,fun(B,B),times_times(B),K5),D3)))
                  & aa(B,$o,Q,aa(B,B,aa(B,fun(B,B),minus_minus(B),X4),aa(B,B,aa(B,fun(B,B),times_times(B),K5),D3))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_939_inf__period_I2_J,axiom,
    ! [B: $tType] :
      ( ( comm_ring(B)
        & dvd(B) )
     => ! [Pa: fun(B,$o),D3: B,Q: fun(B,$o)] :
          ( ! [X2: B,K4: B] :
              ( aa(B,$o,Pa,X2)
            <=> aa(B,$o,Pa,aa(B,B,aa(B,fun(B,B),minus_minus(B),X2),aa(B,B,aa(B,fun(B,B),times_times(B),K4),D3))) )
         => ( ! [X2: B,K4: B] :
                ( aa(B,$o,Q,X2)
              <=> aa(B,$o,Q,aa(B,B,aa(B,fun(B,B),minus_minus(B),X2),aa(B,B,aa(B,fun(B,B),times_times(B),K4),D3))) )
           => ! [X4: B,K5: B] :
                ( ( aa(B,$o,Pa,X4)
                  | aa(B,$o,Q,X4) )
              <=> ( aa(B,$o,Pa,aa(B,B,aa(B,fun(B,B),minus_minus(B),X4),aa(B,B,aa(B,fun(B,B),times_times(B),K5),D3)))
                  | aa(B,$o,Q,aa(B,B,aa(B,fun(B,B),minus_minus(B),X4),aa(B,B,aa(B,fun(B,B),times_times(B),K5),D3))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_940_square__diff__square__factored,axiom,
    ! [B: $tType] :
      ( comm_ring(B)
     => ! [X: B,Y: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Y)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y)) ) ).

% square_diff_square_factored
tff(fact_941_eq__add__iff2,axiom,
    ! [B: $tType] :
      ( ring(B)
     => ! [A2: B,E2: B,C2: B,B2: B,D2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),D2) )
        <=> ( C2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_942_eq__add__iff1,axiom,
    ! [B: $tType] :
      ( ring(B)
     => ! [A2: B,E2: B,C2: B,B2: B,D2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),D2) )
        <=> ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_943_le__add__iff1,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [A2: B,E2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),D2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),E2)),C2)),D2) ) ) ).

% le_add_iff1
tff(fact_944_le__add__iff2,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [A2: B,E2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),D2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)),E2)),D2)) ) ) ).

% le_add_iff2
tff(fact_945_less__add__iff2,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [A2: B,E2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),D2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)),E2)),D2)) ) ) ).

% less_add_iff2
tff(fact_946_less__add__iff1,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [A2: B,E2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),C2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),D2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),E2)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_947_square__diff__one__factored,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),one_one(B)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),one_one(B))),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),one_one(B))) ) ).

% square_diff_one_factored
tff(fact_948_of__nat__diff,axiom,
    ! [B: $tType] :
      ( semiring_1_cancel(B)
     => ! [N: nat,Ma: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
         => ( aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ) ) ).

% of_nat_diff
tff(fact_949_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K4: int] :
            ( aa(int,$o,P1,X2)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K4),D2))) )
       => ( ? [Z3: int] :
            ! [X2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z3)
             => ( aa(int,$o,Pa,X2)
              <=> aa(int,$o,P1,X2) ) )
         => ( ? [X_1: int] : aa(int,$o,P1,X_1)
           => ? [X_13: int] : aa(int,$o,Pa,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_950_plusinfinity,axiom,
    ! [D2: int,P: fun(int,$o),Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K4: int] :
            ( aa(int,$o,P,X2)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K4),D2))) )
       => ( ? [Z3: int] :
            ! [X2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X2)
             => ( aa(int,$o,Pa,X2)
              <=> aa(int,$o,P,X2) ) )
         => ( ? [X_1: int] : aa(int,$o,P,X_1)
           => ? [X_13: int] : aa(int,$o,Pa,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_951_int__minus,axiom,
    ! [N: nat,Ma: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) = 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),Ma)))) ).

% int_minus
tff(fact_952_diff__size__le__size__Diff,axiom,
    ! [B: $tType,M: multiset(B),M7: multiset(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(B),nat,size_size(multiset(B)),M)),aa(multiset(B),nat,size_size(multiset(B)),M7))),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),M7))) ).

% diff_size_le_size_Diff
tff(fact_953_diff__eq__diff__less__eq,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),D2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_954_diff__right__mono,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) ) ) ).

% diff_right_mono
tff(fact_955_diff__left__mono,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2)) ) ) ).

% diff_left_mono
tff(fact_956_diff__mono,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,D2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),D2)) ) ) ) ).

% diff_mono
tff(fact_957_eq__iff__diff__eq__0,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = B2 )
        <=> ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = zero_zero(B) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_958_diff__strict__right__mono,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_959_diff__strict__left__mono,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),A2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_960_diff__eq__diff__less,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),D2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_961_diff__strict__mono,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,D2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),D2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_962_diff__diff__eq,axiom,
    ! [B: $tType] :
      ( cancel2418104881723323429up_add(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ).

% diff_diff_eq
tff(fact_963_add__implies__diff,axiom,
    ! [B: $tType] :
      ( cancel1802427076303600483id_add(B)
     => ! [C2: B,B2: B,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2) = A2 )
         => ( C2 = aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_964_diff__add__eq__diff__diff__swap,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_965_diff__add__eq,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_966_diff__diff__eq2,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_967_add__diff__eq,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_968_eq__diff__eq,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,C2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2) )
        <=> ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_969_diff__eq__eq,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = C2 )
        <=> ( A2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_970_group__cancel_Osub1,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A4: B,K: B,A2: B,B2: B] :
          ( ( A4 = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),A2) )
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A4),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_971_minus__diff__commute,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [B2: B,A2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),B2)),A2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),A2)),B2) ) ).

% minus_diff_commute
tff(fact_972_mult__not__zero,axiom,
    ! [B: $tType] :
      ( mult_zero(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) != zero_zero(B) )
         => ( ( A2 != zero_zero(B) )
            & ( B2 != zero_zero(B) ) ) ) ) ).

% mult_not_zero
tff(fact_973_divisors__zero,axiom,
    ! [B: $tType] :
      ( semiri3467727345109120633visors(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) = zero_zero(B) )
         => ( ( A2 = zero_zero(B) )
            | ( B2 = zero_zero(B) ) ) ) ) ).

% divisors_zero
tff(fact_974_no__zero__divisors,axiom,
    ! [B: $tType] :
      ( semiri3467727345109120633visors(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) != zero_zero(B) ) ) ) ) ).

% no_zero_divisors
tff(fact_975_mult__left__cancel,axiom,
    ! [B: $tType] :
      ( semiri6575147826004484403cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_976_mult__right__cancel,axiom,
    ! [B: $tType] :
      ( semiri6575147826004484403cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_977_crossproduct__eq,axiom,
    ! [B: $tType] :
      ( semiri1453513574482234551roduct(B)
     => ! [W: B,Y: B,X: B,Z: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),W),Y)),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),W),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),X),Y)) )
        <=> ( ( W = X )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_978_crossproduct__noteq,axiom,
    ! [B: $tType] :
      ( semiri1453513574482234551roduct(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) != aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),D2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_979_ring__class_Oring__distribs_I2_J,axiom,
    ! [B: $tType] :
      ( ring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_980_ring__class_Oring__distribs_I1_J,axiom,
    ! [B: $tType] :
      ( ring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_981_comm__semiring__class_Odistrib,axiom,
    ! [B: $tType] :
      ( comm_semiring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_982_distrib__left,axiom,
    ! [B: $tType] :
      ( semiring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ).

% distrib_left
tff(fact_983_distrib__right,axiom,
    ! [B: $tType] :
      ( semiring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ).

% distrib_right
tff(fact_984_combine__common__factor,axiom,
    ! [B: $tType] :
      ( semiring(B)
     => ! [A2: B,E2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),E2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),E2)),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),E2)),C2) ) ).

% combine_common_factor
tff(fact_985_mult_Ocomm__neutral,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),one_one(B)) = A2 ) ).

% mult.comm_neutral
tff(fact_986_comm__monoid__mult__class_Omult__1,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),one_one(B)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_987_diffs0__imp__equal,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma) = zero_zero(nat) )
       => ( Ma = N ) ) ) ).

% diffs0_imp_equal
tff(fact_988_minus__nat_Odiff__0,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),zero_zero(nat)) = Ma ).

% minus_nat.diff_0
tff(fact_989_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_990_zero__induct__lemma,axiom,
    ! [Pa: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,Pa,K)
     => ( ! [N2: nat] :
            ( aa(nat,$o,Pa,aa(nat,nat,suc,N2))
           => aa(nat,$o,Pa,N2) )
       => aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I)) ) ) ).

% zero_induct_lemma
tff(fact_991_diff__less__mono2,axiom,
    ! [Ma: nat,N: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),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),Ma)) ) ) ).

% diff_less_mono2
tff(fact_992_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N)),K) ) ).

% less_imp_diff_less
tff(fact_993_diff__le__mono2,axiom,
    ! [Ma: nat,N: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),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),Ma)) ) ).

% diff_le_mono2
tff(fact_994_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_995_diff__le__self,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),Ma) ).

% diff_le_self
tff(fact_996_diff__le__mono,axiom,
    ! [Ma: nat,N: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L)) ) ).

% diff_le_mono
tff(fact_997_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_998_le__diff__iff,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ) ).

% le_diff_iff
tff(fact_999_eq__diff__iff,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K) )
        <=> ( Ma = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1000_diff__add__inverse2,axiom,
    ! [Ma: 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),Ma),N)),N) = Ma ).

% diff_add_inverse2
tff(fact_1001_diff__add__inverse,axiom,
    ! [N: nat,Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma)),N) = Ma ).

% diff_add_inverse
tff(fact_1002_diff__cancel2,axiom,
    ! [Ma: nat,K: 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),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) ).

% diff_cancel2
tff(fact_1003_Nat_Odiff__cancel,axiom,
    ! [K: nat,Ma: 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),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) ).

% Nat.diff_cancel
tff(fact_1004_Multiset_Odiff__add,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B),Q: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),Q)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),N5)),Q) ).

% Multiset.diff_add
tff(fact_1005_diff__union__cancelL,axiom,
    ! [B: $tType,N5: multiset(B),M: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),N5),M)),N5) = M ).

% diff_union_cancelL
tff(fact_1006_diff__union__cancelR,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)),N5) = M ).

% diff_union_cancelR
tff(fact_1007_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_1008_Suc__mult__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( Ma = N ) ) ).

% Suc_mult_cancel1
tff(fact_1009_mult__of__nat__commute,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [X: nat,Y: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),X)),Y) = aa(B,B,aa(B,fun(B,B),times_times(B),Y),aa(nat,B,semiring_1_of_nat(B),X)) ) ).

% mult_of_nat_commute
tff(fact_1010_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_1011_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_1012_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_1013_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_1014_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_1015_le__square,axiom,
    ! [Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Ma)) ).

% le_square
tff(fact_1016_le__cube,axiom,
    ! [Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Ma))) ).

% le_cube
tff(fact_1017_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N2: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_1018_add__mult__distrib2,axiom,
    ! [K: nat,Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_1019_add__mult__distrib,axiom,
    ! [Ma: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_1020_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_1021_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_1022_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_1023_mult__of__int__commute,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: int,Y: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(int,B,ring_1_of_int(B),X)),Y) = aa(B,B,aa(B,fun(B,B),times_times(B),Y),aa(int,B,ring_1_of_int(B),X)) ) ).

% mult_of_int_commute
tff(fact_1024_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_1025_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_1026_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_1027_power__int__mult__distrib,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B,Y: B,Ma: int] : power_int(B,aa(B,B,aa(B,fun(B,B),times_times(B),X),Y),Ma) = aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,Ma)),power_int(B,Y,Ma)) ) ).

% power_int_mult_distrib
tff(fact_1028_power__int__commutes,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,N: int] : aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,N)),X) = aa(B,B,aa(B,fun(B,B),times_times(B),X),power_int(B,X,N)) ) ).

% power_int_commutes
tff(fact_1029_power__int__mult,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: int,N: int] : power_int(B,X,aa(int,int,aa(int,fun(int,int),times_times(int),Ma),N)) = power_int(B,power_int(B,X,Ma),N) ) ).

% power_int_mult
tff(fact_1030_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ).

% int_ops(6)
tff(fact_1031_mult__eq__if,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) = $ite(Ma = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))),N))) ).

% mult_eq_if
tff(fact_1032_decr__mult__lemma,axiom,
    ! [D2: int,Pa: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,Pa,X2)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,Pa,X4)
             => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1033_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),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_1034_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z4),Z)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_1035_lambda__zero,axiom,
    ! [B: $tType] :
      ( mult_zero(B)
     => ( aTP_Lamp_as(B,B) = aa(B,fun(B,B),times_times(B),zero_zero(B)) ) ) ).

% lambda_zero
tff(fact_1036_lambda__one,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ( aTP_Lamp_at(B,B) = aa(B,fun(B,B),times_times(B),one_one(B)) ) ) ).

% lambda_one
tff(fact_1037_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_1038_zdiff__int__split,axiom,
    ! [Pa: fun(int,$o),X: nat,Y: nat] :
      ( aa(int,$o,Pa,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X)
         => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
         => aa(int,$o,Pa,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_1039_le__iff__diff__le__0,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),zero_zero(B)) ) ) ).

% le_iff_diff_le_0
tff(fact_1040_power__int__minus__mult,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B,N: int] :
          ( ( ( X != zero_zero(B) )
            | ( N != zero_zero(int) ) )
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(B,X,N) ) ) ) ).

% power_int_minus_mult
tff(fact_1041_less__iff__diff__less__0,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),zero_zero(B)) ) ) ).

% less_iff_diff_less_0
tff(fact_1042_le__mult__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A2)),archim6421214686448440834_floor(B,B2))),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))) ) ) ) ).

% le_mult_floor
tff(fact_1043_diff__le__eq,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_1044_le__diff__eq,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_1045_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)),A2) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1046_le__add__diff,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_1047_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1048_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1049_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1050_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1051_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1052_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1053_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1054_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
           => ( ( aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2) = C2 )
            <=> ( B2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1055_add__le__imp__le__diff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [I: B,K: B,N: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),N)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),aa(B,B,aa(B,fun(B,B),minus_minus(B),N),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1056_add__le__add__imp__diff__le,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [I: B,K: B,N: B,J: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),N)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),K))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K)),N)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),K))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),N),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1057_mult__ceiling__le,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A2)),archimedean_ceiling(B,B2))) ) ) ) ).

% mult_ceiling_le
tff(fact_1058_less__diff__eq,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_1059_diff__less__eq,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_1060_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1061_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,uminus_uminus(B),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1062_diff__conv__add__uminus,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,uminus_uminus(B),B2)) ) ).

% diff_conv_add_uminus
tff(fact_1063_group__cancel_Osub2,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [B4: B,K: B,B2: B,A2: B] :
          ( ( B4 = aa(B,B,aa(B,fun(B,B),plus_plus(B),K),B2) )
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),K)),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1064_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_1065_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [B: $tType] :
      ( ordere2520102378445227354miring(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1066_zero__le__mult__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1067_mult__nonneg__nonpos2,axiom,
    ! [B: $tType] :
      ( ordered_semiring_0(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2)),zero_zero(B)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1068_mult__nonpos__nonneg,axiom,
    ! [B: $tType] :
      ( ordered_semiring_0(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1069_mult__nonneg__nonpos,axiom,
    ! [B: $tType] :
      ( ordered_semiring_0(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1070_mult__nonneg__nonneg,axiom,
    ! [B: $tType] :
      ( ordered_semiring_0(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1071_split__mult__neg__le,axiom,
    ! [B: $tType] :
      ( ordered_semiring_0(B)
     => ! [A2: B,B2: B] :
          ( ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B)) ) ) ).

% split_mult_neg_le
tff(fact_1072_mult__le__0__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1073_mult__right__mono,axiom,
    ! [B: $tType] :
      ( ordered_semiring(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_1074_mult__right__mono__neg,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1075_mult__left__mono,axiom,
    ! [B: $tType] :
      ( ordered_semiring(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_1076_mult__nonpos__nonpos,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1077_mult__left__mono__neg,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1078_split__mult__pos__le,axiom,
    ! [B: $tType] :
      ( ordered_ring(B)
     => ! [A2: B,B2: B] :
          ( ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) ) ) ).

% split_mult_pos_le
tff(fact_1079_zero__le__square,axiom,
    ! [B: $tType] :
      ( linordered_ring(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),A2)) ) ).

% zero_le_square
tff(fact_1080_mult__mono_H,axiom,
    ! [B: $tType] :
      ( ordered_semiring(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1081_mult__mono,axiom,
    ! [B: $tType] :
      ( ordered_semiring(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_1082_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [B: $tType] :
      ( linord2810124833399127020strict(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1083_mult__less__cancel__right__disj,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1084_mult__strict__right__mono,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1085_mult__strict__right__mono__neg,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1086_mult__less__cancel__left__disj,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1087_mult__strict__left__mono,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1088_mult__strict__left__mono__neg,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1089_mult__less__cancel__left__pos,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1090_mult__less__cancel__left__neg,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1091_zero__less__mult__pos2,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1092_zero__less__mult__pos,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_1093_zero__less__mult__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1094_mult__pos__neg2,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2)),zero_zero(B)) ) ) ) ).

% mult_pos_neg2
tff(fact_1095_mult__pos__pos,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_1096_mult__pos__neg,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B)) ) ) ) ).

% mult_pos_neg
tff(fact_1097_mult__neg__pos,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B)) ) ) ) ).

% mult_neg_pos
tff(fact_1098_mult__less__0__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B)) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1099_not__square__less__zero,axiom,
    ! [B: $tType] :
      ( linordered_ring(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),A2)),zero_zero(B)) ) ).

% not_square_less_zero
tff(fact_1100_mult__neg__neg,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_1101_add__scale__eq__noteq,axiom,
    ! [B: $tType] :
      ( semiri1453513574482234551roduct(B)
     => ! [R: B,A2: B,B2: B,C2: B,D2: B] :
          ( ( R != zero_zero(B) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),R),C2)) != aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),R),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1102_less__1__mult,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Ma: B,N: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),N)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),N)) ) ) ) ).

% less_1_mult
tff(fact_1103_Suc__to__right,axiom,
    ! [N: nat,Ma: nat] :
      ( ( aa(nat,nat,suc,N) = Ma )
     => ( N = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_1104_diff__less,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),Ma) ) ) ).

% diff_less
tff(fact_1105_Suc__diff__Suc,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N) ) ) ).

% Suc_diff_Suc
tff(fact_1106_diff__less__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),aa(nat,nat,suc,Ma)) ).

% diff_less_Suc
tff(fact_1107_square__eq__1__iff,axiom,
    ! [B: $tType] :
      ( ring_15535105094025558882visors(B)
     => ! [X: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),X),X) = one_one(B) )
        <=> ( ( X = one_one(B) )
            | ( X = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1108_Suc__diff__le,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) ) ) ).

% Suc_diff_le
tff(fact_1109_diff__add__0,axiom,
    ! [N: nat,Ma: 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),Ma)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1110_less__diff__iff,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ) ).

% less_diff_iff
tff(fact_1111_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2)) ) ) ).

% diff_less_mono
tff(fact_1112_add__diff__inverse__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) = Ma ) ) ).

% add_diff_inverse_nat
tff(fact_1113_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_1114_union__diff__assoc,axiom,
    ! [B: $tType,C6: multiset(B),B4: multiset(B),A4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C6),B4) = zero_zero(multiset(B)) )
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),C6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),C6)) ) ) ).

% union_diff_assoc
tff(fact_1115_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1116_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1117_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1118_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1119_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1120_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_1121_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_1122_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_1123_int__le__induct,axiom,
    ! [I: int,K: int,Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,Pa,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,Pa,I2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,Pa,I) ) ) ) ).

% int_le_induct
tff(fact_1124_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ).

% Suc_mult_less_cancel1
tff(fact_1125_diff__Suc__eq__diff__pred,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),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),Ma),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1126_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z4),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_1127_int__less__induct,axiom,
    ! [I: int,K: int,Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,Pa,I2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,Pa,I) ) ) ) ).

% int_less_induct
tff(fact_1128_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% Suc_mult_le_cancel1
tff(fact_1129_mult__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Ma)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)) ).

% mult_Suc
tff(fact_1130_mlex__snd__decrI,axiom,
    ! [A2: nat,A5: nat,B2: nat,B5: nat,N5: nat] :
      ( ( A2 = A5 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N5)),B5)) ) ) ).

% mlex_snd_decrI
tff(fact_1131_mlex__fst__decrI,axiom,
    ! [A2: nat,A5: nat,B2: nat,N5: nat,B5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B5),N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N5)),B5)) ) ) ) ).

% mlex_fst_decrI
tff(fact_1132_mlex__bound,axiom,
    ! [A2: nat,A4: nat,B2: nat,N5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N5)) ) ) ).

% mlex_bound
tff(fact_1133_mlex__leI,axiom,
    ! [A2: nat,A5: nat,B2: nat,B5: nat,N5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N5)),B5)) ) ) ).

% mlex_leI
tff(fact_1134_mult__eq__self__implies__10,axiom,
    ! [Ma: nat,N: nat] :
      ( ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N) )
     => ( ( N = one_one(nat) )
        | ( Ma = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1135_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1136_zmult__eq__1__iff,axiom,
    ! [Ma: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Ma),N) = one_one(int) )
    <=> ( ( ( Ma = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( Ma = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_1137_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Ma: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Ma),N) = one_one(int) )
     => ( ( Ma = one_one(int) )
        | ( Ma = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_1138_list_Osize_I3_J,axiom,
    ! [B: $tType] : aa(list(B),nat,size_size(list(B)),nil(B)) = zero_zero(nat) ).

% list.size(3)
tff(fact_1139_len__greater__imp__nonempty,axiom,
    ! [B: $tType,X: nat,L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(B),nat,size_size(list(B)),L))
     => ( L != nil(B) ) ) ).

% len_greater_imp_nonempty
tff(fact_1140_fails__array__make,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,F: fun(nat,B),H: heap_ext(product_unit)] : ~ time_fails(array(B),array_make(B,N,F),H) ) ).

% fails_array_make
tff(fact_1141_mult__less__le__imp__less,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1142_mult__le__less__imp__less,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1143_mult__right__le__imp__le,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1144_mult__left__le__imp__le,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1145_mult__le__cancel__left__pos,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1146_mult__le__cancel__left__neg,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1147_mult__less__cancel__right,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1148_mult__strict__mono_H,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1149_mult__right__less__imp__less,axiom,
    ! [B: $tType] :
      ( linordered_semiring(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1150_mult__less__cancel__left,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1151_mult__strict__mono,axiom,
    ! [B: $tType] :
      ( linord8928482502909563296strict(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1152_mult__left__less__imp__less,axiom,
    ! [B: $tType] :
      ( linordered_semiring(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1153_mult__le__cancel__right,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1154_mult__le__cancel__left,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1155_sum__squares__ge__zero,axiom,
    ! [B: $tType] :
      ( linordered_ring(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1156_mult__left__le,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_1157_mult__le__one,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),one_one(B)) ) ) ) ) ).

% mult_le_one
tff(fact_1158_mult__right__le__one__le,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Y)),X) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1159_mult__left__le__one__le,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),Y),X)),X) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1160_not__sum__squares__lt__zero,axiom,
    ! [B: $tType] :
      ( linordered_ring(B)
     => ! [X: B,Y: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Y))),zero_zero(B)) ) ).

% not_sum_squares_lt_zero
tff(fact_1161_ex__less__of__nat__mult,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => ? [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N2)),X)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1162_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I))),N) ) ).

% diff_Suc_less
tff(fact_1163_nat__diff__split__asm,axiom,
    ! [Pa: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,Pa,zero_zero(nat)) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ aa(nat,$o,Pa,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_1164_nat__diff__split,axiom,
    ! [Pa: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,Pa,zero_zero(nat)) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => aa(nat,$o,Pa,D5) ) ) ) ).

% nat_diff_split
tff(fact_1165_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1166_power__int__minus__one__diff__commute,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: int,B2: int] : power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A2),B2)) = power_int(B,aa(B,B,uminus_uminus(B),one_one(B)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A2)) ) ).

% power_int_minus_one_diff_commute
tff(fact_1167_n__less__n__mult__m,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Ma)) ) ) ).

% n_less_n_mult_m
tff(fact_1168_n__less__m__mult__n,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)) ) ) ).

% n_less_m_mult_n
tff(fact_1169_one__less__mult,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)) ) ) ).

% one_less_mult
tff(fact_1170_int__induct,axiom,
    ! [Pa: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,Pa,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,Pa,I2)
             => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,Pa,I2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,Pa,I) ) ) ) ).

% int_induct
tff(fact_1171_pos__zmult__eq__1__iff,axiom,
    ! [Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ma)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Ma),N) = one_one(int) )
      <=> ( ( Ma = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1172_frac__def,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_frac(B,X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,X))) ) ).

% frac_def
tff(fact_1173_dbl__dec__def,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [X: B] : neg_numeral_dbl_dec(B,X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),X)),one_one(B)) ) ).

% dbl_dec_def
tff(fact_1174_field__le__mult__one__interval,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( ! [Z2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Z2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),one_one(B))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),Z2),X)),Y) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1175_mult__less__cancel__right2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),C2)
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B)) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1176_mult__less__cancel__right1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),one_one(B)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1177_mult__less__cancel__left2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),C2)
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B)) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1178_mult__less__cancel__left1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),one_one(B)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1179_mult__le__cancel__right2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),C2)
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B)) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1180_mult__le__cancel__right1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),one_one(B)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1181_mult__le__cancel__left2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),C2)
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B)) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1182_mult__le__cancel__left1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),one_one(B)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1183_convex__bound__le,axiom,
    ! [B: $tType] :
      ( linord6961819062388156250ring_1(B)
     => ! [X: B,A2: B,Y: B,U: B,V2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),U)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),V2)
               => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),U),V2) = one_one(B) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),U),X)),aa(B,B,aa(B,fun(B,B),times_times(B),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1184_nz__le__conv__less,axiom,
    ! [K: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),aa(nat,nat,suc,zero_zero(nat)))),Ma) ) ) ).

% nz_le_conv_less
tff(fact_1185_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Ma)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1186_Suc__pred_H,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1187_frac__add,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          archimedean_frac(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),archimedean_frac(B,X)),archimedean_frac(B,Y))),one_one(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),archimedean_frac(B,X)),archimedean_frac(B,Y)),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),archimedean_frac(B,X)),archimedean_frac(B,Y))),one_one(B))) ) ).

% frac_add
tff(fact_1188_add__eq__if,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N) = $ite(Ma = zero_zero(nat),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),Ma),one_one(nat))),N))) ).

% add_eq_if
tff(fact_1189_Suc__n__minus__m__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Ma)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1190_foldr__length__aux,axiom,
    ! [B: $tType,L: list(B),A2: nat] : aa(nat,nat,foldr(B,nat,aTP_Lamp_ar(B,fun(nat,nat)),L),A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(B),nat,size_size(list(B)),L)) ).

% foldr_length_aux
tff(fact_1191_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1192_power__int__add,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: int,N: int] :
          ( ( ( X != zero_zero(B) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),N) != zero_zero(int) ) )
         => ( power_int(B,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ma),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,Ma)),power_int(B,X,N)) ) ) ) ).

% power_int_add
tff(fact_1193_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R3: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1194_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R3: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1195_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B5: int,Q5: int,R3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),R3)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1196_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B5: int,Q5: int,R3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),R3))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B5)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1197_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B5),Q5)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1198_incr__mult__lemma,axiom,
    ! [D2: int,Pa: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,Pa,X2)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,Pa,X4)
             => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1199_ceiling__diff__floor__le__1,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(B,X)),archim6421214686448440834_floor(B,X))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_1200_convex__bound__lt,axiom,
    ! [B: $tType] :
      ( linord715952674999750819strict(B)
     => ! [X: B,A2: B,Y: B,U: B,V2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),U)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),V2)
               => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),U),V2) = one_one(B) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),U),X)),aa(B,B,aa(B,fun(B,B),times_times(B),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1201_ceiling__correct,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,X))),one_one(B))),X)
          & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,X))) ) ) ).

% ceiling_correct
tff(fact_1202_ceiling__unique,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: int,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(int,B,ring_1_of_int(B),Z))
           => ( archimedean_ceiling(B,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_1203_ceiling__eq__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,A2: int] :
          ( ( archimedean_ceiling(B,X) = A2 )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),A2)),one_one(B))),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(int,B,ring_1_of_int(B),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_1204_ceiling__split,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Pa: fun(int,$o),Ta: B] :
          ( aa(int,$o,Pa,archimedean_ceiling(B,Ta))
        <=> ! [I3: int] :
              ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),I3)),one_one(B))),Ta)
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ta),aa(int,B,ring_1_of_int(B),I3)) )
             => aa(int,$o,Pa,I3) ) ) ) ).

% ceiling_split
tff(fact_1205_ceiling__less__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(B,X)),Z)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),Z)),one_one(B))) ) ) ).

% ceiling_less_iff
tff(fact_1206_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1207_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1208_sum__squares__gt__zero__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Y)))
        <=> ( ( X != zero_zero(B) )
            | ( Y != zero_zero(B) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1209_sum__squares__le__zero__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Y))),zero_zero(B))
        <=> ( ( X = zero_zero(B) )
            & ( Y = zero_zero(B) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1210_mult__le__cancel__iff2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: B,X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Z)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),Z),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1211_mult__le__cancel__iff1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: B,X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Z)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1212_array__make,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,X: B] : array_new(B,N,X) = array_make(B,N,aTP_Lamp_au(B,fun(nat,B),X)) ) ).

% array_make
tff(fact_1213_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),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),J),I)),U)),N)) ) ) ).

% nat_less_add_iff2
tff(fact_1214_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(3)
tff(fact_1215_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(4)
tff(fact_1216_diff__mult__distrib2,axiom,
    ! [K: nat,Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_1217_diff__mult__distrib,axiom,
    ! [Ma: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_1218_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).

% diff_commute
tff(fact_1219_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( Ma = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1220_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1221_upd__ureturn,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,A2: array(B)] : heap_Time_bind(array(B),array(B),array_upd(B,I,X,A2),aTP_Lamp_av(array(B),fun(array(B),heap_Time_Heap(array(B))),A2)) = array_upd(B,I,X,A2) ) ).

% upd_ureturn
tff(fact_1222_nat__mult__less__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% nat_mult_less_cancel1
tff(fact_1223_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( Ma = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1224_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( 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),I),U)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),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),J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1225_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( 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),I),U)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),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),I),J)),U)),Ma)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1226_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),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),J),I)),U)),N)) ) ) ).

% nat_le_add_iff2
tff(fact_1227_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma)),N) ) ) ).

% nat_le_add_iff1
tff(fact_1228_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( Ma = 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),J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1229_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1230_mult__less__iff1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: B,X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Z)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1231_sum__squares__eq__zero__iff,axiom,
    ! [B: $tType] :
      ( linord4710134922213307826strict(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Y)) = zero_zero(B) )
        <=> ( ( X = zero_zero(B) )
            & ( Y = zero_zero(B) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1232_nat__mult__le__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% nat_mult_le_cancel1
tff(fact_1233_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma)),N) ) ) ).

% nat_less_add_iff1
tff(fact_1234_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,N))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1235_slice__len,axiom,
    ! [B: $tType,From: nat,To: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(list(B),nat,size_size(list(B)),slice(B,From,To,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From) ) ) ) ).

% slice_len
tff(fact_1236_power__int__minus__left__distrib,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( division_ring(D)
        & one(B)
        & uminus(B) )
     => ! [X: C,A2: D,N: int] :
          ( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
         => ( power_int(D,aa(D,D,uminus_uminus(D),A2),N) = aa(D,D,aa(D,fun(D,D),times_times(D),power_int(D,aa(D,D,uminus_uminus(D),one_one(D)),N)),power_int(D,A2,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_1237_ceiling__divide__lower,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Q2: B,P2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Q2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),P2),Q2)))),one_one(B))),Q2)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_1238_floor__divide__upper,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Q2: B,P2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Q2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),P2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),P2),Q2)))),one_one(B))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_1239_nat__ivt__aux,axiom,
    ! [N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,N))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1240_mult__ceiling__le__Ints,axiom,
    ! [C: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(C) )
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)))),aa(int,C,ring_1_of_int(C),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A2)),archimedean_ceiling(B,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_1241_le__mult__floor__Ints,axiom,
    ! [C: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(C) )
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(int,C,ring_1_of_int(C),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A2)),archim6421214686448440834_floor(B,B2)))),aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_1242_frac__unique__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,A2: B] :
          ( ( archimedean_frac(B,X) = A2 )
        <=> ( aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),A2)),ring_1_Ints(B))
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B)) ) ) ) ).

% frac_unique_iff
tff(fact_1243_abs__idempotent,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : aa(B,B,abs_abs(B),aa(B,B,abs_abs(B),A2)) = aa(B,B,abs_abs(B),A2) ) ).

% abs_idempotent
tff(fact_1244_divide__eq__0__iff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = zero_zero(B) )
        <=> ( ( A2 = zero_zero(B) )
            | ( B2 = zero_zero(B) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_1245_divide__cancel__left,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2) )
        <=> ( ( C2 = zero_zero(B) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_1246_divide__cancel__right,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,C2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) )
        <=> ( ( C2 = zero_zero(B) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_1247_division__ring__divide__zero,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),zero_zero(B)) = zero_zero(B) ) ).

% division_ring_divide_zero
tff(fact_1248_div__0,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% div_0
tff(fact_1249_div__by__0,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),zero_zero(B)) = zero_zero(B) ) ).

% div_by_0
tff(fact_1250_div__by__1,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),one_one(B)) = A2 ) ).

% div_by_1
tff(fact_1251_abs__0__eq,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( ( zero_zero(B) = aa(B,B,abs_abs(B),A2) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% abs_0_eq
tff(fact_1252_abs__eq__0,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( ( aa(B,B,abs_abs(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% abs_eq_0
tff(fact_1253_abs__zero,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ( aa(B,B,abs_abs(B),zero_zero(B)) = zero_zero(B) ) ) ).

% abs_zero
tff(fact_1254_abs__0,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ( aa(B,B,abs_abs(B),zero_zero(B)) = zero_zero(B) ) ) ).

% abs_0
tff(fact_1255_abs__add__abs,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2)) ) ).

% abs_add_abs
tff(fact_1256_abs__1,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ( aa(B,B,abs_abs(B),one_one(B)) = one_one(B) ) ) ).

% abs_1
tff(fact_1257_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)) ).

% nat_mult_div_cancel_disj
tff(fact_1258_abs__minus__cancel,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : aa(B,B,abs_abs(B),aa(B,B,uminus_uminus(B),A2)) = aa(B,B,abs_abs(B),A2) ) ).

% abs_minus_cancel
tff(fact_1259_abs__of__nat,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat] : aa(B,B,abs_abs(B),aa(nat,B,semiring_1_of_nat(B),N)) = aa(nat,B,semiring_1_of_nat(B),N) ) ).

% abs_of_nat
tff(fact_1260_slice__eq__bounds__empty,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] : slice(B,I,I,Xs) = nil(B) ).

% slice_eq_bounds_empty
tff(fact_1261_slice__Nil,axiom,
    ! [B: $tType,Begin: nat,End: nat] : slice(B,Begin,End,nil(B)) = nil(B) ).

% slice_Nil
tff(fact_1262_frac__in__Ints__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(set(B),$o,member(B,archimedean_frac(B,X)),ring_1_Ints(B))
        <=> aa(set(B),$o,member(B,X),ring_1_Ints(B)) ) ) ).

% frac_in_Ints_iff
tff(fact_1263_nonzero__mult__div__cancel__left,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1264_nonzero__mult__div__cancel__right,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1265_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1266_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1267_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1268_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1269_mult__divide__mult__cancel__left__if,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: B,A2: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) = $ite(C2 = zero_zero(B),zero_zero(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1270_zero__eq__1__divide__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( ( zero_zero(B) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1271_one__divide__eq__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1272_eq__divide__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( ( one_one(B) = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2) )
        <=> ( ( A2 != zero_zero(B) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1273_divide__eq__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2) = one_one(B) )
        <=> ( ( A2 != zero_zero(B) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1274_divide__self__if,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),A2) = $ite(A2 = zero_zero(B),zero_zero(B),one_one(B)) ) ).

% divide_self_if
tff(fact_1275_divide__self,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),A2) = one_one(B) ) ) ) ).

% divide_self
tff(fact_1276_one__eq__divide__iff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( ( one_one(B) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) )
        <=> ( ( B2 != zero_zero(B) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1277_divide__eq__1__iff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = one_one(B) )
        <=> ( ( B2 != zero_zero(B) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1278_div__self,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),A2) = one_one(B) ) ) ) ).

% div_self
tff(fact_1279_abs__le__zero__iff,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),zero_zero(B))
        <=> ( A2 = zero_zero(B) ) ) ) ).

% abs_le_zero_iff
tff(fact_1280_abs__le__self__iff,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% abs_le_self_iff
tff(fact_1281_abs__of__nonneg,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,B,abs_abs(B),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_1282_zero__less__abs__iff,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,abs_abs(B),A2))
        <=> ( A2 != zero_zero(B) ) ) ) ).

% zero_less_abs_iff
tff(fact_1283_divide__minus1,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),X),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),X) ) ).

% divide_minus1
tff(fact_1284_abs__neg__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ( aa(B,B,abs_abs(B),aa(B,B,uminus_uminus(B),one_one(B))) = one_one(B) ) ) ).

% abs_neg_one
tff(fact_1285_abs__power__int__minus,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,N: int] : aa(B,B,abs_abs(B),power_int(B,aa(B,B,uminus_uminus(B),A2),N)) = aa(B,B,abs_abs(B),power_int(B,A2,N)) ) ).

% abs_power_int_minus
tff(fact_1286_of__int__abs,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: int] : aa(int,B,ring_1_of_int(B),aa(int,int,abs_abs(int),X)) = aa(B,B,abs_abs(B),aa(int,B,ring_1_of_int(B),X)) ) ).

% of_int_abs
tff(fact_1287_frac__eq__0__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( ( archimedean_frac(B,X) = zero_zero(B) )
        <=> aa(set(B),$o,member(B,X),ring_1_Ints(B)) ) ) ).

% frac_eq_0_iff
tff(fact_1288_slice__complete,axiom,
    ! [B: $tType,Xs: list(B)] : slice(B,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs),Xs) = Xs ).

% slice_complete
tff(fact_1289_zero__le__divide__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_1290_divide__le__0__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% divide_le_0_1_iff
tff(fact_1291_zero__less__divide__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_1292_less__divide__eq__1__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1293_less__divide__eq__1__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1294_divide__less__eq__1__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),one_one(B))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1295_divide__less__eq__1__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),one_one(B))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1296_divide__less__0__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% divide_less_0_1_iff
tff(fact_1297_nonzero__divide__mult__cancel__right,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1298_nonzero__divide__mult__cancel__left,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1299_abs__of__nonpos,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
         => ( aa(B,B,abs_abs(B),A2) = aa(B,B,uminus_uminus(B),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_1300_zero__le__divide__abs__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,abs_abs(B),B2)))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
            | ( B2 = zero_zero(B) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_1301_divide__le__0__abs__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,abs_abs(B),B2))),zero_zero(B))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
            | ( B2 = zero_zero(B) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_1302_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_1303_floor__add2,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          ( ( aa(set(B),$o,member(B,X),ring_1_Ints(B))
            | aa(set(B),$o,member(B,Y),ring_1_Ints(B)) )
         => ( archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(B,X)),archim6421214686448440834_floor(B,Y)) ) ) ) ).

% floor_add2
tff(fact_1304_frac__gt__0__iff,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),archimedean_frac(B,X))
        <=> ~ aa(set(B),$o,member(B,X),ring_1_Ints(B)) ) ) ).

% frac_gt_0_iff
tff(fact_1305_le__divide__eq__1__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1306_le__divide__eq__1__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1307_divide__le__eq__1__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),one_one(B))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1308_divide__le__eq__1__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),one_one(B))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1309_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_1310_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_1311_Ints__abs,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => aa(set(B),$o,member(B,aa(B,B,abs_abs(B),A2)),ring_1_Ints(B)) ) ) ).

% Ints_abs
tff(fact_1312_nonzero__abs__divide,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_1313_abs__div__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,abs_abs(B),X)),Y) = aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% abs_div_pos
tff(fact_1314_abs__ge__self,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,abs_abs(B),A2)) ) ).

% abs_ge_self
tff(fact_1315_abs__le__D1,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% abs_le_D1
tff(fact_1316_abs__eq__0__iff,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ! [A2: B] :
          ( ( aa(B,B,abs_abs(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% abs_eq_0_iff
tff(fact_1317_floor__divide__of__int__eq,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [K: int,L: int] : archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(int,B,ring_1_of_int(B),K)),aa(int,B,ring_1_of_int(B),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).

% floor_divide_of_int_eq
tff(fact_1318_abs__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ( aa(B,B,abs_abs(B),one_one(B)) = one_one(B) ) ) ).

% abs_one
tff(fact_1319_abs__minus__commute,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) = aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2)) ) ).

% abs_minus_commute
tff(fact_1320_add__divide__distrib,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ).

% add_divide_distrib
tff(fact_1321_power__int__abs,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,N: int] : aa(B,B,abs_abs(B),power_int(B,A2,N)) = power_int(B,aa(B,B,abs_abs(B),A2),N) ) ).

% power_int_abs
tff(fact_1322_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_1323_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring(C)
     => ! [X: B,Y: B,C2: C,A2: C,B2: C] :
          ( nO_MATCH(B,C,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C2)
         => ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),A2),B2)),C2) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),times_times(C),A2),C2)),aa(C,C,aa(C,fun(C,C),times_times(C),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_1324_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring(C)
     => ! [X: B,Y: B,A2: C,B2: C,C2: C] :
          ( nO_MATCH(B,C,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A2)
         => ( aa(C,C,aa(C,fun(C,C),times_times(C),A2),aa(C,C,aa(C,fun(C,C),plus_plus(C),B2),C2)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),times_times(C),A2),B2)),aa(C,C,aa(C,fun(C,C),times_times(C),A2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_1325_power__int__divide__distrib,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B,Y: B,Ma: int] : power_int(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),Ma) = aa(B,B,aa(B,fun(B,B),divide_divide(B),power_int(B,X,Ma)),power_int(B,Y,Ma)) ) ).

% power_int_divide_distrib
tff(fact_1326_Ints__0,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => aa(set(B),$o,member(B,zero_zero(B)),ring_1_Ints(B)) ) ).

% Ints_0
tff(fact_1327_Ints__add,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => ( aa(set(B),$o,member(B,B2),ring_1_Ints(B))
           => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),ring_1_Ints(B)) ) ) ) ).

% Ints_add
tff(fact_1328_Ints__mult,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => ( aa(set(B),$o,member(B,B2),ring_1_Ints(B))
           => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),ring_1_Ints(B)) ) ) ) ).

% Ints_mult
tff(fact_1329_Ints__1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => aa(set(B),$o,member(B,one_one(B)),ring_1_Ints(B)) ) ).

% Ints_1
tff(fact_1330_Ints__diff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => ( aa(set(B),$o,member(B,B2),ring_1_Ints(B))
           => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),ring_1_Ints(B)) ) ) ) ).

% Ints_diff
tff(fact_1331_minus__in__Ints__iff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: B] :
          ( aa(set(B),$o,member(B,aa(B,B,uminus_uminus(B),X)),ring_1_Ints(B))
        <=> aa(set(B),$o,member(B,X),ring_1_Ints(B)) ) ) ).

% minus_in_Ints_iff
tff(fact_1332_Ints__minus,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [A2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => aa(set(B),$o,member(B,aa(B,B,uminus_uminus(B),A2)),ring_1_Ints(B)) ) ) ).

% Ints_minus
tff(fact_1333_Ints__of__nat,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] : aa(set(B),$o,member(B,aa(nat,B,semiring_1_of_nat(B),N)),ring_1_Ints(B)) ) ).

% Ints_of_nat
tff(fact_1334_Ints__of__int,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: int] : aa(set(B),$o,member(B,aa(int,B,ring_1_of_int(B),Z)),ring_1_Ints(B)) ) ).

% Ints_of_int
tff(fact_1335_Ints__induct,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Q2: B,Pa: fun(B,$o)] :
          ( aa(set(B),$o,member(B,Q2),ring_1_Ints(B))
         => ( ! [Z2: int] : aa(B,$o,Pa,aa(int,B,ring_1_of_int(B),Z2))
           => aa(B,$o,Pa,Q2) ) ) ) ).

% Ints_induct
tff(fact_1336_Ints__cases,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Q2: B] :
          ( aa(set(B),$o,member(B,Q2),ring_1_Ints(B))
         => ~ ! [Z2: int] : Q2 != aa(int,B,ring_1_of_int(B),Z2) ) ) ).

% Ints_cases
tff(fact_1337_Ints__nonzero__abs__ge1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          ( aa(set(B),$o,member(B,X),ring_1_Ints(B))
         => ( ( X != zero_zero(B) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,abs_abs(B),X)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_1338_Ints__nonzero__abs__less1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          ( aa(set(B),$o,member(B,X),ring_1_Ints(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),X)),one_one(B))
           => ( X = zero_zero(B) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_1339_Ints__eq__abs__less1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(set(B),$o,member(B,X),ring_1_Ints(B))
         => ( aa(set(B),$o,member(B,Y),ring_1_Ints(B))
           => ( ( X = Y )
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y))),one_one(B)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_1340_floor__divide__of__nat__eq,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Ma: nat,N: nat] : archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)) ) ).

% floor_divide_of_nat_eq
tff(fact_1341_abs__ge__zero,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,abs_abs(B),A2)) ) ).

% abs_ge_zero
tff(fact_1342_abs__not__less__zero,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),A2)),zero_zero(B)) ) ).

% abs_not_less_zero
tff(fact_1343_abs__of__pos,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,B,abs_abs(B),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_1344_abs__triangle__ineq,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2))) ) ).

% abs_triangle_ineq
tff(fact_1345_abs__triangle__ineq2__sym,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2))),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_1346_abs__triangle__ineq3,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2)))),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_1347_abs__triangle__ineq2,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2))),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_1348_abs__leI,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_1349_abs__le__D2,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_1350_abs__le__iff,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),B2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_1351_abs__ge__minus__self,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),A2)),aa(B,B,abs_abs(B),A2)) ) ).

% abs_ge_minus_self
tff(fact_1352_divide__right__mono__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1353_divide__nonpos__nonpos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1354_divide__nonpos__nonneg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),zero_zero(B)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1355_divide__nonneg__nonpos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),zero_zero(B)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1356_divide__nonneg__nonneg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1357_zero__le__divide__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1358_divide__right__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ) ) ).

% divide_right_mono
tff(fact_1359_divide__le__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),zero_zero(B))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1360_divide__neg__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_1361_divide__neg__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),zero_zero(B)) ) ) ) ).

% divide_neg_pos
tff(fact_1362_divide__pos__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),zero_zero(B)) ) ) ) ).

% divide_pos_neg
tff(fact_1363_divide__pos__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_1364_divide__less__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),zero_zero(B))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B)) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1365_divide__less__cancel,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) )
            & ( C2 != zero_zero(B) ) ) ) ) ).

% divide_less_cancel
tff(fact_1366_zero__less__divide__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1367_divide__strict__right__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_1368_divide__strict__right__mono__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1369_nonzero__eq__divide__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( ( A2 = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) )
          <=> ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1370_nonzero__divide__eq__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [C2: B,B2: B,A2: B] :
          ( ( C2 != zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) = A2 )
          <=> ( B2 = aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1371_eq__divide__imp,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = B2 )
           => ( A2 = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1372_divide__eq__imp,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [C2: B,B2: B,A2: B] :
          ( ( C2 != zero_zero(B) )
         => ( ( B2 = aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1373_eq__divide__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) )
        <=> $ite(C2 != zero_zero(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = B2,A2 = zero_zero(B)) ) ) ).

% eq_divide_eq
tff(fact_1374_divide__eq__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,C2: B,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) = A2 )
        <=> $ite(C2 != zero_zero(B),B2 = aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2),A2 = zero_zero(B)) ) ) ).

% divide_eq_eq
tff(fact_1375_frac__eq__eq,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Y: B,Z: B,X: B,W: B] :
          ( ( Y != zero_zero(B) )
         => ( ( Z != zero_zero(B) )
           => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y) = aa(B,B,aa(B,fun(B,B),divide_divide(B),W),Z) )
            <=> ( aa(B,B,aa(B,fun(B,B),times_times(B),X),Z) = aa(B,B,aa(B,fun(B,B),times_times(B),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1376_right__inverse__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = one_one(B) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1377_nonzero__minus__divide__right,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,uminus_uminus(B),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1378_nonzero__minus__divide__divide,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,uminus_uminus(B),A2)),aa(B,B,uminus_uminus(B),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1379_abs__zmult__eq__1,axiom,
    ! [Ma: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ma),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),Ma) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_1380_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1381_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1382_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1383_power__int__one__over,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,N: int] : power_int(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),X),N) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),power_int(B,X,N)) ) ).

% power_int_one_over
tff(fact_1384_Ints__double__eq__0__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [A2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2) = zero_zero(B) )
          <=> ( A2 = zero_zero(B) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_1385_dense__eq0__I,axiom,
    ! [B: $tType] :
      ( ( ordere166539214618696060dd_abs(B)
        & dense_linorder(B) )
     => ! [X: B] :
          ( ! [E4: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),E4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),X)),E4) )
         => ( X = zero_zero(B) ) ) ) ).

% dense_eq0_I
tff(fact_1386_abs__eq__mult,axiom,
    ! [B: $tType] :
      ( ordered_ring_abs(B)
     => ! [A2: B,B2: B] :
          ( ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
              | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
              | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),zero_zero(B)) ) )
         => ( aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_1387_abs__mult__pos,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,abs_abs(B),Y)),X) = aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),times_times(B),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_1388_abs__minus__le__zero,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(B,B,abs_abs(B),A2))),zero_zero(B)) ) ).

% abs_minus_le_zero
tff(fact_1389_eq__abs__iff_H,axiom,
    ! [B: $tType] :
      ( linordered_ring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,abs_abs(B),B2) )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(B,B,uminus_uminus(B),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1390_abs__eq__iff_H,axiom,
    ! [B: $tType] :
      ( linordered_ring(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,abs_abs(B),A2) = B2 )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(B,B,uminus_uminus(B),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1391_abs__diff__triangle__ineq,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B,C2: B,D2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D2)))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),C2))),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_1392_abs__triangle__ineq4,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2))) ) ).

% abs_triangle_ineq4
tff(fact_1393_abs__diff__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,A2: B,R: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),A2))),R)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),R)),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),R)) ) ) ) ).

% abs_diff_le_iff
tff(fact_1394_abs__if,axiom,
    ! [B: $tType] :
      ( abs_if(B)
     => ! [A2: B] :
          aa(B,B,abs_abs(B),A2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)),aa(B,B,uminus_uminus(B),A2),A2) ) ).

% abs_if
tff(fact_1395_abs__of__neg,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,B,abs_abs(B),A2) = aa(B,B,uminus_uminus(B),A2) ) ) ) ).

% abs_of_neg
tff(fact_1396_abs__if__raw,axiom,
    ! [B: $tType] :
      ( abs_if(B)
     => ! [X4: B] :
          aa(B,B,abs_abs(B),X4) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),zero_zero(B)),aa(B,B,uminus_uminus(B),X4),X4) ) ).

% abs_if_raw
tff(fact_1397_abs__diff__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,A2: B,R: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),A2))),R)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),R)),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),R)) ) ) ) ).

% abs_diff_less_iff
tff(fact_1398_div__positive,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ) ) ).

% div_positive
tff(fact_1399_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = zero_zero(B) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1400_divide__nonpos__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),zero_zero(B)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1401_divide__nonpos__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1402_divide__nonneg__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1403_divide__nonneg__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),zero_zero(B)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1404_divide__le__cancel,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1405_frac__less2,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B,W: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),W)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),W),Z)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1406_frac__less,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B,W: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),W)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),W),Z)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_1407_frac__le,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,X: B,W: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),W)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),W),Z)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_1408_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1409_divide__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),A2)
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2),aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_1410_less__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))) ) ) ) ).

% less_divide_eq
tff(fact_1411_neg__divide__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1412_neg__less__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1413_pos__divide__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1414_pos__less__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1415_mult__imp__div__pos__less,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,X: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1416_mult__imp__less__div__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,Z: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y)),X)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1417_divide__strict__left__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1418_divide__strict__left__mono__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1419_less__divide__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1420_divide__less__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),one_one(B))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) )
            | ( A2 = zero_zero(B) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1421_add__divide__eq__if__simps_I2_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,Z: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Z)),B2) = $ite(Z = zero_zero(B),B2,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1422_add__divide__eq__if__simps_I1_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,Z: B] :
          aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),Z)) = $ite(Z = zero_zero(B),A2,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1423_add__frac__eq,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Y: B,Z: B,X: B,W: B] :
          ( ( Y != zero_zero(B) )
         => ( ( Z != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),aa(B,B,aa(B,fun(B,B),divide_divide(B),W),Z)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),W),Y))),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1424_add__frac__num,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Y: B,X: B,Z: B] :
          ( ( Y != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),Z) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1425_add__num__frac,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Y: B,Z: B,X: B] :
          ( ( Y != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1426_add__divide__eq__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Z: B,X: B,Y: B] :
          ( ( Z != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),Z)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1427_divide__add__eq__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Z: B,X: B,Y: B] :
          ( ( Z != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z)),Y) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1428_divide__diff__eq__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Z: B,X: B,Y: B] :
          ( ( Z != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z)),Y) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1429_diff__divide__eq__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Z: B,X: B,Y: B] :
          ( ( Z != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),Z)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1430_diff__frac__eq,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Y: B,Z: B,X: B,W: B] :
          ( ( Y != zero_zero(B) )
         => ( ( Z != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),aa(B,B,aa(B,fun(B,B),divide_divide(B),W),Z)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),W),Y))),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1431_add__divide__eq__if__simps_I4_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,Z: B] :
          aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),Z)) = $ite(Z = zero_zero(B),A2,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1432_less__half__sum,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),one_one(B)))) ) ) ).

% less_half_sum
tff(fact_1433_gt__half__sum,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),one_one(B)))),B2) ) ) ).

% gt_half_sum
tff(fact_1434_eq__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 = aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) )
        <=> $ite(C2 != zero_zero(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) = aa(B,B,uminus_uminus(B),B2),A2 = zero_zero(B)) ) ) ).

% eq_minus_divide_eq
tff(fact_1435_minus__divide__eq__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,C2: B,A2: B] :
          ( ( aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) = A2 )
        <=> $ite(C2 != zero_zero(B),aa(B,B,uminus_uminus(B),B2) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2),A2 = zero_zero(B)) ) ) ).

% minus_divide_eq_eq
tff(fact_1436_nonzero__neg__divide__eq__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ( aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) = C2 )
          <=> ( aa(B,B,uminus_uminus(B),A2) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1437_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,C2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ( C2 = aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) )
          <=> ( aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2) = aa(B,B,uminus_uminus(B),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1438_divide__eq__minus__1__iff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = aa(B,B,uminus_uminus(B),one_one(B)) )
        <=> ( ( B2 != zero_zero(B) )
            & ( A2 = aa(B,B,uminus_uminus(B),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1439_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_1440_nat__mult__div__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1441_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),W))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_1442_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1443_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1444_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1445_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1446_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1447_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1448_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_1449_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1450_zdiv__mono2__neg,axiom,
    ! [A2: int,B5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B5)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1451_zdiv__mono1__neg,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1452_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1453_zdiv__mono2,axiom,
    ! [A2: int,B5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B5)) ) ) ) ).

% zdiv_mono2
tff(fact_1454_zdiv__mono1,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2)) ) ) ).

% zdiv_mono1
tff(fact_1455_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1456_power__int__diff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B,Ma: int,N: int] :
          ( ( ( X != zero_zero(B) )
            | ( Ma != N ) )
         => ( power_int(B,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ma),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),power_int(B,X,Ma)),power_int(B,X,N)) ) ) ) ).

% power_int_diff
tff(fact_1457_Ints__odd__nonzero,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [A2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),A2)),A2) != zero_zero(B) ) ) ) ).

% Ints_odd_nonzero
tff(fact_1458_abs__add__one__gt__zero,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,abs_abs(B),X))) ) ).

% abs_add_one_gt_zero
tff(fact_1459_of__int__leD,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: int,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(int,B,ring_1_of_int(B),N))),X)
         => ( ( N = zero_zero(int) )
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),X) ) ) ) ).

% of_int_leD
tff(fact_1460_of__int__lessD,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: int,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),aa(int,B,ring_1_of_int(B),N))),X)
         => ( ( N = zero_zero(int) )
            | aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),X) ) ) ) ).

% of_int_lessD
tff(fact_1461_divide__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),A2)
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1462_le__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))) ) ) ) ).

% le_divide_eq
tff(fact_1463_divide__left__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1464_neg__divide__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1465_neg__le__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1466_pos__divide__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1467_pos__le__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1468_mult__imp__div__pos__le,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,X: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1469_mult__imp__le__div__pos,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,Z: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),Z),Y)),X)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1470_divide__left__mono__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),zero_zero(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1471_le__divide__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1472_divide__le__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),one_one(B))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) )
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) )
            | ( A2 = zero_zero(B) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1473_frac__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,Z: B,X: B,W: B] :
          ( ( Y != zero_zero(B) )
         => ( ( Z != zero_zero(B) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),aa(B,B,aa(B,fun(B,B),divide_divide(B),W),Z))
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),W),Y))),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))),zero_zero(B)) ) ) ) ) ).

% frac_le_eq
tff(fact_1474_frac__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Y: B,Z: B,X: B,W: B] :
          ( ( Y != zero_zero(B) )
         => ( ( Z != zero_zero(B) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y)),aa(B,B,aa(B,fun(B,B),divide_divide(B),W),Z))
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),Z)),aa(B,B,aa(B,fun(B,B),times_times(B),W),Y))),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))),zero_zero(B)) ) ) ) ) ).

% frac_less_eq
tff(fact_1475_pos__minus__divide__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1476_pos__less__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1477_neg__minus__divide__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1478_neg__less__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1479_minus__divide__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),A2)
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_1480_less__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))) ) ) ) ).

% less_minus_divide_eq
tff(fact_1481_minus__divide__add__eq__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Z: B,X: B,Y: B] :
          ( ( Z != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z))),Y) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1482_add__divide__eq__if__simps_I3_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,Z: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Z))),B2) = $ite(Z = zero_zero(B),B2,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1483_add__divide__eq__if__simps_I6_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,Z: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Z))),B2) = $ite(Z = zero_zero(B),aa(B,B,uminus_uminus(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_1484_add__divide__eq__if__simps_I5_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,Z: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Z)),B2) = $ite(Z = zero_zero(B),aa(B,B,uminus_uminus(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_1485_minus__divide__diff__eq__iff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Z: B,X: B,Y: B] :
          ( ( Z != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Z))),Y) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),X)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1486_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),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),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_1487_div__geq,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),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),Ma),N)),N)) ) ) ) ).

% div_geq
tff(fact_1488_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A4),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1489_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1490_Ints__odd__less__0,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),A2)),A2)),zero_zero(B))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ) ).

% Ints_odd_less_0
tff(fact_1491_scaling__mono,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [U: B,V2: B,R: B,S2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),V2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),R)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),R),S2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),U),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),R),aa(B,B,aa(B,fun(B,B),minus_minus(B),V2),U))),S2))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_1492_pos__minus__divide__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_1493_pos__le__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_1494_neg__minus__divide__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_1495_neg__le__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_1496_minus__divide__le__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),A2)
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_1497_le__minus__divide__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,uminus_uminus(B),B2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B))) ) ) ) ).

% le_minus_divide_eq
tff(fact_1498_nat__approx__posE,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [E2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),E2)
         => ~ ! [N2: nat] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,N2)))),E2) ) ) ).

% nat_approx_posE
tff(fact_1499_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      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),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) ).

% nat_abs_int_diff
tff(fact_1500_split__zdiv,axiom,
    ! [Pa: fun(int,$o),N: int,K: int] :
      ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,Pa,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,Pa,I3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,Pa,I3) ) ) ) ) ).

% split_zdiv
tff(fact_1501_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1502_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1503_frac__neg,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          archimedean_frac(B,aa(B,B,uminus_uminus(B),X)) = $ite(aa(set(B),$o,member(B,X),ring_1_Ints(B)),zero_zero(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),archimedean_frac(B,X))) ) ).

% frac_neg
tff(fact_1504_nat__intermed__int__val,axiom,
    ! [Ma: nat,N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,Ma)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,N))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
                & ( aa(nat,int,F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1505_floor__divide__lower,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Q2: B,P2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Q2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),P2),Q2)))),Q2)),P2) ) ) ).

% floor_divide_lower
tff(fact_1506_ceiling__divide__upper,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Q2: B,P2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Q2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),P2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),P2),Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_1507_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),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),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_1508_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),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),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_1509_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1510_div__add__self1__no__field,axiom,
    ! [B: $tType,C: $tType] :
      ( ( euclid4440199948858584721cancel(C)
        & field(B) )
     => ! [X: B,B2: C,A2: C] :
          ( nO_MATCH(B,C,X,B2)
         => ( ( B2 != zero_zero(C) )
           => ( aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),B2),A2)),B2) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),divide_divide(C),A2),B2)),one_one(C)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_1511_div__add__self2__no__field,axiom,
    ! [B: $tType,C: $tType] :
      ( ( euclid4440199948858584721cancel(C)
        & field(B) )
     => ! [X: B,B2: C,A2: C] :
          ( nO_MATCH(B,C,X,B2)
         => ( ( B2 != zero_zero(C) )
           => ( aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),A2),B2)),B2) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),divide_divide(C),A2),B2)),one_one(C)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_1512_div__mult__self1,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_1513_div__mult__self2,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_1514_div__mult__self3,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,C2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)),A2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_1515_div__mult__self4,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,C2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)),A2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_1516_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1517_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1518_div__mult__mult1__if,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) = $ite(C2 = zero_zero(B),zero_zero(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_1519_div__mult__mult2,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_1520_div__mult__mult1,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_1521_div__minus1__right,axiom,
    ! [B: $tType] :
      ( euclid8851590272496341667cancel(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),A2) ) ).

% div_minus1_right
tff(fact_1522_div__by__Suc__0,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(nat,nat,suc,zero_zero(nat))) = Ma ).

% div_by_Suc_0
tff(fact_1523_div__less,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1524_div__mult__self__is__m,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)),N) = Ma ) ) ).

% div_mult_self_is_m
tff(fact_1525_div__mult__self1__is__m,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Ma)),N) = Ma ) ) ).

% div_mult_self1_is_m
tff(fact_1526_div__le__dividend,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),Ma) ).

% div_le_dividend
tff(fact_1527_div__le__mono,axiom,
    ! [Ma: nat,N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K)) ) ).

% div_le_mono
tff(fact_1528_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1529_Suc__div__le__mono,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Ma)),N)) ).

% Suc_div_le_mono
tff(fact_1530_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1531_div__times__less__eq__dividend,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),N)),Ma) ).

% div_times_less_eq_dividend
tff(fact_1532_times__div__less__eq__dividend,axiom,
    ! [N: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N))),Ma) ).

% times_div_less_eq_dividend
tff(fact_1533_div__mult2__eq_H,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B,Ma: nat,N: nat] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,semiring_1_of_nat(B),Ma))),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% div_mult2_eq'
tff(fact_1534_div__le__mono2,axiom,
    ! [Ma: nat,N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Ma)) ) ) ).

% div_le_mono2
tff(fact_1535_div__greater__zero__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% div_greater_zero_iff
tff(fact_1536_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),Q2)),N)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1537_div__eq__dividend__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) = Ma )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1538_div__less__dividend,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),Ma) ) ) ).

% div_less_dividend
tff(fact_1539_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1540_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_1541_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_1542_div__add__self2,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ).

% div_add_self2
tff(fact_1543_div__add__self1,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ).

% div_add_self1
tff(fact_1544_div__if,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | ( N = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),N)) ) ).

% div_if
tff(fact_1545_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q2)),N) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1546_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q2)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1547_split__div,axiom,
    ! [Pa: fun(nat,$o),Ma: nat,N: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(nat,$o,Pa,zero_zero(nat)) )
        & ( ( N != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),N)
             => ( ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J3) )
               => aa(nat,$o,Pa,I3) ) ) ) ) ) ).

% split_div
tff(fact_1548_dividend__less__div__times,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),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),Ma),N)),N))) ) ).

% dividend_less_div_times
tff(fact_1549_dividend__less__times__div,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),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),Ma),N)))) ) ).

% dividend_less_times_div
tff(fact_1550_le__div__geq,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),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),Ma),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_1551_split__div_H,axiom,
    ! [Pa: fun(nat,$o),Ma: nat,N: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N))
    <=> ( ( ( N = zero_zero(nat) )
          & aa(nat,$o,Pa,zero_zero(nat)) )
        | ? [Q6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q6)),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q6)))
            & aa(nat,$o,Pa,Q6) ) ) ) ).

% split_div'
tff(fact_1552_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1553_bits__div__by__1,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),one_one(B)) = A2 ) ).

% bits_div_by_1
tff(fact_1554_bits__div__by__0,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),zero_zero(B)) = zero_zero(B) ) ).

% bits_div_by_0
tff(fact_1555_bits__div__0,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% bits_div_0
tff(fact_1556_int__power__div__base,axiom,
    ! [Ma: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),Ma)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1557_gbinomial__absorption_H,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,B,gbinomial(B,A2),K) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,semiring_1_of_nat(B),K))),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_1558_pochhammer__absorb__comp,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [R: B,K: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),R),aa(nat,B,semiring_1_of_nat(B),K))),comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),R),K)) = aa(B,B,aa(B,fun(B,B),times_times(B),R),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),R)),one_one(B)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_1559_power_Opower__eq__if,axiom,
    ! [B: $tType,One: B,Times: fun(B,fun(B,B)),P2: B,Ma: nat] :
      power2(B,One,Times,P2,Ma) = $ite(Ma = zero_zero(nat),One,aa(B,B,aa(B,fun(B,B),Times,P2),power2(B,One,Times,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))) ).

% power.power_eq_if
tff(fact_1560_fact__reduce,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( semiring_char_0_fact(B,N) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_1561_power__one,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),one_one(B)),N) = one_one(B) ) ).

% power_one
tff(fact_1562_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [X: nat,B2: nat,W: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),X) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),B2)),W) )
        <=> ( X = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1563_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [B2: nat,W: nat,X: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),B2)),W) = aa(nat,B,semiring_1_of_nat(B),X) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1564_of__nat__power,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ma),N)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),Ma)),N) ) ).

% of_nat_power
tff(fact_1565_power__one__right,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B] : aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),one_one(nat)) = A2 ) ).

% power_one_right
tff(fact_1566_of__nat__fact,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : aa(nat,B,semiring_1_of_nat(B),semiring_char_0_fact(nat,N)) = semiring_char_0_fact(B,N) ) ).

% of_nat_fact
tff(fact_1567_gbinomial__1,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [A2: B] : aa(nat,B,gbinomial(B,A2),one_one(nat)) = A2 ) ).

% gbinomial_1
tff(fact_1568_pochhammer__1,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B] : comm_s3205402744901411588hammer(B,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_1569_power__inject__exp,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma) = aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N) )
          <=> ( Ma = N ) ) ) ) ).

% power_inject_exp
tff(fact_1570_power__0__Suc,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),aa(nat,nat,suc,N)) = zero_zero(B) ) ).

% power_0_Suc
tff(fact_1571_power__Suc0__right,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B] : aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_1572_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_1573_fact__0,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ( semiring_char_0_fact(B,zero_zero(nat)) = one_one(B) ) ) ).

% fact_0
tff(fact_1574_gbinomial__0_I1_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [A2: B] : aa(nat,B,gbinomial(B,A2),zero_zero(nat)) = one_one(B) ) ).

% gbinomial_0(1)
tff(fact_1575_gbinomial__Suc0,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [A2: B] : aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_1576_pochhammer__0,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B] : comm_s3205402744901411588hammer(B,A2,zero_zero(nat)) = one_one(B) ) ).

% pochhammer_0
tff(fact_1577_fact__1,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ( semiring_char_0_fact(B,one_one(nat)) = one_one(B) ) ) ).

% fact_1
tff(fact_1578_pochhammer__Suc0,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B] : comm_s3205402744901411588hammer(B,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% pochhammer_Suc0
tff(fact_1579_power__int__of__nat,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & power(B) )
     => ! [X: B,N: nat] : power_int(B,X,aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N) ) ).

% power_int_of_nat
tff(fact_1580_of__int__power,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: int,N: nat] : aa(int,B,ring_1_of_int(B),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),Z)),N) ) ).

% of_int_power
tff(fact_1581_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [B2: int,W: nat,X: int] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),B2)),W) = aa(int,B,ring_1_of_int(B),X) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1582_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [X: int,B2: int,W: nat] :
          ( ( aa(int,B,ring_1_of_int(B),X) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),B2)),W) )
        <=> ( X = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1583_power__strict__increasing__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: B,X: nat,Y: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1584_minus__one__mult__self,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)) = one_one(B) ) ).

% minus_one_mult_self
tff(fact_1585_left__minus__one__mult__self,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_1586_power__eq__0__iff,axiom,
    ! [B: $tType] :
      ( semiri2026040879449505780visors(B)
     => ! [A2: B,N: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N) = zero_zero(B) )
        <=> ( ( A2 = zero_zero(B) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% power_eq_0_iff
tff(fact_1587_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: nat,W: nat,X: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),B2)),W)),aa(nat,B,semiring_1_of_nat(B),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),X) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1588_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: nat,B2: nat,W: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1589_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: nat,B2: nat,W: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1590_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: nat,W: nat,X: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),B2)),W)),aa(nat,B,semiring_1_of_nat(B),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),X) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1591_fact__Suc__0,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ( semiring_char_0_fact(B,aa(nat,nat,suc,zero_zero(nat))) = one_one(B) ) ) ).

% fact_Suc_0
tff(fact_1592_fact__Suc,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : semiring_char_0_fact(B,aa(nat,nat,suc,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,N))),semiring_char_0_fact(B,N)) ) ).

% fact_Suc
tff(fact_1593_power__strict__decreasing__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: B,Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),one_one(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1594_power__mono__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,B2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N))
              <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1595_power__increasing__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: B,X: nat,Y: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ).

% power_increasing_iff
tff(fact_1596_zero__less__power__abs__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,abs_abs(B),A2)),N))
        <=> ( ( A2 != zero_zero(B) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1597_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [B2: int,W: nat,X: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),B2)),W)),aa(int,B,ring_1_of_int(B),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),X) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1598_of__int__power__le__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: int,B2: int,W: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1599_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [B2: int,W: nat,X: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),B2)),W)),aa(int,B,ring_1_of_int(B),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),X) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1600_of__int__power__less__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: int,B2: int,W: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(int,B,ring_1_of_int(B),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1601_power__decreasing__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [B2: B,Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),one_one(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1602_of__nat__zero__less__power__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(nat,B,semiring_1_of_nat(B),X)),N))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1603_pochhammer__of__nat,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [X: nat,N: nat] : comm_s3205402744901411588hammer(B,aa(nat,B,semiring_1_of_nat(B),X),N) = aa(nat,B,semiring_1_of_nat(B),comm_s3205402744901411588hammer(nat,X,N)) ) ).

% pochhammer_of_nat
tff(fact_1604_pochhammer__fact,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & comm_semiring_1(B) )
     => ! [N: nat] : semiring_char_0_fact(B,N) = comm_s3205402744901411588hammer(B,one_one(B),N) ) ).

% pochhammer_fact
tff(fact_1605_of__nat__gbinomial,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [N: nat,K: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,gbinomial(nat,N),K)) = aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),N)),K) ) ).

% of_nat_gbinomial
tff(fact_1606_gbinomial__pochhammer,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),K) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),K)),comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),A2),K))),semiring_char_0_fact(B,K)) ) ).

% gbinomial_pochhammer
tff(fact_1607_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [K: nat,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),K)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,semiring_1_of_nat(B),K))),K)),aa(nat,B,gbinomial(B,A2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_1608_fact__mono__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,Ma)),semiring_char_0_fact(nat,N)) ) ).

% fact_mono_nat
tff(fact_1609_fact__ge__self,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),semiring_char_0_fact(nat,N)) ).

% fact_ge_self
tff(fact_1610_power__not__zero,axiom,
    ! [B: $tType] :
      ( semiri2026040879449505780visors(B)
     => ! [A2: B,N: nat] :
          ( ( A2 != zero_zero(B) )
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N) != zero_zero(B) ) ) ) ).

% power_not_zero
tff(fact_1611_fact__nonzero,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semiri3467727345109120633visors(B) )
     => ! [N: nat] : semiring_char_0_fact(B,N) != zero_zero(B) ) ).

% fact_nonzero
tff(fact_1612_pochhammer__same,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & comm_ring_1(B)
        & semiri3467727345109120633visors(B) )
     => ! [N: nat] : comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),N)),N) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),semiring_char_0_fact(B,N)) ) ).

% pochhammer_same
tff(fact_1613_Ints__power,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [A2: B,N: nat] :
          ( aa(set(B),$o,member(B,A2),ring_1_Ints(B))
         => aa(set(B),$o,member(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),ring_1_Ints(B)) ) ) ).

% Ints_power
tff(fact_1614_gbinomial__pochhammer_H,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),K) = aa(B,B,aa(B,fun(B,B),divide_divide(B),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(nat,B,semiring_1_of_nat(B),K))),one_one(B)),K)),semiring_char_0_fact(B,K)) ) ).

% gbinomial_pochhammer'
tff(fact_1615_gbinomial__index__swap,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),K)),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),N))),one_one(B))),K)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),K))),one_one(B))),N)) ) ).

% gbinomial_index_swap
tff(fact_1616_gbinomial__negated__upper,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),K) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),K)),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,semiring_1_of_nat(B),K)),A2)),one_one(B))),K)) ) ).

% gbinomial_negated_upper
tff(fact_1617_fact__less__mono__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,Ma)),semiring_char_0_fact(nat,N)) ) ) ).

% fact_less_mono_nat
tff(fact_1618_zero__le__power,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ).

% zero_le_power
tff(fact_1619_power__mono,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,B2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N)) ) ) ) ).

% power_mono
tff(fact_1620_zero__less__power,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ).

% zero_less_power
tff(fact_1621_gbinomial__Suc__Suc,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,nat,suc,K)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,gbinomial(B,A2),K)),aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_1622_one__le__power,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ).

% one_le_power
tff(fact_1623_left__right__inverse__power,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [X: B,Y: B,N: nat] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),X),Y) = one_one(B) )
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),N)) = one_one(B) ) ) ) ).

% left_right_inverse_power
tff(fact_1624_power__0,axiom,
    ! [B: $tType] :
      ( power(B)
     => ! [A2: B] : aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),zero_zero(nat)) = one_one(B) ) ).

% power_0
tff(fact_1625_power__one__over,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2)),N) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ).

% power_one_over
tff(fact_1626_power__add,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B,Ma: nat,N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ).

% power_add
tff(fact_1627_fact__ge__zero,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),semiring_char_0_fact(B,N)) ) ).

% fact_ge_zero
tff(fact_1628_gbinomial__of__nat__symmetric,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),N)),K) = aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_1629_fact__gt__zero,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),semiring_char_0_fact(B,N)) ) ).

% fact_gt_zero
tff(fact_1630_fact__not__neg,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),semiring_char_0_fact(B,N)),zero_zero(B)) ) ).

% fact_not_neg
tff(fact_1631_fact__ge__1,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),semiring_char_0_fact(B,N)) ) ).

% fact_ge_1
tff(fact_1632_fact__mono,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),semiring_char_0_fact(B,Ma)),semiring_char_0_fact(B,N)) ) ) ).

% fact_mono
tff(fact_1633_pochhammer__pos,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),comm_s3205402744901411588hammer(B,X,N)) ) ) ).

% pochhammer_pos
tff(fact_1634_pochhammer__neq__0__mono,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,Ma: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(B,A2,Ma) != zero_zero(B) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
           => ( comm_s3205402744901411588hammer(B,A2,N) != zero_zero(B) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1635_pochhammer__eq__0__mono,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,N: nat,Ma: nat] :
          ( ( comm_s3205402744901411588hammer(B,A2,N) = zero_zero(B) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
           => ( comm_s3205402744901411588hammer(B,A2,Ma) = zero_zero(B) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1636_gbinomial__minus,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,aa(B,B,uminus_uminus(B),A2)),K) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),K)),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(nat,B,semiring_1_of_nat(B),K))),one_one(B))),K)) ) ).

% gbinomial_minus
tff(fact_1637_power_Opower_Opower__0,axiom,
    ! [B: $tType,One: B,Times: fun(B,fun(B,B)),A2: B] : power2(B,One,Times,A2,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_1638_gbinomial__addition__formula,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,K)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),aa(nat,nat,suc,K))),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),K)) ) ).

% gbinomial_addition_formula
tff(fact_1639_gbinomial__absorb__comp,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(nat,B,semiring_1_of_nat(B),K))),aa(nat,B,gbinomial(B,A2),K)) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_1640_gbinomial__mult__1_H,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,A2),K)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),K)),aa(nat,B,gbinomial(B,A2),K))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,K))),aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_1641_gbinomial__mult__1,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,gbinomial(B,A2),K)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),K)),aa(nat,B,gbinomial(B,A2),K))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,K))),aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_1642_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N)) ).

% fact_ge_Suc_0_nat
tff(fact_1643_pochhammer__minus,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [B2: B,K: nat] : comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),B2),K) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),K)),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),aa(nat,B,semiring_1_of_nat(B),K))),one_one(B)),K)) ) ).

% pochhammer_minus
tff(fact_1644_pochhammer__minus_H,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [B2: B,K: nat] : comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),aa(nat,B,semiring_1_of_nat(B),K))),one_one(B)),K) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),K)),comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),B2),K)) ) ).

% pochhammer_minus'
tff(fact_1645_power__less__imp__less__base,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_1646_power__le__one,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),one_one(B)) ) ) ) ).

% power_le_one
tff(fact_1647_power__le__imp__le__base,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,N))),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),aa(nat,nat,suc,N)))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_1648_power__inject__base,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat,B2: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,N)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),aa(nat,nat,suc,N)) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1649_power__gt1__lemma,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))) ) ) ).

% power_gt1_lemma
tff(fact_1650_power__less__power__Suc,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))) ) ) ).

% power_less_power_Suc
tff(fact_1651_power__0__left,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] :
          aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),N) = $ite(N = zero_zero(nat),one_one(B),zero_zero(B)) ) ).

% power_0_left
tff(fact_1652_power__gt1,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,N))) ) ) ).

% power_gt1
tff(fact_1653_power__less__imp__less__exp,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1654_power__strict__increasing,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat,N5: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N5)) ) ) ) ).

% power_strict_increasing
tff(fact_1655_power__minus,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [A2: B,N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),A2)),N) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ).

% power_minus
tff(fact_1656_power__increasing,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat,N5: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N5)) ) ) ) ).

% power_increasing
tff(fact_1657_zero__power,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),N) = zero_zero(B) ) ) ) ).

% zero_power
tff(fact_1658_zero__le__power__abs,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,abs_abs(B),A2)),N)) ) ).

% zero_le_power_abs
tff(fact_1659_fact__less__mono,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),semiring_char_0_fact(B,Ma)),semiring_char_0_fact(B,N)) ) ) ) ).

% fact_less_mono
tff(fact_1660_pochhammer__nonneg,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),comm_s3205402744901411588hammer(B,X,N)) ) ) ).

% pochhammer_nonneg
tff(fact_1661_pochhammer__0__left,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [N: nat] :
          comm_s3205402744901411588hammer(B,zero_zero(B),N) = $ite(N = zero_zero(nat),one_one(B),zero_zero(B)) ) ).

% pochhammer_0_left
tff(fact_1662_Suc__times__gbinomial,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,K))),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,nat,suc,K))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,B,gbinomial(B,A2),K)) ) ).

% Suc_times_gbinomial
tff(fact_1663_gbinomial__absorption,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,K))),aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,K))) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),K)) ) ).

% gbinomial_absorption
tff(fact_1664_gbinomial__trinomial__revision,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,Ma: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,A2),Ma)),aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),Ma)),K)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,A2),K)),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),aa(nat,B,semiring_1_of_nat(B),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_1665_power__Suc__less,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ) ).

% power_Suc_less
tff(fact_1666_power__Suc__le__self,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,N))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_1667_power__Suc__less__one,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,N))),one_one(B)) ) ) ) ).

% power_Suc_less_one
tff(fact_1668_power__strict__decreasing,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat,N5: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N5)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1669_power__decreasing,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat,N5: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N5)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ) ) ).

% power_decreasing
tff(fact_1670_power__eq__imp__eq__base,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat,B2: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N) = aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1671_power__eq__iff__eq__base,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat,A2: B,B2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),B2)
             => ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N) = aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1672_power__le__imp__le__exp,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1673_self__le__power,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ) ).

% self_le_power
tff(fact_1674_one__less__power,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ) ).

% one_less_power
tff(fact_1675_power__diff,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B,N: nat,Ma: nat] :
          ( ( A2 != zero_zero(B) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
           => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ) ) ).

% power_diff
tff(fact_1676_pochhammer__rec,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] : comm_s3205402744901411588hammer(B,A2,aa(nat,nat,suc,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B)),N)) ) ).

% pochhammer_rec
tff(fact_1677_pochhammer__Suc,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] : comm_s3205402744901411588hammer(B,A2,aa(nat,nat,suc,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),comm_s3205402744901411588hammer(B,A2,N)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(nat,B,semiring_1_of_nat(B),N))) ) ).

% pochhammer_Suc
tff(fact_1678_pochhammer__rec_H,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Z: B,N: nat] : comm_s3205402744901411588hammer(B,Z,aa(nat,nat,suc,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),aa(nat,B,semiring_1_of_nat(B),N))),comm_s3205402744901411588hammer(B,Z,N)) ) ).

% pochhammer_rec'
tff(fact_1679_pochhammer__eq__0__iff,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,N: nat] :
          ( ( comm_s3205402744901411588hammer(B,A2,N) = zero_zero(B) )
        <=> ? [K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),N)
              & ( A2 = aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),K2)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_1680_pochhammer__of__nat__eq__0__iff,axiom,
    ! [B: $tType] :
      ( ( ring_char_0(B)
        & idom(B) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),N)),K) = zero_zero(B) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_1681_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [B: $tType] :
      ( idom(B)
     => ! [N: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K)
         => ( comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),N)),K) = zero_zero(B) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_1682_gbinomial__rec,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,nat,suc,K)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,A2),K)),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_1683_gbinomial__factors,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,nat,suc,K)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,K)))),aa(nat,B,gbinomial(B,A2),K)) ) ).

% gbinomial_factors
tff(fact_1684_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [B: $tType] :
      ( ( ring_char_0(B)
        & idom(B) )
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( comm_s3205402744901411588hammer(B,aa(B,B,uminus_uminus(B),aa(nat,B,semiring_1_of_nat(B),N)),K) != zero_zero(B) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_1685_pochhammer__product_H,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Z: B,N: nat,Ma: nat] : comm_s3205402744901411588hammer(B,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma)) = aa(B,B,aa(B,fun(B,B),times_times(B),comm_s3205402744901411588hammer(B,Z,N)),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),aa(nat,B,semiring_1_of_nat(B),N)),Ma)) ) ).

% pochhammer_product'
tff(fact_1686_power__minus_H,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: B,N: nat] :
          ( nO_MATCH(B,B,one_one(B),X)
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),X)),N) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)) ) ) ) ).

% power_minus'
tff(fact_1687_power__strict__mono,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: B,B2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N)) ) ) ) ) ).

% power_strict_mono
tff(fact_1688_power__diff__power__eq,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,Ma: nat,N: nat] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1689_power__eq__if,axiom,
    ! [B: $tType] :
      ( power(B)
     => ! [P2: B,Ma: nat] :
          aa(nat,B,aa(B,fun(nat,B),power_power(B),P2),Ma) = $ite(Ma = zero_zero(nat),one_one(B),aa(B,B,aa(B,fun(B,B),times_times(B),P2),aa(nat,B,aa(B,fun(nat,B),power_power(B),P2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1690_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1691_power__minus__mult,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [N: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A2) = aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N) ) ) ) ).

% power_minus_mult
tff(fact_1692_gbinomial__reduce__nat,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,B,gbinomial(B,A2),K) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_1693_pochhammer__product,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Ma: nat,N: nat,Z: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( comm_s3205402744901411588hammer(B,Z,N) = aa(B,B,aa(B,fun(B,B),times_times(B),comm_s3205402744901411588hammer(B,Z,Ma)),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))) ) ) ) ).

% pochhammer_product
tff(fact_1694_fact__num__eq__if,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [Ma: nat] :
          semiring_char_0_fact(B,Ma) = $ite(Ma = zero_zero(nat),one_one(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Ma)),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_1695_gbinomial__code,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] :
          aa(nat,B,gbinomial(B,A2),K) = $ite(K = zero_zero(nat),one_one(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),set_fo6178422350223883121st_nat(B,aTP_Lamp_aw(B,fun(nat,fun(B,B)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(B))),semiring_char_0_fact(B,K))) ) ).

% gbinomial_code
tff(fact_1696_pochhammer__code,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] :
          comm_s3205402744901411588hammer(B,A2,N) = $ite(N = zero_zero(nat),one_one(B),set_fo6178422350223883121st_nat(B,aTP_Lamp_ax(B,fun(nat,fun(B,B)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(B))) ) ).

% pochhammer_code
tff(fact_1697_slice__nth,axiom,
    ! [B: $tType,From: nat,To: nat,Xs: list(B),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From))
         => ( aa(nat,B,nth(B,slice(B,From,To,Xs)),I) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_1698_split__pos__lemma,axiom,
    ! [K: int,Pa: fun(int,fun(int,$o)),N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),Pa,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),Pa,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1699_split__neg__lemma,axiom,
    ! [K: int,Pa: fun(int,fun(int,$o)),N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),Pa,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),Pa,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1700_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B4),N) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_1701_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),
                $ite(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B4),N) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N)) ) ) ).

% verit_le_mono_div
tff(fact_1702_le__divide__eq__numeral_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [W: num,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),B2),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),zero_zero(B))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_1703_numeral__eq__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [Ma: num,N: num] :
          ( ( aa(num,B,numeral_numeral(B),Ma) = aa(num,B,numeral_numeral(B),N) )
        <=> ( Ma = N ) ) ) ).

% numeral_eq_iff
tff(fact_1704_int__eq__iff__numeral,axiom,
    ! [Ma: nat,V2: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Ma) = aa(num,int,numeral_numeral(int),V2) )
    <=> ( Ma = aa(num,nat,numeral_numeral(nat),V2) ) ) ).

% int_eq_iff_numeral
tff(fact_1705_nat__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_1706_power__int__mult__numeral,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: num,N: num] : power_int(B,power_int(B,X,aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),N)) = power_int(B,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),N))) ) ).

% power_int_mult_numeral
tff(fact_1707_numeral__le__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: num,N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),N) ) ) ).

% numeral_le_iff
tff(fact_1708_numeral__less__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Ma: num,N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),N) ) ) ).

% numeral_less_iff
tff(fact_1709_numeral__plus__numeral,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)) ) ).

% numeral_plus_numeral
tff(fact_1710_add__numeral__left,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [V2: num,W: num,Z: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),V2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),W)),Z)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W))),Z) ) ).

% add_numeral_left
tff(fact_1711_numeral__times__numeral,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),N)) ) ).

% numeral_times_numeral
tff(fact_1712_mult__numeral__left__semiring__numeral,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [V2: num,W: num,Z: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),Z)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1713_neg__numeral__eq__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Ma: num,N: num] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)) )
        <=> ( Ma = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_1714_power__zero__numeral,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [K: num] : aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(B) ) ).

% power_zero_numeral
tff(fact_1715_mod__self,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),A2) = zero_zero(B) ) ).

% mod_self
tff(fact_1716_mod__by__0,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),zero_zero(B)) = A2 ) ).

% mod_by_0
tff(fact_1717_mod__0,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% mod_0
tff(fact_1718_bits__mod__0,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% bits_mod_0
tff(fact_1719_power__add__numeral,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B,Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),Ma))),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N))) ) ).

% power_add_numeral
tff(fact_1720_power__add__numeral2,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [A2: B,Ma: num,N: num,B2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),Ma))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)))),B2) ) ).

% power_add_numeral2
tff(fact_1721_of__nat__numeral,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: num] : aa(nat,B,semiring_1_of_nat(B),aa(num,nat,numeral_numeral(nat),N)) = aa(num,B,numeral_numeral(B),N) ) ).

% of_nat_numeral
tff(fact_1722_mod__add__self1,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ).

% mod_add_self1
tff(fact_1723_mod__add__self2,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ).

% mod_add_self2
tff(fact_1724_abs__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num] : aa(B,B,abs_abs(B),aa(num,B,numeral_numeral(B),N)) = aa(num,B,numeral_numeral(B),N) ) ).

% abs_numeral
tff(fact_1725_of__int__eq__numeral__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Z: int,N: num] :
          ( ( aa(int,B,ring_1_of_int(B),Z) = aa(num,B,numeral_numeral(B),N) )
        <=> ( Z = aa(num,int,numeral_numeral(int),N) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_1726_of__int__numeral,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [K: num] : aa(int,B,ring_1_of_int(B),aa(num,int,numeral_numeral(int),K)) = aa(num,B,numeral_numeral(B),K) ) ).

% of_int_numeral
tff(fact_1727_power__int__add__numeral,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,aa(num,int,numeral_numeral(int),Ma))),power_int(B,X,aa(num,int,numeral_numeral(int),N))) = power_int(B,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N))) ) ).

% power_int_add_numeral
tff(fact_1728_power__int__add__numeral2,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: num,N: num,B2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,aa(num,int,numeral_numeral(int),Ma))),aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_1729_floor__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num] : archim6421214686448440834_floor(B,aa(num,B,numeral_numeral(B),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% floor_numeral
tff(fact_1730_ceiling__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num] : archimedean_ceiling(B,aa(num,B,numeral_numeral(B),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% ceiling_numeral
tff(fact_1731_power__int__numeral,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & power(B) )
     => ! [X: B,N: num] : power_int(B,X,aa(num,int,numeral_numeral(int),N)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),N)) ) ).

% power_int_numeral
tff(fact_1732_distrib__right__numeral,axiom,
    ! [B: $tType] :
      ( ( numeral(B)
        & semiring(B) )
     => ! [A2: B,B2: B,V2: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),aa(num,B,numeral_numeral(B),V2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),V2))),aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(num,B,numeral_numeral(B),V2))) ) ).

% distrib_right_numeral
tff(fact_1733_distrib__left__numeral,axiom,
    ! [B: $tType] :
      ( ( numeral(B)
        & semiring(B) )
     => ! [V2: num,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),C2)) ) ).

% distrib_left_numeral
tff(fact_1734_neg__numeral__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num,N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N),Ma) ) ) ).

% neg_numeral_le_iff
tff(fact_1735_left__diff__distrib__numeral,axiom,
    ! [B: $tType] :
      ( ( numeral(B)
        & ring(B) )
     => ! [A2: B,B2: B,V2: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),aa(num,B,numeral_numeral(B),V2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),V2))),aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(num,B,numeral_numeral(B),V2))) ) ).

% left_diff_distrib_numeral
tff(fact_1736_right__diff__distrib__numeral,axiom,
    ! [B: $tType] :
      ( ( numeral(B)
        & ring(B) )
     => ! [V2: num,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),V2)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_1737_neg__numeral__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num,N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),N),Ma) ) ) ).

% neg_numeral_less_iff
tff(fact_1738_semiring__norm_I168_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [V2: num,W: num,Y: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),Y)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_1739_add__neg__numeral__simps_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_1740_mult__neg__numeral__simps_I3_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1741_mult__neg__numeral__simps_I2_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(num,B,numeral_numeral(B),N)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1742_mult__neg__numeral__simps_I1_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),times_times(num),Ma),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1743_diff__numeral__simps_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(num,B,numeral_numeral(B),N)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N))) ) ).

% diff_numeral_simps(3)
tff(fact_1744_diff__numeral__simps_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)) ) ).

% diff_numeral_simps(2)
tff(fact_1745_mod__mult__self2__is__0,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),B2) = zero_zero(B) ) ).

% mod_mult_self2_is_0
tff(fact_1746_mod__mult__self1__is__0,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2)),B2) = zero_zero(B) ) ).

% mod_mult_self1_is_0
tff(fact_1747_mod__by__1,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),one_one(B)) = zero_zero(B) ) ).

% mod_by_1
tff(fact_1748_bits__mod__by__1,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),one_one(B)) = zero_zero(B) ) ).

% bits_mod_by_1
tff(fact_1749_mod__mult__self1,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,C2: B,B2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ).

% mod_mult_self1
tff(fact_1750_mod__mult__self2,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ).

% mod_mult_self2
tff(fact_1751_mod__mult__self3,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [C2: B,B2: B,A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)),A2)),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ).

% mod_mult_self3
tff(fact_1752_mod__mult__self4,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [B2: B,C2: B,A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)),A2)),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ).

% mod_mult_self4
tff(fact_1753_mod__div__trivial,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)),B2) = zero_zero(B) ) ).

% mod_div_trivial
tff(fact_1754_bits__mod__div__trivial,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)),B2) = zero_zero(B) ) ).

% bits_mod_div_trivial
tff(fact_1755_abs__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num] : aa(B,B,abs_abs(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(num,B,numeral_numeral(B),N) ) ).

% abs_neg_numeral
tff(fact_1756_power__int__mult__distrib__numeral1,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [W: num,Y: B,Ma: int] : power_int(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),Y),Ma) = aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,aa(num,B,numeral_numeral(B),W),Ma)),power_int(B,Y,Ma)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_1757_power__int__mult__distrib__numeral2,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B,W: num,Ma: int] : power_int(B,aa(B,B,aa(B,fun(B,B),times_times(B),X),aa(num,B,numeral_numeral(B),W)),Ma) = aa(B,B,aa(B,fun(B,B),times_times(B),power_int(B,X,Ma)),power_int(B,aa(num,B,numeral_numeral(B),W),Ma)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_1758_mod__by__Suc__0,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1759_nat__neg__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_1760_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,Ma: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),Ma) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Ma = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1761_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_1762_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1763_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),aa(num,nat,numeral_numeral(nat),V3)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_1764_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( aa(int,nat,nat2,Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_1765_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = aa(int,nat,nat2,Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_1766_dbl__simps_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_dbl(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),K))) = aa(B,B,uminus_uminus(B),neg_numeral_dbl(B,aa(num,B,numeral_numeral(B),K))) ) ).

% dbl_simps(1)
tff(fact_1767_le__divide__eq__numeral1_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(num,B,numeral_numeral(B),W)))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),W))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1768_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,W: num,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(num,B,numeral_numeral(B),W))),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1769_eq__divide__eq__numeral1_I1_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,W: num] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(num,B,numeral_numeral(B),W)) )
        <=> $ite(aa(num,B,numeral_numeral(B),W) != zero_zero(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),W)) = B2,A2 = zero_zero(B)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1770_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,W: num,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(num,B,numeral_numeral(B),W)) = A2 )
        <=> $ite(aa(num,B,numeral_numeral(B),W) != zero_zero(B),B2 = aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),W)),A2 = zero_zero(B)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1771_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,W: num,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(num,B,numeral_numeral(B),W))),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1772_less__divide__eq__numeral1_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(num,B,numeral_numeral(B),W)))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),W))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1773_mod__minus1__right,axiom,
    ! [B: $tType] :
      ( euclid8851590272496341667cancel(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,uminus_uminus(B),one_one(B))) = zero_zero(B) ) ).

% mod_minus1_right
tff(fact_1774_of__int__numeral__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num,Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),N)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_1775_of__int__le__numeral__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int,N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Z)),aa(num,B,numeral_numeral(B),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),N)) ) ) ).

% of_int_le_numeral_iff
tff(fact_1776_of__int__numeral__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num,Z: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),N)),aa(int,B,ring_1_of_int(B),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_1777_of__int__less__numeral__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Z: int,N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),Z)),aa(num,B,numeral_numeral(B),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),N)) ) ) ).

% of_int_less_numeral_iff
tff(fact_1778_numeral__le__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),V2)),X) ) ) ).

% numeral_le_floor
tff(fact_1779_floor__less__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(B,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(num,B,numeral_numeral(B),V2)) ) ) ).

% floor_less_numeral
tff(fact_1780_ceiling__le__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(num,B,numeral_numeral(B),V2)) ) ) ).

% ceiling_le_numeral
tff(fact_1781_numeral__less__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),V2)),X) ) ) ).

% numeral_less_ceiling
tff(fact_1782_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [X: num,N: nat,Y: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N) = aa(nat,B,semiring_1_of_nat(B),Y) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1783_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [Y: nat,X: num,N: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),Y) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1784_Suc__mod__mult__self1,axiom,
    ! [Ma: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Ma)),N) ).

% Suc_mod_mult_self1
tff(fact_1785_Suc__mod__mult__self2,axiom,
    ! [Ma: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Ma)),N) ).

% Suc_mod_mult_self2
tff(fact_1786_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),Ma))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Ma)),N) ).

% Suc_mod_mult_self3
tff(fact_1787_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),Ma))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Ma)),N) ).

% Suc_mod_mult_self4
tff(fact_1788_floor__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num] : archim6421214686448440834_floor(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_neg_numeral
tff(fact_1789_ceiling__add__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] : archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(num,B,numeral_numeral(B),V2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(B,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_add_numeral
tff(fact_1790_floor__diff__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] : archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(num,B,numeral_numeral(B),V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(B,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_diff_numeral
tff(fact_1791_ceiling__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num] : archimedean_ceiling(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_neg_numeral
tff(fact_1792_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N))),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1793_ceiling__diff__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] : archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(num,B,numeral_numeral(B),V2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(B,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_diff_numeral
tff(fact_1794_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Y: int,X: num,N: nat] :
          ( ( aa(int,B,ring_1_of_int(B),Y) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_1795_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [X: num,N: nat,Y: int] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N) = aa(int,B,ring_1_of_int(B),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_1796_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1797_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1798_le__divide__eq__numeral1_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_1799_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,W: num,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))),B2) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_1800_eq__divide__eq__numeral1_I2_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B,W: num] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))) )
        <=> $ite(aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)) != zero_zero(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))) = B2,A2 = zero_zero(B)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1801_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,W: num,A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))) = A2 )
        <=> $ite(aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)) != zero_zero(B),B2 = aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),A2 = zero_zero(B)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1802_less__divide__eq__numeral1_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_1803_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,W: num,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))),B2) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_1804_dbl__dec__simps_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_dbl_dec(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),K))) = aa(B,B,uminus_uminus(B),neg_numeral_dbl_inc(B,aa(num,B,numeral_numeral(B),K))) ) ).

% dbl_dec_simps(1)
tff(fact_1805_dbl__inc__simps_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_dbl_inc(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),K))) = aa(B,B,uminus_uminus(B),neg_numeral_dbl_dec(B,aa(num,B,numeral_numeral(B),K))) ) ).

% dbl_inc_simps(1)
tff(fact_1806_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_1807_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_1808_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_1809_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_1810_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_1811_numeral__less__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),V2)),one_one(B))),X) ) ) ).

% numeral_less_floor
tff(fact_1812_floor__le__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),V2)),one_one(B))) ) ) ).

% floor_le_numeral
tff(fact_1813_ceiling__less__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(B,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),V2)),one_one(B))) ) ) ).

% ceiling_less_numeral
tff(fact_1814_numeral__le__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),V2)),one_one(B))),X) ) ) ).

% numeral_le_ceiling
tff(fact_1815_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [I: num,N: nat,X: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),I)),N)),aa(nat,B,semiring_1_of_nat(B),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1816_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: nat,I: num,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,semiring_1_of_nat(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),I)),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1817_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: nat,I: num,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,semiring_1_of_nat(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),I)),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1818_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [I: num,N: nat,X: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),I)),N)),aa(nat,B,semiring_1_of_nat(B),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1819_neg__numeral__le__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),X) ) ) ).

% neg_numeral_le_floor
tff(fact_1820_floor__less__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(B,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_1821_ceiling__le__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(B,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_1822_neg__numeral__less__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),X) ) ) ).

% neg_numeral_less_ceiling
tff(fact_1823_of__int__le__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: int,X: num,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_1824_numeral__power__le__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: num,N: nat,A2: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N)),aa(int,B,ring_1_of_int(B),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_1825_numeral__power__less__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: num,N: nat,A2: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N)),aa(int,B,ring_1_of_int(B),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_1826_of__int__less__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: int,X: num,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),X)),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_1827_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Y: int,X: num,N: nat] :
          ( ( aa(int,B,ring_1_of_int(B),Y) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X))),N) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_1828_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [X: num,N: nat,Y: int] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X))),N) = aa(int,B,ring_1_of_int(B),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_1829_neg__numeral__less__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),one_one(B))),X) ) ) ).

% neg_numeral_less_floor
tff(fact_1830_floor__le__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),one_one(B))) ) ) ).

% floor_le_neg_numeral
tff(fact_1831_ceiling__less__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(B,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),one_one(B))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1832_neg__numeral__le__ceiling,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [V2: num,X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),one_one(B))),X) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1833_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: int,X: num,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X))),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_1834_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: num,N: nat,A2: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X))),N)),aa(int,B,ring_1_of_int(B),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_1835_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: num,N: nat,A2: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X))),N)),aa(int,B,ring_1_of_int(B),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_1836_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: int,X: num,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X))),N))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_1837_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_1838_nat__numeral__as__int,axiom,
    ! [X4: num] : aa(num,nat,numeral_numeral(nat),X4) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X4)) ).

% nat_numeral_as_int
tff(fact_1839_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_1840_of__nat__mod,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_mod
tff(fact_1841_mod__add__eq,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,C2: B,B2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),C2)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),C2))),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) ) ).

% mod_add_eq
tff(fact_1842_mod__add__cong,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,C2: B,A5: B,B2: B,B5: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A5),C2) )
         => ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B5),C2) )
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A5),B5)),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1843_mod__add__left__eq,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,C2: B,B2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),C2)),B2)),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) ) ).

% mod_add_left_eq
tff(fact_1844_mod__add__right__eq,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),C2))),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) ) ).

% mod_add_right_eq
tff(fact_1845_zero__neq__numeral,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: num] : zero_zero(B) != aa(num,B,numeral_numeral(B),N) ) ).

% zero_neq_numeral
tff(fact_1846_mod__less__eq__dividend,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)),Ma) ).

% mod_less_eq_dividend
tff(fact_1847_neg__numeral__neq__numeral,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Ma: num,N: num] : aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma)) != aa(num,B,numeral_numeral(B),N) ) ).

% neg_numeral_neq_numeral
tff(fact_1848_numeral__neq__neg__numeral,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [Ma: num,N: num] : aa(num,B,numeral_numeral(B),Ma) != aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_1849_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_1850_Ints__numeral,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: num] : aa(set(B),$o,member(B,aa(num,B,numeral_numeral(B),N)),ring_1_Ints(B)) ) ).

% Ints_numeral
tff(fact_1851_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_1852_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_1853_of__int__neg__numeral,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [K: num] : aa(int,B,ring_1_of_int(B),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),K)) ) ).

% of_int_neg_numeral
tff(fact_1854_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1855_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1856_mod__eqE,axiom,
    ! [B: $tType] :
      ( euclid8851590272496341667cancel(B)
     => ! [A2: B,C2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),C2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),C2) )
         => ~ ! [D6: B] : B2 != aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),D6)) ) ) ).

% mod_eqE
tff(fact_1857_mod__eq__self__iff__div__eq__0,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = A2 )
        <=> ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = zero_zero(B) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1858_div__add1__eq,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),C2)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),C2))),C2)) ) ).

% div_add1_eq
tff(fact_1859_nth__rotate1,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,B,nth(B,rotate1(B,Xs)),N) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N)),aa(list(B),nat,size_size(list(B)),Xs))) ) ) ).

% nth_rotate1
tff(fact_1860_list__eq__iff__nth__eq,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
           => ( aa(nat,B,nth(B,Xs),I3) = aa(nat,B,nth(B,Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1861_Skolem__list__nth,axiom,
    ! [B: $tType,K: nat,Pa: fun(nat,fun(B,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
         => ? [X_12: B] : aa(B,$o,aa(nat,fun(B,$o),Pa,I3),X_12) )
    <=> ? [Xs3: list(B)] :
          ( ( aa(list(B),nat,size_size(list(B)),Xs3) = K )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
             => aa(B,$o,aa(nat,fun(B,$o),Pa,I3),aa(nat,B,nth(B,Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1862_nth__equalityI,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Xs))
           => ( aa(nat,B,nth(B,Xs),I2) = aa(nat,B,nth(B,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1863_obtain__list__from__elements,axiom,
    ! [B: $tType,N: nat,Pa: fun(B,fun(nat,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
         => ? [Li: B] : aa(nat,$o,aa(B,fun(nat,$o),Pa,Li),I2) )
     => ~ ! [L2: list(B)] :
            ( ( aa(list(B),nat,size_size(list(B)),L2) = N )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),N)
                 => aa(nat,$o,aa(B,fun(nat,$o),Pa,aa(nat,B,nth(B,L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_1864_mod__Suc,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Ma)),N) = $ite(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)) = N,zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N))) ).

% mod_Suc
tff(fact_1865_mod__less__divisor,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)),N) ) ).

% mod_less_divisor
tff(fact_1866_gcd__nat__induct,axiom,
    ! [Pa: fun(nat,fun(nat,$o)),Ma: nat,N: nat] :
      ( ! [M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),Pa,M2),zero_zero(nat))
     => ( ! [M2: nat,N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),Pa,N2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M2),N2))
             => aa(nat,$o,aa(nat,fun(nat,$o),Pa,M2),N2) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),Pa,Ma),N) ) ) ).

% gcd_nat_induct
tff(fact_1867_zero__le__numeral,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(num,B,numeral_numeral(B),N)) ) ).

% zero_le_numeral
tff(fact_1868_not__numeral__le__zero,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),N)),zero_zero(B)) ) ).

% not_numeral_le_zero
tff(fact_1869_mod__Suc__le__divisor,axiom,
    ! [Ma: nat,N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(nat,nat,suc,N))),N) ).

% mod_Suc_le_divisor
tff(fact_1870_zero__less__numeral,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(num,B,numeral_numeral(B),N)) ) ).

% zero_less_numeral
tff(fact_1871_not__numeral__less__zero,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),N)),zero_zero(B)) ) ).

% not_numeral_less_zero
tff(fact_1872_one__le__numeral,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(num,B,numeral_numeral(B),N)) ) ).

% one_le_numeral
tff(fact_1873_nat__power__less__imp__less,axiom,
    ! [I: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ) ) ).

% nat_power_less_imp_less
tff(fact_1874_mod__eq__0D,axiom,
    ! [Ma: nat,D2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),D2) = zero_zero(nat) )
     => ? [Q4: nat] : Ma = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q4) ) ).

% mod_eq_0D
tff(fact_1875_zmod__le__nonneg__dividend,axiom,
    ! [Ma: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ma)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ma),K)),Ma) ) ).

% zmod_le_nonneg_dividend
tff(fact_1876_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1877_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% neg_mod_bound
tff(fact_1878_not__numeral__less__one,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),N)),one_one(B)) ) ).

% not_numeral_less_one
tff(fact_1879_le__mod__geq,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),N) ) ) ).

% le_mod_geq
tff(fact_1880_not__numeral__le__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num,N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) ) ).

% not_numeral_le_neg_numeral
tff(fact_1881_neg__numeral__le__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num,N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(num,B,numeral_numeral(B),N)) ) ).

% neg_numeral_le_numeral
tff(fact_1882_zero__neq__neg__numeral,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [N: num] : zero_zero(B) != aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)) ) ).

% zero_neq_neg_numeral
tff(fact_1883_one__plus__numeral__commute,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(num,B,numeral_numeral(B),X)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),X)),one_one(B)) ) ).

% one_plus_numeral_commute
tff(fact_1884_zmod__eq__0D,axiom,
    ! [Ma: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ma),D2) = zero_zero(int) )
     => ? [Q4: int] : Ma = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0D
tff(fact_1885_zmod__eq__0__iff,axiom,
    ! [Ma: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ma),D2) = zero_zero(int) )
    <=> ? [Q6: int] : Ma = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q6) ) ).

% zmod_eq_0_iff
tff(fact_1886_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_1887_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_1888_neg__numeral__less__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num,N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(num,B,numeral_numeral(B),N)) ) ).

% neg_numeral_less_numeral
tff(fact_1889_not__numeral__less__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num,N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) ) ).

% not_numeral_less_neg_numeral
tff(fact_1890_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),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_1891_numeral__neq__neg__one,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [N: num] : aa(num,B,numeral_numeral(B),N) != aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% numeral_neq_neg_one
tff(fact_1892_one__neq__neg__numeral,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [N: num] : one_one(B) != aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)) ) ).

% one_neq_neg_numeral
tff(fact_1893_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1894_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1895_div__mult1__eq,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),C2))),C2)) ) ).

% div_mult1_eq
tff(fact_1896_cancel__div__mod__rules_I2_J,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [B2: B,A2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2))),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2))),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_1897_cancel__div__mod__rules_I1_J,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),B2)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2))),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_1898_mod__div__decomp,axiom,
    ! [B: $tType] :
      ( semiring_modulo(B)
     => ! [A2: B,B2: B] : A2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),B2)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)) ) ).

% mod_div_decomp
tff(fact_1899_div__mult__mod__eq,axiom,
    ! [B: $tType] :
      ( semiring_modulo(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),B2)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_1900_mod__div__mult__eq,axiom,
    ! [B: $tType] :
      ( semiring_modulo(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_1901_mod__mult__div__eq,axiom,
    ! [B: $tType] :
      ( semiring_modulo(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_1902_mult__div__mod__eq,axiom,
    ! [B: $tType] :
      ( semiring_modulo(B)
     => ! [B2: B,A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2))),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_1903_mod__le__divisor,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)),N) ) ).

% mod_le_divisor
tff(fact_1904_fact__mod,axiom,
    ! [B: $tType] :
      ( ( linordered_semidom(B)
        & semidom_modulo(B) )
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),semiring_char_0_fact(B,N)),semiring_char_0_fact(B,Ma)) = zero_zero(B) ) ) ) ).

% fact_mod
tff(fact_1905_neg__numeral__le__zero,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))),zero_zero(B)) ) ).

% neg_numeral_le_zero
tff(fact_1906_not__zero__le__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) ) ).

% not_zero_le_neg_numeral
tff(fact_1907_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K)) ) ).

% power_gt_expt
tff(fact_1908_not__zero__less__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) ) ).

% not_zero_less_neg_numeral
tff(fact_1909_neg__numeral__less__zero,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))),zero_zero(B)) ) ).

% neg_numeral_less_zero
tff(fact_1910_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)) ) ).

% nat_one_le_power
tff(fact_1911_not__one__le__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))) ) ).

% not_one_le_neg_numeral
tff(fact_1912_not__numeral__le__neg__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% not_numeral_le_neg_one
tff(fact_1913_neg__numeral__le__neg__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% neg_numeral_le_neg_one
tff(fact_1914_neg__one__le__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),Ma)) ) ).

% neg_one_le_numeral
tff(fact_1915_neg__numeral__le__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),one_one(B)) ) ).

% neg_numeral_le_one
tff(fact_1916_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1917_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1918_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% neg_mod_conj
tff(fact_1919_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_1920_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_1921_eq__divide__eq__numeral_I1_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [W: num,B2: B,C2: B] :
          ( ( aa(num,B,numeral_numeral(B),W) = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) )
        <=> $ite(C2 != zero_zero(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2) = B2,aa(num,B,numeral_numeral(B),W) = zero_zero(B)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1922_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,C2: B,W: num] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) = aa(num,B,numeral_numeral(B),W) )
        <=> $ite(C2 != zero_zero(B),B2 = aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2),aa(num,B,numeral_numeral(B),W) = zero_zero(B)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1923_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),N) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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_1924_mod__eq__nat2E,axiom,
    ! [Ma: nat,Q2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
       => ~ ! [S3: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_1925_mod__eq__nat1E,axiom,
    ! [Ma: nat,Q2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
       => ~ ! [S3: nat] : Ma != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_1926_neg__numeral__less__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),one_one(B)) ) ).

% neg_numeral_less_one
tff(fact_1927_neg__one__less__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),Ma)) ) ).

% neg_one_less_numeral
tff(fact_1928_not__numeral__less__neg__one,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% not_numeral_less_neg_one
tff(fact_1929_not__one__less__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))) ) ).

% not_one_less_neg_numeral
tff(fact_1930_not__neg__one__less__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1931_div__less__mono,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A4),N) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B4),N) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N)) ) ) ) ) ).

% div_less_mono
tff(fact_1932_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_1933_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))) ).

% zmod_zminus1_eq_if
tff(fact_1934_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_1935_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A4),N) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B4),N) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1936_mod__mult2__eq,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),Q2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)) ).

% mod_mult2_eq
tff(fact_1937_div__mod__decomp,axiom,
    ! [A4: nat,N: nat] : A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),N)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A4),N)) ).

% div_mod_decomp
tff(fact_1938_fact__le__power,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),semiring_char_0_fact(B,N)),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N))) ) ).

% fact_le_power
tff(fact_1939_div__mod__decomp__int,axiom,
    ! [A4: int,N: int] : A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),N)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A4),N)) ).

% div_mod_decomp_int
tff(fact_1940_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R)) ) ).

% fact_div_fact_le_pow
tff(fact_1941_mod__mult2__eq_H,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B,Ma: nat,N: nat] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,semiring_1_of_nat(B),Ma))),aa(nat,B,semiring_1_of_nat(B),N)))),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(nat,B,semiring_1_of_nat(B),Ma))) ) ).

% mod_mult2_eq'
tff(fact_1942_split__mod,axiom,
    ! [Pa: fun(nat,$o),Ma: nat,N: nat] :
      ( aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(nat,$o,Pa,Ma) )
        & ( ( N != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),N)
             => ( ( Ma = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J3) )
               => aa(nat,$o,Pa,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1943_less__divide__eq__numeral_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [W: num,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),W)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),B2),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(num,B,numeral_numeral(B),W)),zero_zero(B))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1944_divide__less__eq__numeral_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),aa(num,B,numeral_numeral(B),W))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),B2),aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(num,B,numeral_numeral(B),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1945_field__char__0__class_Oof__nat__div,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N)))),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% field_char_0_class.of_nat_div
tff(fact_1946_eq__divide__eq__numeral_I2_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [W: num,B2: B,C2: B] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) )
        <=> $ite(C2 != zero_zero(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2) = B2,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)) = zero_zero(B)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1947_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [B2: B,C2: B,W: num] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)) )
        <=> $ite(C2 != zero_zero(B),B2 = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)) = zero_zero(B)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_1948_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1949_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_1950_array__of__list__make,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B)] : array_of_list(B,Xs) = array_make(B,aa(list(B),nat,size_size(list(B)),Xs),nth(B,Xs)) ) ).

% array_of_list_make
tff(fact_1951_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_1952_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),C2)
         => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2))),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1953_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: $tType,F: fun(nat,fun(B,B)),A2: nat,B2: nat,Acc: B] :
      set_fo6178422350223883121st_nat(B,F,A2,B2,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc,set_fo6178422350223883121st_nat(B,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(B,B,aa(nat,fun(B,B),F,A2),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_1954_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [B: $tType,X: fun(nat,fun(B,B)),Xa2: nat,Xb: nat,Xc: B,Y: B] :
      ( ( set_fo6178422350223883121st_nat(B,X,Xa2,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa2),Xc,set_fo6178422350223883121st_nat(B,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(B,B,aa(nat,fun(B,B),X,Xa2),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_1955_divide__le__eq__numeral_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),aa(num,B,numeral_numeral(B),W))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),B2),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(num,B,numeral_numeral(B),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1956_le__divide__eq__numeral_I1_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [W: num,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),W)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),B2),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),W)),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),W)),zero_zero(B))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1957_Suc__times__mod__eq,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N))),Ma) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1958_divide__less__eq__numeral_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),B2),aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_1959_less__divide__eq__numeral_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [W: num,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),B2),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),zero_zero(B))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_1960_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1961_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1962_split__zmod,axiom,
    ! [Pa: fun(int,$o),N: int,K: int] :
      ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,Pa,N) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,Pa,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,Pa,J3) ) ) ) ) ).

% split_zmod
tff(fact_1963_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1964_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),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_1965_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1966_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_1967_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_1968_divide__le__eq__numeral_I2_J,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [B2: B,C2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))
        <=> $ite(
              aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
              aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),
              $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),zero_zero(B)),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),C2)),B2),aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_1969_sorted__in__between,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [I: nat,J: nat,L: list(B),X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
             => ( sorted_wrt(B,ord_less_eq(B),L)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,L),I)),X)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(nat,B,nth(B,L),J))
                   => ~ ! [K4: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K4)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K4),J)
                           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,L),K4)),X)
                             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(nat,B,nth(B,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K4),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_1970_diff__numeral__special_I6_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),one_one(B))) = aa(num,B,numeral_numeral(B),inc(Ma)) ) ).

% diff_numeral_special(6)
tff(fact_1971_diff__numeral__special_I5_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),N)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_1972_add__neg__numeral__special_I5_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_1973_add__neg__numeral__special_I6_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),inc(Ma))) ) ).

% add_neg_numeral_special(6)
tff(fact_1974_nth__rotate,axiom,
    ! [B: $tType,N: nat,Xs: list(B),Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,B,nth(B,aa(list(B),list(B),rotate(B,Ma),Xs)),N) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),aa(list(B),nat,size_size(list(B)),Xs))) ) ) ).

% nth_rotate
tff(fact_1975_nth__Cons__pos,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),N) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_1976_length__rotate,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),rotate(B,N),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_rotate
tff(fact_1977_nth__Cons__0,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_1978_rotate__length01,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat))
     => ( aa(list(B),list(B),rotate(B,N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_1979_rotate__id,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Xs)) = zero_zero(nat) )
     => ( aa(list(B),list(B),rotate(B,N),Xs) = Xs ) ) ).

% rotate_id
tff(fact_1980_nth__Cons__numeral,axiom,
    ! [B: $tType,X: B,Xs: list(B),V2: num] : aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_1981_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_1982_sorted2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B,Zs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Zs)))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
            & sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_1983_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_1984_list__tail__coinc,axiom,
    ! [B: $tType,N1: B,R1: list(B),N22: B,R22: list(B)] :
      ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),N1),R1) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),N22),R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_1985_sorted__wrt__true,axiom,
    ! [B: $tType,Xs: list(B)] : sorted_wrt(B,aTP_Lamp_ay(B,fun(B,$o)),Xs) ).

% sorted_wrt_true
tff(fact_1986_sorted1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B] : sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ) ).

% sorted1
tff(fact_1987_mergesort__by__rel__merge__induct,axiom,
    ! [B: $tType,C: $tType,Pa: fun(list(B),fun(list(C),$o)),R2: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C)] :
      ( ! [Xs2: list(B)] : aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs2),nil(C))
     => ( ! [Ys3: list(C)] : aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,nil(B)),Ys3)
       => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C)] :
              ( aa(C,$o,aa(B,fun(C,$o),R2,X2),Y2)
             => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs2),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys3))
               => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys3)) ) )
         => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C)] :
                ( ~ aa(C,$o,aa(B,fun(C,$o),R2,X2),Y2)
               => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),Ys3)
                 => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys3)) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1988_list__induct__first2,axiom,
    ! [B: $tType,Pa: fun(list(B),$o),Xs: list(B)] :
      ( aa(list(B),$o,Pa,nil(B))
     => ( ! [X2: B] : aa(list(B),$o,Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))
       => ( ! [X1: B,X23: B,Xs2: list(B)] :
              ( aa(list(B),$o,Pa,Xs2)
             => aa(list(B),$o,Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),Xs2))) )
         => aa(list(B),$o,Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_1989_list__2pre__induct,axiom,
    ! [B: $tType,C: $tType,Pa: fun(list(B),fun(list(C),$o)),W1: list(B),W22: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,nil(B)),nil(C))
     => ( ! [E4: B,W12: list(B),W23: list(C)] :
            ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,W12),W23)
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E4),W12)),W23) )
       => ( ! [E4: C,W13: list(B),W24: list(C)] :
              ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,W13),W24)
             => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,W13),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),E4),W24)) )
         => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,W1),W22) ) ) ) ).

% list_2pre_induct
tff(fact_1990_neq__NilE,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( L != nil(B) )
     => ~ ! [X2: B,Xs2: list(B)] : L != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) ) ).

% neq_NilE
tff(fact_1991_rotate__rotate,axiom,
    ! [B: $tType,Ma: nat,N: nat,Xs: list(B)] : aa(list(B),list(B),rotate(B,Ma),aa(list(B),list(B),rotate(B,N),Xs)) = aa(list(B),list(B),rotate(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),Xs) ).

% rotate_rotate
tff(fact_1992_strict__sorted__imp__sorted,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_1993_sorted0,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => sorted_wrt(B,ord_less_eq(B),nil(B)) ) ).

% sorted0
tff(fact_1994_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_1995_list__induct4,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,Xs: list(B),Ys: list(C),Zs: list(D),Ws: list(E),Pa: fun(list(B),fun(list(C),fun(list(D),fun(list(E),$o))))] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( ( aa(list(C),nat,size_size(list(C)),Ys) = aa(list(D),nat,size_size(list(D)),Zs) )
       => ( ( aa(list(D),nat,size_size(list(D)),Zs) = aa(list(E),nat,size_size(list(E)),Ws) )
         => ( aa(list(E),$o,aa(list(D),fun(list(E),$o),aa(list(C),fun(list(D),fun(list(E),$o)),aa(list(B),fun(list(C),fun(list(D),fun(list(E),$o))),Pa,nil(B)),nil(C)),nil(D)),nil(E))
           => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C),Z2: D,Zs2: list(D),W2: E,Ws2: list(E)] :
                  ( ( aa(list(B),nat,size_size(list(B)),Xs2) = aa(list(C),nat,size_size(list(C)),Ys3) )
                 => ( ( aa(list(C),nat,size_size(list(C)),Ys3) = aa(list(D),nat,size_size(list(D)),Zs2) )
                   => ( ( aa(list(D),nat,size_size(list(D)),Zs2) = aa(list(E),nat,size_size(list(E)),Ws2) )
                     => ( aa(list(E),$o,aa(list(D),fun(list(E),$o),aa(list(C),fun(list(D),fun(list(E),$o)),aa(list(B),fun(list(C),fun(list(D),fun(list(E),$o))),Pa,Xs2),Ys3),Zs2),Ws2)
                       => aa(list(E),$o,aa(list(D),fun(list(E),$o),aa(list(C),fun(list(D),fun(list(E),$o)),aa(list(B),fun(list(C),fun(list(D),fun(list(E),$o))),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys3)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),Z2),Zs2)),aa(list(E),list(E),aa(E,fun(list(E),list(E)),cons(E),W2),Ws2)) ) ) ) )
             => aa(list(E),$o,aa(list(D),fun(list(E),$o),aa(list(C),fun(list(D),fun(list(E),$o)),aa(list(B),fun(list(C),fun(list(D),fun(list(E),$o))),Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_1996_list__induct3,axiom,
    ! [C: $tType,B: $tType,D: $tType,Xs: list(B),Ys: list(C),Zs: list(D),Pa: fun(list(B),fun(list(C),fun(list(D),$o)))] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( ( aa(list(C),nat,size_size(list(C)),Ys) = aa(list(D),nat,size_size(list(D)),Zs) )
       => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),Pa,nil(B)),nil(C)),nil(D))
         => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C),Z2: D,Zs2: list(D)] :
                ( ( aa(list(B),nat,size_size(list(B)),Xs2) = aa(list(C),nat,size_size(list(C)),Ys3) )
               => ( ( aa(list(C),nat,size_size(list(C)),Ys3) = aa(list(D),nat,size_size(list(D)),Zs2) )
                 => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),Pa,Xs2),Ys3),Zs2)
                   => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys3)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),Z2),Zs2)) ) ) )
           => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),Pa,Xs),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_1997_list__induct2,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Pa: fun(list(B),fun(list(C),$o))] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,nil(B)),nil(C))
       => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C)] :
              ( ( aa(list(B),nat,size_size(list(B)),Xs2) = aa(list(C),nat,size_size(list(C)),Ys3) )
             => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs2),Ys3)
               => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys3)) ) )
         => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_1998_length__Cons,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_Cons
tff(fact_1999_Suc__length__conv,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( aa(nat,nat,suc,N) = aa(list(B),nat,size_size(list(B)),Xs) )
    <=> ? [Y5: B,Ys4: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
          & ( aa(list(B),nat,size_size(list(B)),Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_2000_length__Suc__conv,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y5: B,Ys4: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
          & ( aa(list(B),nat,size_size(list(B)),Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_2001_impossible__Cons,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))
     => ( Xs != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Ys) ) ) ).

% impossible_Cons
tff(fact_2002_rotate__conv__mod,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),list(B),rotate(B,N),Xs) = aa(list(B),list(B),rotate(B,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_2003_sorted__wrt__nth__less,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),Xs: list(B),I: nat,J: nat] :
      ( sorted_wrt(B,Pa,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
         => aa(B,$o,aa(B,fun(B,$o),Pa,aa(nat,B,nth(B,Xs),I)),aa(nat,B,nth(B,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_2004_sorted__wrt__iff__nth__less,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),Xs: list(B)] :
      ( sorted_wrt(B,Pa,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(B),nat,size_size(list(B)),Xs))
           => aa(B,$o,aa(B,fun(B,$o),Pa,aa(nat,B,nth(B,Xs),I3)),aa(nat,B,nth(B,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_2005_sorted__wrt01,axiom,
    ! [B: $tType,Xs: list(B),Pa: fun(B,fun(B,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat))
     => sorted_wrt(B,Pa,Xs) ) ).

% sorted_wrt01
tff(fact_2006_length__compl__induct,axiom,
    ! [B: $tType,Pa: fun(list(B),$o),L: list(B)] :
      ( aa(list(B),$o,Pa,nil(B))
     => ( ! [E4: B,L2: list(B)] :
            ( ! [Ll: list(B)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Ll)),aa(list(B),nat,size_size(list(B)),L2))
               => aa(list(B),$o,Pa,Ll) )
           => aa(list(B),$o,Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E4),L2)) )
       => aa(list(B),$o,Pa,L) ) ) ).

% length_compl_induct
tff(fact_2007_Suc__le__length__iff,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(B),nat,size_size(list(B)),Xs))
    <=> ? [X3: B,Ys4: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(B),nat,size_size(list(B)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_2008_list__decomp__1,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),L) = one_one(nat) )
     => ? [A3: B] : L = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B)) ) ).

% list_decomp_1
tff(fact_2009_sorted__iff__nth__mono__less,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(B),nat,size_size(list(B)),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),I3)),aa(nat,B,nth(B,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_2010_sorted01,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat))
         => sorted_wrt(B,ord_less_eq(B),Xs) ) ) ).

% sorted01
tff(fact_2011_numeral__inc,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [X: num] : aa(num,B,numeral_numeral(B),inc(X)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),X)),one_one(B)) ) ).

% numeral_inc
tff(fact_2012_list_Osize_I4_J,axiom,
    ! [B: $tType,X21: B,X222: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_2013_nth__Cons_H,axiom,
    ! [B: $tType,X: B,Xs: list(B),N: nat] :
      aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),N) = $ite(N = zero_zero(nat),X,aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% nth_Cons'
tff(fact_2014_sorted__iff__nth__Suc,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(B),nat,size_size(list(B)),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),I3)),aa(nat,B,nth(B,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_2015_sorted__iff__nth__mono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(B),nat,size_size(list(B)),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),I3)),aa(nat,B,nth(B,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_2016_sorted__nth__mono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),I: nat,J: nat] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),I)),aa(nat,B,nth(B,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_2017_nth__non__equal__first__eq,axiom,
    ! [B: $tType,X: B,Y: B,Xs: list(B),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),N) = Y )
      <=> ( ( aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2018_slice__Cons,axiom,
    ! [B: $tType,Begin: nat,End: nat,X: B,Xs: list(B)] :
      slice(B,Begin,End,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
        aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),slice(B,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)),
        slice(B,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_2019_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ).

% upto_aux_rec
tff(fact_2020_round__neg__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [N: num] : archimedean_round(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).

% round_neg_numeral
tff(fact_2021_shift__def,axiom,
    ! [C: $tType,B: $tType,Lab: fun(list(B),C),K: B,X4: list(B)] : bNF_Greatest_shift(B,C,Lab,K,X4) = aa(list(B),C,Lab,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),K),X4)) ).

% shift_def
tff(fact_2022_upto__rec__numeral_I4_J,axiom,
    ! [Ma: num,N: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),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),Ma))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_2023_upto__rec__numeral_I3_J,axiom,
    ! [Ma: num,N: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma)),aa(num,int,numeral_numeral(int),N)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),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),Ma))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),one_one(int)),aa(num,int,numeral_numeral(int),N))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_2024_upto__rec__numeral_I2_J,axiom,
    ! [Ma: num,N: num] :
      upto(aa(num,int,numeral_numeral(int),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Ma)),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),Ma)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Ma)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_2025_diff__numeral__special_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),one_one(B)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2026_round__of__int,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [N: int] : archimedean_round(B,aa(int,B,ring_1_of_int(B),N)) = N ) ).

% round_of_int
tff(fact_2027_one__eq__numeral__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: num] :
          ( ( one_one(B) = aa(num,B,numeral_numeral(B),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_2028_numeral__eq__one__iff,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: num] :
          ( ( aa(num,B,numeral_numeral(B),N) = one_one(B) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_2029_round__0,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ( archimedean_round(B,zero_zero(B)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2030_round__numeral,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [N: num] : archimedean_round(B,aa(num,B,numeral_numeral(B),N)) = aa(num,int,numeral_numeral(int),N) ) ).

% round_numeral
tff(fact_2031_round__1,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ( archimedean_round(B,one_one(B)) = one_one(int) ) ) ).

% round_1
tff(fact_2032_round__of__nat,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [N: nat] : archimedean_round(B,aa(nat,B,semiring_1_of_nat(B),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% round_of_nat
tff(fact_2033_neg__one__eq__numeral__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [N: num] :
          ( ( aa(B,B,uminus_uminus(B),one_one(B)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2034_numeral__eq__neg__one__iff,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [N: num] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)) = aa(B,B,uminus_uminus(B),one_one(B)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2035_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% Suc_numeral
tff(fact_2036_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_2037_not__neg__one__le__neg__numeral__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma)))
        <=> ( Ma != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2038_neg__numeral__less__neg__one__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Ma: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),one_one(B)))
        <=> ( Ma != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2039_one__plus__numeral,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(num,B,numeral_numeral(B),N)) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_2040_numeral__plus__one,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),N)),one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_2041_numeral__le__one__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),N)),one_one(B))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N),one2) ) ) ).

% numeral_le_one_iff
tff(fact_2042_one__less__numeral__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [N: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(num,B,numeral_numeral(B),N))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),N) ) ) ).

% one_less_numeral_iff
tff(fact_2043_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = 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),J),I)),one_one(int))) ).

% length_upto
tff(fact_2044_diff__numeral__special_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_2045_upto__rec__numeral_I1_J,axiom,
    ! [Ma: num,N: num] :
      upto(aa(num,int,numeral_numeral(int),Ma),aa(num,int,numeral_numeral(int),N)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Ma)),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),Ma)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Ma)),one_one(int)),aa(num,int,numeral_numeral(int),N))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_2046_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2047_sorted__upto,axiom,
    ! [Ma: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(Ma,N)) ).

% sorted_upto
tff(fact_2048_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).

% add_One_commute
tff(fact_2049_num__induct,axiom,
    ! [Pa: fun(num,$o),X: num] :
      ( aa(num,$o,Pa,one2)
     => ( ! [X2: num] :
            ( aa(num,$o,Pa,X2)
           => aa(num,$o,Pa,inc(X2)) )
       => aa(num,$o,Pa,X) ) ) ).

% num_induct
tff(fact_2050_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),X),one2)
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_2051_mult__numeral__1,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),one2)),A2) = A2 ) ).

% mult_numeral_1
tff(fact_2052_mult__numeral__1__right,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(num,B,numeral_numeral(B),one2)) = A2 ) ).

% mult_numeral_1_right
tff(fact_2053_numeral__One,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ( aa(num,B,numeral_numeral(B),one2) = one_one(B) ) ) ).

% numeral_One
tff(fact_2054_divide__numeral__1,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_2055_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_2056_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_2057_round__mono,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(B,X)),archimedean_round(B,Y)) ) ) ).

% round_mono
tff(fact_2058_floor__le__round,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),archimedean_round(B,X)) ) ).

% floor_le_round
tff(fact_2059_mult__1s__ring__1_I2_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),one2))) = aa(B,B,uminus_uminus(B),B2) ) ).

% mult_1s_ring_1(2)
tff(fact_2060_mult__1s__ring__1_I1_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),one2))),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% mult_1s_ring_1(1)
tff(fact_2061_uminus__numeral__One,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),one2)) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% uminus_numeral_One
tff(fact_2062_ceiling__ge__round,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(B,X)),archimedean_ceiling(B,X)) ) ).

% ceiling_ge_round
tff(fact_2063_cong__exp__iff__simps_I1_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),N)),aa(num,B,numeral_numeral(B),one2)) = zero_zero(B) ) ).

% cong_exp_iff_simps(1)
tff(fact_2064_numeral__1__eq__Suc__0,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% numeral_1_eq_Suc_0
tff(fact_2065_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_2066_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_2067_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa2),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)),nil(int)) ) ) ).

% upto.elims
tff(fact_2068_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),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_2069_round__diff__minimal,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Z: B,Ma: int] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),Z),aa(int,B,ring_1_of_int(B),archimedean_round(B,Z))))),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),Z),aa(int,B,ring_1_of_int(B),Ma)))) ) ).

% round_diff_minimal
tff(fact_2070_pochhammer__double,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Z: B,N: nat] : comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),comm_s3205402744901411588hammer(B,Z,N))),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),N)) ) ).

% pochhammer_double
tff(fact_2071_fact__double,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [N: nat] : semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N))),semiring_char_0_fact(B,N)) ) ).

% fact_double
tff(fact_2072_round__unique,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,Y: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))),aa(int,B,ring_1_of_int(B),Y))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),Y)),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))))
           => ( archimedean_round(B,X) = Y ) ) ) ) ).

% round_unique
tff(fact_2073_divmod__digit__1_I1_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2)))
             => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2)))),one_one(B)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2074_round__altdef,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          archimedean_round(B,X) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),archimedean_frac(B,X)),archimedean_ceiling(B,X),archim6421214686448440834_floor(B,X)) ) ).

% round_altdef
tff(fact_2075_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = 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,one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2076_round__unique_H,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B,N: int] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(int,B,ring_1_of_int(B),N)))),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))
         => ( archimedean_round(B,X) = N ) ) ) ).

% round_unique'
tff(fact_2077_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa(num,num,bit0,X22) = aa(num,num,bit0,Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_2078_semiring__norm_I6_J,axiom,
    ! [Ma: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,Ma)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)) ).

% semiring_norm(6)
tff(fact_2079_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = aa(num,num,bit0,one2) ).

% semiring_norm(2)
tff(fact_2080_num__double,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,one2)),N) = aa(num,num,bit0,N) ).

% num_double
tff(fact_2081_dbl__simps_I5_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_dbl(B,aa(num,B,numeral_numeral(B),K)) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,K)) ) ).

% dbl_simps(5)
tff(fact_2082_one__add__one,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)) ) ) ).

% one_add_one
tff(fact_2083_one__mod__two__eq__one,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = one_one(B) ) ) ).

% one_mod_two_eq_one
tff(fact_2084_bits__one__mod__two__eq__one,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = one_one(B) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_2085_zero__eq__power2,axiom,
    ! [B: $tType] :
      ( semiri2026040879449505780visors(B)
     => ! [A2: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% zero_eq_power2
tff(fact_2086_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,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_2087_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,one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_2088_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% Suc_1
tff(fact_2089_add__self__div__2,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Ma)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Ma ).

% add_self_div_2
tff(fact_2090_dbl__simps_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl(B,one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)) ) ) ).

% dbl_simps(3)
tff(fact_2091_one__div__two__eq__zero,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = zero_zero(B) ) ) ).

% one_div_two_eq_zero
tff(fact_2092_bits__1__div__2,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = zero_zero(B) ) ) ).

% bits_1_div_2
tff(fact_2093_add__neg__numeral__special_I9_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_2094_diff__numeral__special_I10_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),one_one(B))),one_one(B)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2095_diff__numeral__special_I11_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(B,B,uminus_uminus(B),one_one(B))) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2096_not__mod__2__eq__0__eq__1,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) != zero_zero(B) )
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = one_one(B) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2097_not__mod__2__eq__1__eq__0,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) != one_one(B) )
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = zero_zero(B) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2098_minus__1__div__2__eq,axiom,
    ! [B: $tType] :
      ( euclid8789492081693882211th_nat(B)
     => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% minus_1_div_2_eq
tff(fact_2099_power2__less__eq__zero__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(B))
        <=> ( A2 = zero_zero(B) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_2100_power2__eq__iff__nonneg,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2101_zero__less__power2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
        <=> ( A2 != zero_zero(B) ) ) ) ).

% zero_less_power2
tff(fact_2102_minus__1__mod__2__eq,axiom,
    ! [B: $tType] :
      ( euclid8789492081693882211th_nat(B)
     => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = one_one(B) ) ) ).

% minus_1_mod_2_eq
tff(fact_2103_bits__minus__1__mod__2__eq,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = one_one(B) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2104_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2105_add__self__mod__2,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Ma)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2106_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_2107_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_2108_dbl__simps_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl(B,aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ).

% dbl_simps(4)
tff(fact_2109_power__minus1__even,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = one_one(B) ) ).

% power_minus1_even
tff(fact_2110_one__less__floor,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(B,X))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),X) ) ) ).

% one_less_floor
tff(fact_2111_floor__le__one,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(B,X)),one_one(int))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ).

% floor_le_one
tff(fact_2112_mod2__gr__0,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2113_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != aa(num,num,bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_2114_numeral__Bit0,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [N: num] : aa(num,B,numeral_numeral(B),aa(num,num,bit0,N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),N)),aa(num,B,numeral_numeral(B),N)) ) ).

% numeral_Bit0
tff(fact_2115_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_2116_numeral__code_I2_J,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [N: num] :
          aa(num,B,numeral_numeral(B),aa(num,num,bit0,N)) = $let(
            m2: B,
            m2:= aa(num,B,numeral_numeral(B),N),
            aa(B,B,aa(B,fun(B,B),plus_plus(B),m2),m2) ) ) ).

% numeral_code(2)
tff(fact_2117_cong__exp__iff__simps_I2_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num,Q2: num] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,N))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Q2))) = zero_zero(B) )
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),N)),aa(num,B,numeral_numeral(B),Q2)) = zero_zero(B) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2118_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2119_power__numeral__even,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [Z: B,W: num] :
          aa(nat,B,aa(B,fun(nat,B),power_power(B),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,W))) = $let(
            w: B,
            w:= aa(nat,B,aa(B,fun(nat,B),power_power(B),Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(B,B,aa(B,fun(B,B),times_times(B),w),w) ) ) ).

% power_numeral_even
tff(fact_2120_left__add__twice,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_2121_mult__2__right,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [Z: B] : aa(B,B,aa(B,fun(B,B),times_times(B),Z),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),Z) ) ).

% mult_2_right
tff(fact_2122_mult__2,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [Z: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Z) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),Z) ) ).

% mult_2
tff(fact_2123_zero__power2,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(B) ) ) ).

% zero_power2
tff(fact_2124_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_2125_one__power2,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),one_one(B)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(B) ) ) ).

% one_power2
tff(fact_2126_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,one2)) ).

% nat_1_add_1
tff(fact_2127_power2__nat__le__imp__le,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% power2_nat_le_imp_le
tff(fact_2128_power2__nat__le__eq__le,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% power2_nat_le_eq_le
tff(fact_2129_self__le__ge2__pow,axiom,
    ! [K: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Ma)) ) ).

% self_le_ge2_pow
tff(fact_2130_half__gt__zero,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) ) ) ).

% half_gt_zero
tff(fact_2131_half__gt__zero__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_2132_zero__le__power2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% zero_le_power2
tff(fact_2133_power2__eq__imp__eq,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: B,Y: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2134_power2__le__imp__le,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_2135_power2__less__0,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(B)) ) ).

% power2_less_0
tff(fact_2136_exp__add__not__zero__imp__right,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) != zero_zero(B) )
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) != zero_zero(B) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2137_exp__add__not__zero__imp__left,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) != zero_zero(B) )
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma) != zero_zero(B) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2138_sum__power2__eq__zero__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(B) )
        <=> ( ( X = zero_zero(B) )
            & ( Y = zero_zero(B) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2139_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [N: nat,Ma: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) != zero_zero(B) )
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) != zero_zero(B) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2140_list__decomp__2,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),L) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
     => ? [A3: B,B3: B] : L = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),nil(B))) ) ).

% list_decomp_2
tff(fact_2141_div__exp__eq,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,Ma: nat,N: nat] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N))) ) ).

% div_exp_eq
tff(fact_2142_power2__eq__1__iff,axiom,
    ! [B: $tType] :
      ( ring_15535105094025558882visors(B)
     => ! [A2: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(B) )
        <=> ( ( A2 = one_one(B) )
            | ( A2 = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_2143_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2144_less__2__cases,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2145_abs__le__square__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),X)),aa(B,B,abs_abs(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% abs_le_square_iff
tff(fact_2146_abs__square__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(B) )
        <=> ( aa(B,B,abs_abs(B),X) = one_one(B) ) ) ) ).

% abs_square_eq_1
tff(fact_2147_nat__induct2,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( aa(nat,$o,Pa,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,Pa,N2)
             => aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
         => aa(nat,$o,Pa,N) ) ) ) ).

% nat_induct2
tff(fact_2148_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_2149_diff__le__diff__pow,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N))) ) ).

% diff_le_diff_pow
tff(fact_2150_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_2151_divmod__digit__0_I2_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2))),B2)
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2)) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2152_power2__less__imp__less,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_2153_sum__power2__le__zero__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(B))
        <=> ( ( X = zero_zero(B) )
            & ( Y = zero_zero(B) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2154_sum__power2__ge__zero,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% sum_power2_ge_zero
tff(fact_2155_bits__stable__imp__add__self,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = A2 )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) = zero_zero(B) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2156_sum__power2__gt__zero__iff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        <=> ( ( X != zero_zero(B) )
            | ( Y != zero_zero(B) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2157_not__sum__power2__lt__zero,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(B)) ) ).

% not_sum_power2_lt_zero
tff(fact_2158_power2__sum,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [X: B,Y: B] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_sum
tff(fact_2159_square__le__1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),one_one(B))),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(B)) ) ) ) ).

% square_le_1
tff(fact_2160_zero__le__even__power_H,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ).

% zero_le_even_power'
tff(fact_2161_power2__le__iff__abs__le,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),X)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2162_abs__square__le__1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),X)),one_one(B)) ) ) ).

% abs_square_le_1
tff(fact_2163_abs__square__less__1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,abs_abs(B),X)),one_one(B)) ) ) ).

% abs_square_less_1
tff(fact_2164_div__exp__mod__exp__eq,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat,Ma: nat] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_2165_nat__bit__induct,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,Pa,N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => aa(nat,$o,Pa,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa(nat,$o,Pa,N2)
             => aa(nat,$o,Pa,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,one2))),N2))) )
         => aa(nat,$o,Pa,N) ) ) ) ).

% nat_bit_induct
tff(fact_2166_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% div_2_gt_zero
tff(fact_2167_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),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,one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2168_fact__code,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : semiring_char_0_fact(B,N) = aa(nat,B,semiring_1_of_nat(B),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_2169_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2170_divmod__digit__0_I1_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2))),B2)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2171_power2__diff,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [X: B,Y: B] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_diff
tff(fact_2172_odd__0__le__power__imp__0__le,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2173_odd__power__less__zero,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),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,one2))),N)))),zero_zero(B)) ) ) ).

% odd_power_less_zero
tff(fact_2174_mult__exp__mod__exp__eq,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2175_power__minus1__odd,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),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,one2))),N))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% power_minus1_odd
tff(fact_2176_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2177_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2178_int__bit__induct,axiom,
    ! [Pa: fun(int,$o),K: int] :
      ( aa(int,$o,Pa,zero_zero(int))
     => ( aa(int,$o,Pa,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K4: int] :
              ( aa(int,$o,Pa,K4)
             => ( ( K4 != zero_zero(int) )
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),times_times(int),K4),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ) )
         => ( ! [K4: int] :
                ( aa(int,$o,Pa,K4)
               => ( ( K4 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K4),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
           => aa(int,$o,Pa,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2179_round__def,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : archimedean_round(B,X) = archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_2180_mod__double__modulus,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X)
           => ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),X),Ma) )
              | ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),X),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),X),Ma)),Ma) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2181_divmod__digit__1_I2_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2)))
             => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2))),B2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2182_of__int__round__le,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_round(B,X))),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))) ) ).

% of_int_round_le
tff(fact_2183_of__int__round__ge,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))),aa(int,B,ring_1_of_int(B),archimedean_round(B,X))) ) ).

% of_int_round_ge
tff(fact_2184_of__int__round__gt,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))),aa(int,B,ring_1_of_int(B),archimedean_round(B,X))) ) ).

% of_int_round_gt
tff(fact_2185_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_2186_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_2187_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_2188_of__int__round__abs__le,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(int,B,ring_1_of_int(B),archimedean_round(B,X))),X))),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) ) ).

% of_int_round_abs_le
tff(fact_2189_set__bit__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),zero_zero(nat)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_2190_unset__bit__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),zero_zero(nat)),A2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_2191_set__bit__Suc,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),aa(nat,nat,suc,N)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),N),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_2192_flip__bit__Suc,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),aa(nat,nat,suc,N)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),N),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_2193_unset__bit__Suc,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),aa(nat,nat,suc,N)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),N),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_2194_signed__take__bit__rec,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat,A2: B] :
          aa(B,B,bit_ri4674362597316999326ke_bit(B,N),A2) = $ite(N = zero_zero(nat),aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,bit_ri4674362597316999326ke_bit(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))))) ) ).

% signed_take_bit_rec
tff(fact_2195_time__array__map__entry,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,F: fun(B,B),P2: array(B),H: heap_ext(product_unit)] :
          time_time(array(B),array_map_entry(B,I,F,P2),H) = $ite(time_fails(array(B),array_map_entry(B,I,F,P2),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% time_array_map_entry
tff(fact_2196_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2197_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2198_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2199_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_2200_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_2201_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_2202_signed__take__bit__of__0,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] : aa(B,B,bit_ri4674362597316999326ke_bit(B,N),zero_zero(B)) = zero_zero(B) ) ).

% signed_take_bit_of_0
tff(fact_2203_signed__take__bit__of__minus__1,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] : aa(B,B,bit_ri4674362597316999326ke_bit(B,N),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% signed_take_bit_of_minus_1
tff(fact_2204_signed__take__bit__Suc__1,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] : aa(B,B,bit_ri4674362597316999326ke_bit(B,aa(nat,nat,suc,N)),one_one(B)) = one_one(B) ) ).

% signed_take_bit_Suc_1
tff(fact_2205_signed__take__bit__numeral__of__1,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [K: num] : aa(B,B,bit_ri4674362597316999326ke_bit(B,aa(num,nat,numeral_numeral(nat),K)),one_one(B)) = one_one(B) ) ).

% signed_take_bit_numeral_of_1
tff(fact_2206_signed__take__bit__0,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B] : aa(B,B,bit_ri4674362597316999326ke_bit(B,zero_zero(nat)),A2) = aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_2207_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_2208_unset__bit__nat__def,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),Ma),N) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Ma),aa(nat,int,semiring_1_of_nat(int),N))) ).

% unset_bit_nat_def
tff(fact_2209_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),K) ).

% unset_bit_less_eq
tff(fact_2210_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)) ).

% set_bit_greater_eq
tff(fact_2211_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2212_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2213_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2214_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2215_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2216_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2217_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2218_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2219_signed__take__bit__Suc,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat,A2: B] : aa(B,B,bit_ri4674362597316999326ke_bit(B,aa(nat,nat,suc,N)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,bit_ri4674362597316999326ke_bit(B,N),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2220_time__array__swap,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,P2: array(B),H: heap_ext(product_unit),F: fun(B,B)] :
          time_time(B,array_swap(B,I,X,P2),H) = $ite(time_fails(array(B),array_map_entry(B,I,F,P2),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% time_array_swap
tff(fact_2221_binomial__code,axiom,
    ! [N: nat,K: nat] :
      aa(nat,nat,binomial(N),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),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,one2))),K)),aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_2222_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_2223_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,N),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),bit_concat_bit(N,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))) ).

% concat_bit_Suc
tff(fact_2224_even__succ__mod__exp,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2225_even__succ__div__exp,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2226_take__bit__rec,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] :
          aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) = $ite(N = zero_zero(nat),zero_zero(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))) ) ).

% take_bit_rec
tff(fact_2227_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_2228_nat__dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),one_one(nat))
    <=> ( Ma = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2229_int__dvd__int__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ).

% int_dvd_int_iff
tff(fact_2230_dvd__0__right,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),zero_zero(B)) ) ).

% dvd_0_right
tff(fact_2231_dvd__0__left__iff,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),zero_zero(B)),A2)
        <=> ( A2 = zero_zero(B) ) ) ) ).

% dvd_0_left_iff
tff(fact_2232_dvd__add__triv__left__iff,axiom,
    ! [B: $tType] :
      ( comm_s4317794764714335236cancel(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2233_dvd__add__triv__right__iff,axiom,
    ! [B: $tType] :
      ( comm_s4317794764714335236cancel(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2234_dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Ma = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_2235_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_2236_take__bit__of__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),zero_zero(B)) = zero_zero(B) ) ).

% take_bit_of_0
tff(fact_2237_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2238_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),one_one(int))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2239_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_2240_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_2241_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : bit_concat_bit(N,K,zero_zero(int)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) ).

% concat_bit_of_zero_2
tff(fact_2242_dvd__mult__cancel__left,axiom,
    ! [B: $tType] :
      ( idom(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( ( C2 = zero_zero(B) )
            | aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2243_dvd__mult__cancel__right,axiom,
    ! [B: $tType] :
      ( idom(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( ( C2 = zero_zero(B) )
            | aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2244_dvd__times__left__cancel__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2245_dvd__times__right__cancel__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2246_dvd__add__times__triv__right__iff,axiom,
    ! [B: $tType] :
      ( comm_s4317794764714335236cancel(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2247_dvd__add__times__triv__left__iff,axiom,
    ! [B: $tType] :
      ( comm_s4317794764714335236cancel(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2248_unit__prod,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),one_one(B)) ) ) ) ).

% unit_prod
tff(fact_2249_div__add,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),B2)
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_2250_unit__div,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ).

% unit_div
tff(fact_2251_unit__div__1__unit,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2)),one_one(B)) ) ) ).

% unit_div_1_unit
tff(fact_2252_unit__div__1__div__1,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_2253_dvd__imp__mod__0,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),A2) = zero_zero(B) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2254_take__bit__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),zero_zero(nat)),A2) = zero_zero(B) ) ).

% take_bit_0
tff(fact_2255_take__bit__Suc__1,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,suc,N)),one_one(B)) = one_one(B) ) ).

% take_bit_Suc_1
tff(fact_2256_take__bit__numeral__1,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [L: num] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(num,nat,numeral_numeral(nat),L)),one_one(B)) = one_one(B) ) ).

% take_bit_numeral_1
tff(fact_2257_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_2258_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2259_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K) ) ).

% binomial_eq_0_iff
tff(fact_2260_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2261_of__nat__nat__take__bit__eq,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat,K: int] : aa(nat,B,semiring_1_of_nat(B),aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) = aa(int,B,ring_1_of_int(B),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_2262_semiring__norm_I7_J,axiom,
    ! [Ma: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,Ma)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)) ).

% semiring_norm(7)
tff(fact_2263_semiring__norm_I9_J,axiom,
    ! [Ma: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,Ma)),aa(num,num,bit0,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)) ).

% semiring_norm(9)
tff(fact_2264_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2265_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),N)),K) ) ).

% dvd_nat_abs_iff
tff(fact_2266_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nat_abs_dvd_iff
tff(fact_2267_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_concat_bit(N,K,L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_2268_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_concat_bit(N,K,L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_2269_dbl__inc__simps_I5_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_dbl_inc(B,aa(num,B,numeral_numeral(B),K)) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,K)) ) ).

% dbl_inc_simps(5)
tff(fact_2270_unit__mult__div__div,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_2271_unit__div__mult__self,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_2272_take__bit__of__1__eq__0__iff,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat] :
          ( ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),one_one(B)) = zero_zero(B) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_2273_pow__divides__pow__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [N: nat,A2: B,B2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2274_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) ) ).

% zero_less_binomial_iff
tff(fact_2275_semiring__norm_I3_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit0,N)) = aa(num,num,bit1,N) ).

% semiring_norm(3)
tff(fact_2276_semiring__norm_I4_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% semiring_norm(4)
tff(fact_2277_semiring__norm_I5_J,axiom,
    ! [Ma: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,Ma)),one2) = aa(num,num,bit1,Ma) ).

% semiring_norm(5)
tff(fact_2278_semiring__norm_I8_J,axiom,
    ! [Ma: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,Ma)),one2) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),one2)) ).

% semiring_norm(8)
tff(fact_2279_semiring__norm_I10_J,axiom,
    ! [Ma: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,Ma)),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),Ma),N)),one2)) ).

% semiring_norm(10)
tff(fact_2280_semiring__norm_I16_J,axiom,
    ! [Ma: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,Ma)),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),Ma),N)),aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),Ma),N)))) ).

% semiring_norm(16)
tff(fact_2281_odd__add,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))
        <=> ~ ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
            <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2) ) ) ) ).

% odd_add
tff(fact_2282_even__add,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_add
tff(fact_2283_dbl__inc__simps_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl_inc(B,one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_2284_even__plus__one__iff,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B)))
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2) ) ) ).

% even_plus_one_iff
tff(fact_2285_even__diff,axiom,
    ! [B: $tType] :
      ( ring_parity(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ).

% even_diff
tff(fact_2286_zero__le__power__eq__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2287_power__less__zero__eq,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),zero_zero(B))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ) ).

% power_less_zero_eq
tff(fact_2288_power__less__zero__eq__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(B))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2289_neg__one__even__power,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N) = one_one(B) ) ) ) ).

% neg_one_even_power
tff(fact_2290_neg__one__odd__power,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ) ).

% neg_one_odd_power
tff(fact_2291_even__of__nat,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,B,semiring_1_of_nat(B),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ) ) ).

% even_of_nat
tff(fact_2292_even__take__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2))
        <=> ( ( N = zero_zero(nat) )
            | aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_2293_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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 ) ) ).

% odd_Suc_minus_one
tff(fact_2294_even__diff__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) ) ) ).

% even_diff_nat
tff(fact_2295_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Ma: nat,V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Ma)))),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Ma)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2296_mod__Suc__eq__mod__add3,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_2297_Suc__div__eq__add3__div__numeral,axiom,
    ! [Ma: 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,Ma)))),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,one2))),Ma)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2298_div__Suc__eq__div__add3,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_2299_dbl__dec__simps_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_dbl_dec(B,aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_2300_even__succ__div__two,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_2301_odd__succ__div__two,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),one_one(B)) ) ) ) ).

% odd_succ_div_two
tff(fact_2302_even__succ__div__2,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),A2)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_2303_zero__less__power__eq__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( A2 != zero_zero(B) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2304_even__power,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% even_power
tff(fact_2305_take__bit__Suc__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,suc,zero_zero(nat))),A2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_0
tff(fact_2306_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2307_odd__two__times__div__two__succ,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))),one_one(B)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2308_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)),one_one(B)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2309_power__le__zero__eq__numeral,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,W: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(B))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W))
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A2 = zero_zero(B) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2310_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2311_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2312_of__nat__dvd__iff,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ) ).

% of_nat_dvd_iff
tff(fact_2313_take__bit__of__nat,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,Ma: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(nat,B,semiring_1_of_nat(B),Ma)) = aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),Ma)) ) ).

% take_bit_of_nat
tff(fact_2314_subset__divisors__dvd,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_az(B,fun(B,$o),A2))),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_az(B,fun(B,$o),B2)))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_2315_strict__subset__divisors__dvd,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_az(B,fun(B,$o),A2))),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_az(B,fun(B,$o),B2)))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_2316_take__bit__add,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat,A2: B,B2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2)),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),B2))) = aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ).

% take_bit_add
tff(fact_2317_take__bit__tightened,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B,B2: B,Ma: nat] :
          ( ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) = aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
           => ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),A2) = aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_2318_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,Ma: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),Ma)),Ma) ).

% take_bit_nat_less_eq_self
tff(fact_2319_take__bit__tightened__less__eq__nat,axiom,
    ! [Ma: nat,N: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Ma),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2320_dvd__field__iff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
        <=> ( ( A2 = zero_zero(B) )
           => ( B2 = zero_zero(B) ) ) ) ) ).

% dvd_field_iff
tff(fact_2321_dvd__0__left,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),zero_zero(B)),A2)
         => ( A2 = zero_zero(B) ) ) ) ).

% dvd_0_left
tff(fact_2322_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_2323_dvd__add,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_2324_dvd__add__left__iff,axiom,
    ! [B: $tType] :
      ( comm_s4317794764714335236cancel(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_2325_dvd__add__right__iff,axiom,
    ! [B: $tType] :
      ( comm_s4317794764714335236cancel(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_2326_division__decomp,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
         => ? [B6: B,C7: B] :
              ( ( A2 = aa(B,B,aa(B,fun(B,B),times_times(B),B6),C7) )
              & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B6),B2)
              & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C7),C2) ) ) ) ).

% division_decomp
tff(fact_2327_dvd__productE,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [P2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),P2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
         => ~ ! [X2: B,Y2: B] :
                ( ( P2 = aa(B,B,aa(B,fun(B,B),times_times(B),X2),Y2) )
               => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X2),A2)
                 => ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Y2),B2) ) ) ) ) ).

% dvd_productE
tff(fact_2328_one__dvd,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),one_one(B)),A2) ) ).

% one_dvd
tff(fact_2329_unit__imp__dvd,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2) ) ) ).

% unit_imp_dvd
tff(fact_2330_dvd__unit__imp__unit,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_2331_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa(num,num,bit0,X22) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_2332_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_2333_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_2334_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_2335_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_2336_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2337_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2338_dvd__diff__nat,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) ) ) ).

% dvd_diff_nat
tff(fact_2339_uminus__dvd__conv_I2_J,axiom,
    ! [D2: int,Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),Ta)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,uminus_uminus(int),Ta)) ) ).

% uminus_dvd_conv(2)
tff(fact_2340_uminus__dvd__conv_I1_J,axiom,
    ! [D2: int,Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),Ta)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,uminus_uminus(int),D2)),Ta) ) ).

% uminus_dvd_conv(1)
tff(fact_2341_zdvd__zdiffD,axiom,
    ! [K: int,Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),minus_minus(int),Ma),N))
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),N)
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Ma) ) ) ).

% zdvd_zdiffD
tff(fact_2342_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),A2)
       => ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_2343_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_2344_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_2345_concat__bit__assoc,axiom,
    ! [N: nat,K: int,Ma: nat,L: int,R: int] : bit_concat_bit(N,K,bit_concat_bit(Ma,L,R)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N),bit_concat_bit(N,K,L),R) ).

% concat_bit_assoc
tff(fact_2346_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),K)
     => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2347_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ).

% binomial_symmetric
tff(fact_2348_take__bit__tightened__less__eq__int,axiom,
    ! [Ma: nat,N: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Ma),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2349_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2350_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ).

% take_bit_nonnegative
tff(fact_2351_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_2352_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_2353_choose__mult__lemma,axiom,
    ! [Ma: nat,R: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),R)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),R)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),R)),Ma)) ).

% choose_mult_lemma
tff(fact_2354_not__is__unit__0,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),zero_zero(B)),one_one(B)) ) ).

% not_is_unit_0
tff(fact_2355_signed__take__bit__take__bit,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [Ma: nat,N: nat,A2: B] :
          aa(B,B,bit_ri4674362597316999326ke_bit(B,Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2)) = aa(B,B,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma),aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),bit_ri4674362597316999326ke_bit(B,Ma)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_2356_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)) ) ) ) ).

% pinf(9)
tff(fact_2357_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z2),X4)
         => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))
          <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)) ) ) ) ).

% pinf(10)
tff(fact_2358_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)) ) ) ) ).

% minf(9)
tff(fact_2359_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S2: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Z2)
         => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2))
          <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S2)) ) ) ) ).

% minf(10)
tff(fact_2360_is__unit__mult__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),one_one(B))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
            & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B)) ) ) ) ).

% is_unit_mult_iff
tff(fact_2361_dvd__mult__unit__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2362_mult__unit__dvd__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2)
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2363_dvd__mult__unit__iff_H,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2364_mult__unit__dvd__iff_H,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2)
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2365_unit__mult__left__cancel,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_2366_unit__mult__right__cancel,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( ( aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_2367_dvd__div__eq__0__iff,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2)
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = zero_zero(B) )
          <=> ( A2 = zero_zero(B) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2368_div__plus__div__distrib__dvd__right,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),B2)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2369_div__plus__div__distrib__dvd__left,axiom,
    ! [B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),A2)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),C2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2370_unit__div__cancel,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_2371_div__unit__dvd__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2)
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2372_dvd__div__unit__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2373_mod__0__imp__dvd,axiom,
    ! [B: $tType] :
      ( semiring_modulo(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = zero_zero(B) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_2374_dvd__eq__mod__eq__0,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),A2) = zero_zero(B) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2375_mod__eq__0__iff__dvd,axiom,
    ! [B: $tType] :
      ( semidom_modulo(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = zero_zero(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2376_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != aa(num,num,bit0,X23)
       => ~ ! [X33: num] : Y != aa(num,num,bit1,X33) ) ) ).

% num.exhaust
tff(fact_2377_take__bit__unset__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,Ma: nat,A2: B] :
          aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),Ma),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2),aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_2378_take__bit__set__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,Ma: nat,A2: B] :
          aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),Ma),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2),aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_2379_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(N),R)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R)) ) ).

% binomial_le_pow
tff(fact_2380_take__bit__flip__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,Ma: nat,A2: B] :
          aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),Ma),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2),aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_2381_binomial__gbinomial,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [N: nat,K: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(N),K)) = aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),N)),K) ) ).

% binomial_gbinomial
tff(fact_2382_dvd__pos__nat,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma) ) ) ).

% dvd_pos_nat
tff(fact_2383_nat__dvd__not__less,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
       => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),Ma) ) ) ).

% nat_dvd_not_less
tff(fact_2384_dvd__power__le,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [X: B,Y: B,N: nat,Ma: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),Ma)) ) ) ) ).

% dvd_power_le
tff(fact_2385_power__le__dvd,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat,B2: B,Ma: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),B2) ) ) ) ).

% power_le_dvd
tff(fact_2386_le__imp__power__dvd,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Ma: nat,N: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ).

% le_imp_power_dvd
tff(fact_2387_dvd__minus__self,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ) ).

% dvd_minus_self
tff(fact_2388_zdvd__antisym__nonneg,axiom,
    ! [Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ma)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),N)
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),N),Ma)
           => ( Ma = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2389_zdvd__not__zless,axiom,
    ! [Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ma)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ma),N)
       => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),N),Ma) ) ) ).

% zdvd_not_zless
tff(fact_2390_less__eq__dvd__minus,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) ) ) ).

% less_eq_dvd_minus
tff(fact_2391_dvd__diffD1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),N) ) ) ) ).

% dvd_diffD1
tff(fact_2392_dvd__diffD,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),N)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Ma) ) ) ) ).

% dvd_diffD
tff(fact_2393_zdvd__mult__cancel,axiom,
    ! [K: int,Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N))
     => ( ( K != zero_zero(int) )
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),N) ) ) ).

% zdvd_mult_cancel
tff(fact_2394_zdvd__mono,axiom,
    ! [K: int,Ma: int,Ta: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Ma),Ta)
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ta)) ) ) ).

% zdvd_mono
tff(fact_2395_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D6: nat,X2: nat,Y2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),D6) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)),D6) ) ) ) ).

% bezout_add_nat
tff(fact_2396_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,X: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D2) ) )
         => ? [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y2)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_2397_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D6: nat,X2: nat,Y2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)) = D6 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)) = D6 ) ) ) ).

% bezout1_nat
tff(fact_2398_zdvd__reduce,axiom,
    ! [K: int,N: int,Ma: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),N) ) ).

% zdvd_reduce
tff(fact_2399_zdvd__period,axiom,
    ! [A2: int,D2: int,X: int,Ta: int,C2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),D2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Ta))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),Ta)) ) ) ).

% zdvd_period
tff(fact_2400_fact__dvd,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [N: nat,Ma: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),semiring_char_0_fact(B,N)),semiring_char_0_fact(B,Ma)) ) ) ).

% fact_dvd
tff(fact_2401_take__bit__eq__0__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] :
          ( ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) = zero_zero(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_2402_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = aa(num,num,bit0,inc(X)) ).

% inc.simps(3)
tff(fact_2403_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa(num,num,bit0,X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_2404_nat__dvd__iff,axiom,
    ! [Z: int,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,Z)),Ma)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),Ma)),Ma = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_2405_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K)) ) ).

% zero_less_binomial
tff(fact_2406_take__bit__signed__take__bit,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [Ma: nat,N: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
         => ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),aa(B,B,bit_ri4674362597316999326ke_bit(B,N),A2)) = aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2407_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_2408_choose__mult,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),Ma)),aa(nat,nat,binomial(Ma),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),K))) ) ) ) ).

% choose_mult
tff(fact_2409_unity__coeff__ex,axiom,
    ! [B: $tType] :
      ( ( dvd(B)
        & semiring_0(B) )
     => ! [Pa: fun(B,$o),L: B] :
          ( ? [X3: B] : aa(B,$o,Pa,aa(B,B,aa(B,fun(B,B),times_times(B),L),X3))
        <=> ? [X3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),L),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),zero_zero(B)))
              & aa(B,$o,Pa,X3) ) ) ) ).

% unity_coeff_ex
tff(fact_2410_unit__dvdE,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ~ ( ( A2 != zero_zero(B) )
             => ! [C4: B] : B2 != aa(B,B,aa(B,fun(B,B),times_times(B),A2),C4) ) ) ) ).

% unit_dvdE
tff(fact_2411_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_2412_inf__period_I3_J,axiom,
    ! [B: $tType] :
      ( ( comm_ring(B)
        & dvd(B) )
     => ! [D2: B,D3: B,Ta: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),D3)
         => ! [X4: B,K5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Ta))
            <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X4),aa(B,B,aa(B,fun(B,B),times_times(B),K5),D3))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_2413_inf__period_I4_J,axiom,
    ! [B: $tType] :
      ( ( comm_ring(B)
        & dvd(B) )
     => ! [D2: B,D3: B,Ta: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),D3)
         => ! [X4: B,K5: B] :
              ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Ta))
            <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X4),aa(B,B,aa(B,fun(B,B),times_times(B),K5),D3))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_2414_dvd__div__eq__mult,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
           => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2) = C2 )
            <=> ( B2 = aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2415_div__dvd__iff__mult,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( ( B2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2)
            <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2416_dvd__div__iff__mult,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,B2: B,A2: B] :
          ( ( C2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2))
            <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2417_dvd__div__div__eq__mult,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,C2: B,B2: B,D2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( C2 != zero_zero(B) )
           => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
             => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),D2)
               => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),D2),C2) )
                <=> ( aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2) = aa(B,B,aa(B,fun(B,B),times_times(B),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2418_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) != zero_zero(int) )
     => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_2419_unit__div__eq__0__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = zero_zero(B) )
          <=> ( A2 = zero_zero(B) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2420_numeral__Bit1,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [N: num] : aa(num,B,numeral_numeral(B),aa(num,num,bit1,N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),N)),aa(num,B,numeral_numeral(B),N))),one_one(B)) ) ).

% numeral_Bit1
tff(fact_2421_unit__eq__div1,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = C2 )
          <=> ( A2 = aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_2422_unit__eq__div2,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( ( A2 = aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),B2) )
          <=> ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2423_div__mult__unit2,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2)
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2424_unit__div__commute,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_2425_unit__div__mult__swap,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2426_is__unit__div__mult2__eq,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B))
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2427_unit__imp__mod__eq__0,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = zero_zero(B) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2428_is__unit__power__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),one_one(B))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2429_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_2430_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) ) ) ).

% dvd_imp_le
tff(fact_2431_dvd__mult__cancel,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ) ).

% dvd_mult_cancel
tff(fact_2432_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2433_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D6: nat,X2: nat,Y2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),A2)
          & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),D6) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2434_take__bit__Suc__bit1,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat,K: num] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,suc,N)),aa(num,B,numeral_numeral(B),aa(num,num,bit1,K))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(num,B,numeral_numeral(B),K))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),one_one(B)) ) ).

% take_bit_Suc_bit1
tff(fact_2435_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),N)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),N) ) ) ).

% zdvd_imp_le
tff(fact_2436_fact__fact__dvd__fact,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: nat,N: nat] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),semiring_char_0_fact(B,K)),semiring_char_0_fact(B,N))),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N))) ) ).

% fact_fact_dvd_fact
tff(fact_2437_mod__greater__zero__iff__not__dvd,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),Ma) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2438_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_2439_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,Ma: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q2) )
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2440_of__int__divide__in__Ints,axiom,
    ! [B: $tType] :
      ( idom_divide(B)
     => ! [B2: int,A2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),A2)
         => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(int,B,ring_1_of_int(B),A2)),aa(int,B,ring_1_of_int(B),B2))),ring_1_Ints(B)) ) ) ).

% of_int_divide_in_Ints
tff(fact_2441_numeral__code_I3_J,axiom,
    ! [B: $tType] :
      ( numeral(B)
     => ! [N: num] :
          aa(num,B,numeral_numeral(B),aa(num,num,bit1,N)) = $let(
            m2: B,
            m2:= aa(num,B,numeral_numeral(B),N),
            aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),m2),m2)),one_one(B)) ) ) ).

% numeral_code(3)
tff(fact_2442_dvd__fact,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),semiring_char_0_fact(nat,N)) ) ) ).

% dvd_fact
tff(fact_2443_power__numeral__odd,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [Z: B,W: num] :
          aa(nat,B,aa(B,fun(nat,B),power_power(B),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = $let(
            w: B,
            w:= aa(nat,B,aa(B,fun(nat,B),power_power(B),Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_2444_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K) ) ) ).

% even_nat_iff
tff(fact_2445_stable__imp__take__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,N: nat] :
          ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = A2 )
         => ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) = $ite(aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2),zero_zero(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)),one_one(B))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2446_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_ba(nat,fun(nat,$o)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_2447_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_2448_even__zero,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),zero_zero(B)) ) ).

% even_zero
tff(fact_2449_is__unitE,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ~ ( ( A2 != zero_zero(B) )
             => ! [B3: B] :
                  ( ( B3 != zero_zero(B) )
                 => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B3),one_one(B))
                   => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2) = B3 )
                     => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),B3) = A2 )
                       => ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B3) = one_one(B) )
                         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2) != aa(B,B,aa(B,fun(B,B),times_times(B),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2450_is__unit__div__mult__cancel__left,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2451_is__unit__div__mult__cancel__right,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2452_odd__even__add,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) ) ) ) ).

% odd_even_add
tff(fact_2453_odd__one,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),one_one(B)) ) ).

% odd_one
tff(fact_2454_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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,one2)))) ) ).

% odd_mod_4_div_2
tff(fact_2455_cong__exp__iff__simps_I3_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num,Q2: num] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,N))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Q2))) != zero_zero(B) ) ).

% cong_exp_iff_simps(3)
tff(fact_2456_dvd__power__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [X: B,Ma: nat,N: nat] :
          ( ( X != zero_zero(B) )
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N))
          <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),one_one(B))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ) ) ).

% dvd_power_iff
tff(fact_2457_dvd__power,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [N: nat,X: B] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
            | ( X = one_one(B) ) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)) ) ) ).

% dvd_power
tff(fact_2458_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_2459_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_2460_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),aa(nat,nat,binomial(N),K)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_2461_dvd__mult__cancel2,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Ma)),Ma)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2462_dvd__mult__cancel1,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)),Ma)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2463_choose__dvd,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),semiring_char_0_fact(B,K)),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),semiring_char_0_fact(B,N)) ) ) ).

% choose_dvd
tff(fact_2464_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))))) ) ).

% even_even_mod_4_iff
tff(fact_2465_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),Ma))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),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),R),Ma)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_2466_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),R))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_2467_zdvd__mult__cancel1,axiom,
    ! [Ma: int,N: int] :
      ( ( Ma != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ma),N)),Ma)
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2468_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_2469_power__dvd__imp__le,axiom,
    ! [I: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% power_dvd_imp_le
tff(fact_2470_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_2471_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_2472_binomial__ge__n__over__k__pow__k,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,semiring_1_of_nat(B),N)),aa(nat,B,semiring_1_of_nat(B),K))),K)),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(N),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2473_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2474_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = 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),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2475_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% binomial_le_pow2
tff(fact_2476_even__iff__mod__2__eq__zero,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = zero_zero(B) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2477_cong__exp__iff__simps_I7_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Q2: num,N: num] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),one2)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Q2))) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,N))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Q2))) )
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),N)),aa(num,B,numeral_numeral(B),Q2)) = zero_zero(B) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2478_cong__exp__iff__simps_I11_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num,Q2: num] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Ma))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Q2))) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),one2)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Q2))) )
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),Q2)) = zero_zero(B) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2479_odd__iff__mod__2__eq__one,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
        <=> ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = one_one(B) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2480_power__mono__odd,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat,A2: B,B2: B] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N)) ) ) ) ).

% power_mono_odd
tff(fact_2481_odd__pos,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% odd_pos
tff(fact_2482_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_2483_Suc__mod__eq__add3__mod,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Ma)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Ma)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2484_Suc__div__eq__add3__div,axiom,
    ! [Ma: 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,Ma)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Ma)),N) ).

% Suc_div_eq_add3_div
tff(fact_2485_even__unset__bit__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),Ma),A2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
            | ( Ma = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2486_even__set__bit__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),Ma),A2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
            & ( Ma != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2487_even__flip__bit__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),Ma),A2))
        <=> ~ ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
            <=> ( Ma = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2488_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_diff_iff
tff(fact_2489_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_add_abs_iff
tff(fact_2490_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_abs_add_iff
tff(fact_2491_dvd__power__iff__le,axiom,
    ! [K: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% dvd_power_iff_le
tff(fact_2492_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),Ma)),Ma)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),Ma) ) ).

% take_bit_nat_less_self_iff
tff(fact_2493_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2494_binomial__fact,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(N),K)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),semiring_char_0_fact(B,N)),aa(B,B,aa(B,fun(B,B),times_times(B),semiring_char_0_fact(B,K)),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ) ).

% binomial_fact
tff(fact_2495_fact__binomial,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),semiring_char_0_fact(B,K)),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(N),K))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),semiring_char_0_fact(B,N)),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_binomial
tff(fact_2496_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_2497_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2498_oddE,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ~ ! [B3: B] : A2 != aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B3)),one_one(B)) ) ) ).

% oddE
tff(fact_2499_mod2__eq__if,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = $ite(aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2),zero_zero(B),one_one(B)) ) ).

% mod2_eq_if
tff(fact_2500_parity__cases,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [A2: B] :
          ( ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) != zero_zero(B) ) )
         => ~ ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
             => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) != one_one(B) ) ) ) ) ).

% parity_cases
tff(fact_2501_zero__le__even__power,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat,A2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)) ) ) ).

% zero_le_even_power
tff(fact_2502_zero__le__odd__power,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat,A2: B] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))
          <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_2503_zero__le__power__eq,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2504_minus__one__power__iff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [N: nat] :
          aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),N) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N),one_one(B),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% minus_one_power_iff
tff(fact_2505_power__mono__even,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat,A2: B,B2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N)) ) ) ) ).

% power_mono_even
tff(fact_2506_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% choose_two
tff(fact_2507_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2508_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2509_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2510_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) )
     => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_2511_zero__less__power__eq,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N))
        <=> ( ( N = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
              & ( A2 != zero_zero(B) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2512_take__bit__Suc__minus__1__eq,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,suc,N)),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),one_one(B)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_2513_take__bit__Suc,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,suc,N)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_2514_take__bit__numeral__minus__1__eq,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [K: num] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(num,nat,numeral_numeral(nat),K)),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(B)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_2515_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ).

% take_bit_int_less_eq
tff(fact_2516_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ).

% take_bit_int_greater_eq
tff(fact_2517_even__mask__div__iff_H,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)),one_one(B))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ).

% even_mask_div_iff'
tff(fact_2518_power__le__zero__eq,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),zero_zero(B))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
                & ( A2 = zero_zero(B) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2519_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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,one2)))) ) ).

% even_mod_4_div_2
tff(fact_2520_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2521_even__mask__div__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)),one_one(B))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)))
        <=> ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) = zero_zero(B) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ) ).

% even_mask_div_iff
tff(fact_2522_even__mult__exp__div__exp__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,Ma: nat,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
            | ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) = zero_zero(B) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
              & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2523_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2524_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2525_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_2526_flip__bit__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),zero_zero(nat)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa($o,B,zero_neq_one_of_bool(B),aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_2527_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_2528_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_2529_nat__of__bool,axiom,
    ! [Pa: $o] : aa(int,nat,nat2,aa($o,int,zero_neq_one_of_bool(int),(Pa))) = aa($o,nat,zero_neq_one_of_bool(nat),(Pa)) ).

% nat_of_bool
tff(fact_2530_of__bool__less__eq__iff,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [Pa: $o,Q: $o] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa($o,B,zero_neq_one_of_bool(B),(Pa))),aa($o,B,zero_neq_one_of_bool(B),(Q)))
        <=> ( (Pa)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_2531_of__bool__eq__0__iff,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ! [Pa: $o] :
          ( ( aa($o,B,zero_neq_one_of_bool(B),(Pa)) = zero_zero(B) )
        <=> ~ (Pa) ) ) ).

% of_bool_eq_0_iff
tff(fact_2532_of__bool__eq_I1_J,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ( aa($o,B,zero_neq_one_of_bool(B),$false) = zero_zero(B) ) ) ).

% of_bool_eq(1)
tff(fact_2533_of__bool__eq__1__iff,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ! [Pa: $o] :
          ( ( aa($o,B,zero_neq_one_of_bool(B),(Pa)) = one_one(B) )
        <=> (Pa) ) ) ).

% of_bool_eq_1_iff
tff(fact_2534_of__bool__eq_I2_J,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ( aa($o,B,zero_neq_one_of_bool(B),$true) = one_one(B) ) ) ).

% of_bool_eq(2)
tff(fact_2535_of__nat__of__bool,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Pa: $o] : aa(nat,B,semiring_1_of_nat(B),aa($o,nat,zero_neq_one_of_bool(nat),(Pa))) = aa($o,B,zero_neq_one_of_bool(B),(Pa)) ) ).

% of_nat_of_bool
tff(fact_2536_of__int__of__bool,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Pa: $o] : aa(int,B,ring_1_of_int(B),aa($o,int,zero_neq_one_of_bool(int),(Pa))) = aa($o,B,zero_neq_one_of_bool(B),(Pa)) ) ).

% of_int_of_bool
tff(fact_2537_zero__less__of__bool__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Pa: $o] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa($o,B,zero_neq_one_of_bool(B),(Pa)))
        <=> (Pa) ) ) ).

% zero_less_of_bool_iff
tff(fact_2538_of__bool__less__one__iff,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Pa: $o] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa($o,B,zero_neq_one_of_bool(B),(Pa))),one_one(B))
        <=> ~ (Pa) ) ) ).

% of_bool_less_one_iff
tff(fact_2539_of__bool__not__iff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Pa: $o] : aa($o,B,zero_neq_one_of_bool(B),~ (Pa)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa($o,B,zero_neq_one_of_bool(B),(Pa))) ) ).

% of_bool_not_iff
tff(fact_2540_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa($o,nat,zero_neq_one_of_bool(nat),N != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_2541_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2542_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,N) )
    <=> ( pred_numeral(K) = N ) ) ).

% eq_numeral_Suc
tff(fact_2543_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( N = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2544_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_2545_rec__nat__numeral,axiom,
    ! [B: $tType,A2: B,F: fun(nat,fun(B,B)),V2: num] :
      aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),A2),F),aa(num,nat,numeral_numeral(nat),V2)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(B,B,aa(nat,fun(B,B),F,pv),aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),A2),F),pv)) ) ).

% rec_nat_numeral
tff(fact_2546_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),N) ) ).

% less_numeral_Suc
tff(fact_2547_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_2548_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_2549_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),N) ) ).

% le_numeral_Suc
tff(fact_2550_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_2551_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),N) ).

% diff_numeral_Suc
tff(fact_2552_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2553_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_2554_rec__nat__add__eq__if,axiom,
    ! [B: $tType,A2: B,F: fun(nat,fun(B,B)),V2: num,N: nat] :
      aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),A2),F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(B,B,aa(nat,fun(B,B),F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),N)),aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),A2),F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),N))) ) ).

% rec_nat_add_eq_if
tff(fact_2555_take__bit__of__1,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),one_one(B)) = aa($o,B,zero_neq_one_of_bool(B),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_2556_of__bool__half__eq__0,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [B2: $o] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa($o,B,zero_neq_one_of_bool(B),(B2))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = zero_zero(B) ) ).

% of_bool_half_eq_0
tff(fact_2557_bits__1__div__exp,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa($o,B,zero_neq_one_of_bool(B),N = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_2558_one__div__2__pow__eq,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa($o,B,zero_neq_one_of_bool(B),N = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_2559_take__bit__of__2,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_2560_one__mod__2__pow__eq,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa($o,B,zero_neq_one_of_bool(B),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_2561_dvd__antisym,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),Ma)
       => ( Ma = N ) ) ) ).

% dvd_antisym
tff(fact_2562_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
          & ( A2 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_2563_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( A2 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_2564_gcd__nat_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2) ) ).

% gcd_nat.strict_implies_order
tff(fact_2565_gcd__nat_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_2566_gcd__nat_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
          & ( A2 != B2 ) )
        | ( A2 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_2567_gcd__nat_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_2568_gcd__nat_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_2569_gcd__nat_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
          & ( B2 != C2 ) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_2570_gcd__nat_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
          & ( B2 != C2 ) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_2571_gcd__nat_Oantisym,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
       => ( A2 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_2572_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),A2)
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_2573_gcd__nat_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2) ) ) ).

% gcd_nat.eq_iff
tff(fact_2574_gcd__nat_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2) ) ) ).

% gcd_nat.trans
tff(fact_2575_gcd__nat_Orefl,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),A2) ).

% gcd_nat.refl
tff(fact_2576_gcd__nat_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
          & ( B2 != A2 ) ) ) ).

% gcd_nat.asym
tff(fact_2577_subset__CollectI,axiom,
    ! [B: $tType,B4: set(B),A4: set(B),Q: fun(B,$o),Pa: fun(B,$o)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),B4)
           => ( aa(B,$o,Q,X2)
             => aa(B,$o,Pa,X2) ) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),B4),Q))),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa))) ) ) ).

% subset_CollectI
tff(fact_2578_subset__Collect__iff,axiom,
    ! [B: $tType,B4: set(B),A4: set(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa)))
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),B4)
           => aa(B,$o,Pa,X3) ) ) ) ).

% subset_Collect_iff
tff(fact_2579_zero__less__eq__of__bool,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Pa: $o] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa($o,B,zero_neq_one_of_bool(B),(Pa))) ) ).

% zero_less_eq_of_bool
tff(fact_2580_of__bool__less__eq__one,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Pa: $o] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa($o,B,zero_neq_one_of_bool(B),(Pa))),one_one(B)) ) ).

% of_bool_less_eq_one
tff(fact_2581_of__bool__def,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ! [P2: $o] :
          aa($o,B,zero_neq_one_of_bool(B),(P2)) = $ite((P2),one_one(B),zero_zero(B)) ) ).

% of_bool_def
tff(fact_2582_split__of__bool,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ! [Pa: fun(B,$o),P2: $o] :
          ( aa(B,$o,Pa,aa($o,B,zero_neq_one_of_bool(B),(P2)))
        <=> ( ( (P2)
             => aa(B,$o,Pa,one_one(B)) )
            & ( ~ (P2)
             => aa(B,$o,Pa,zero_zero(B)) ) ) ) ) ).

% split_of_bool
tff(fact_2583_split__of__bool__asm,axiom,
    ! [B: $tType] :
      ( zero_neq_one(B)
     => ! [Pa: fun(B,$o),P2: $o] :
          ( aa(B,$o,Pa,aa($o,B,zero_neq_one_of_bool(B),(P2)))
        <=> ~ ( ( (P2)
                & ~ aa(B,$o,Pa,one_one(B)) )
              | ( ~ (P2)
                & ~ aa(B,$o,Pa,zero_zero(B)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_2584_numeral__eq__Suc,axiom,
    ! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_2585_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2586_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_2587_bits__induct,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Pa: fun(B,$o),A2: B] :
          ( ! [A3: B] :
              ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = A3 )
             => aa(B,$o,Pa,A3) )
         => ( ! [A3: B,B3: $o] :
                ( aa(B,$o,Pa,A3)
               => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa($o,B,zero_neq_one_of_bool(B),(B3))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A3))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) = A3 )
                 => aa(B,$o,Pa,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa($o,B,zero_neq_one_of_bool(B),(B3))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A3))) ) )
           => aa(B,$o,Pa,A2) ) ) ) ).

% bits_induct
tff(fact_2588_exp__div__exp__eq,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat] :
          aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa(B,B,
            aa(B,fun(B,B),times_times(B),
              aa($o,B,zero_neq_one_of_bool(B),
                ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma) != zero_zero(B) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma) ))),
            aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N))) ) ).

% exp_div_exp_eq
tff(fact_2589_take__bit__numeral__bit1,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [L: num,K: num] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(num,nat,numeral_numeral(nat),L)),aa(num,B,numeral_numeral(B),aa(num,num,bit1,K))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),pred_numeral(L)),aa(num,B,numeral_numeral(B),K))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),one_one(B)) ) ).

% take_bit_numeral_bit1
tff(fact_2590_set__decode__0,axiom,
    ! [X: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),nat_set_decode(X))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X) ) ).

% set_decode_0
tff(fact_2591_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bb(nat,fun(nat,$o),X)) ).

% set_decode_def
tff(fact_2592_mask__numeral,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: num] : aa(nat,B,bit_se2239418461657761734s_mask(B),aa(num,nat,numeral_numeral(nat),N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,B,bit_se2239418461657761734s_mask(B),pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_2593_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% and_int_unfold
tff(fact_2594_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = aa(nat,int,bit_se2239418461657761734s_mask(int),N) )
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2595_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = $ite(
        L = zero_zero(int),
        K,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_2596_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ).

% mask_nat_positive_iff
tff(fact_2597_bit_Oconj__zero__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),X),zero_zero(B)) = zero_zero(B) ) ).

% bit.conj_zero_right
tff(fact_2598_bit_Oconj__zero__left,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),zero_zero(B)),X) = zero_zero(B) ) ).

% bit.conj_zero_left
tff(fact_2599_zero__and__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),zero_zero(B)),A2) = zero_zero(B) ) ).

% zero_and_eq
tff(fact_2600_and__zero__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),zero_zero(B)) = zero_zero(B) ) ).

% and_zero_eq
tff(fact_2601_sgn__0,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ( aa(B,B,sgn_sgn(B),zero_zero(B)) = zero_zero(B) ) ) ).

% sgn_0
tff(fact_2602_sgn__1,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ( aa(B,B,sgn_sgn(B),one_one(B)) = one_one(B) ) ) ).

% sgn_1
tff(fact_2603_power__int__sgn,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,N: int] : aa(B,B,sgn_sgn(B),power_int(B,A2,N)) = power_int(B,aa(B,B,sgn_sgn(B),A2),N) ) ).

% power_int_sgn
tff(fact_2604_sgn__greater,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,sgn_sgn(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% sgn_greater
tff(fact_2605_sgn__less,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,sgn_sgn(B),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% sgn_less
tff(fact_2606_and_Oleft__neutral,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(B,B,uminus_uminus(B),one_one(B))),A2) = A2 ) ).

% and.left_neutral
tff(fact_2607_and_Oright__neutral,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),aa(B,B,uminus_uminus(B),one_one(B))) = A2 ) ).

% and.right_neutral
tff(fact_2608_bit_Oconj__one__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),X),aa(B,B,uminus_uminus(B),one_one(B))) = X ) ).

% bit.conj_one_right
tff(fact_2609_mask__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ( aa(nat,B,bit_se2239418461657761734s_mask(B),zero_zero(nat)) = zero_zero(B) ) ) ).

% mask_0
tff(fact_2610_mask__eq__0__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] :
          ( ( aa(nat,B,bit_se2239418461657761734s_mask(B),N) = zero_zero(B) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_2611_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_2612_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_2613_sgn__pos,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,B,sgn_sgn(B),A2) = one_one(B) ) ) ) ).

% sgn_pos
tff(fact_2614_and__numerals_I2_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = one_one(B) ) ).

% and_numerals(2)
tff(fact_2615_and__numerals_I8_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),one_one(B)) = one_one(B) ) ).

% and_numerals(8)
tff(fact_2616_abs__sgn__eq__1,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,abs_abs(B),aa(B,B,sgn_sgn(B),A2)) = one_one(B) ) ) ) ).

% abs_sgn_eq_1
tff(fact_2617_mask__Suc__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ( aa(nat,B,bit_se2239418461657761734s_mask(B),aa(nat,nat,suc,zero_zero(nat))) = one_one(B) ) ) ).

% mask_Suc_0
tff(fact_2618_sgn__mult__self__eq,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,sgn_sgn(B),A2)),aa(B,B,sgn_sgn(B),A2)) = aa($o,B,zero_neq_one_of_bool(B),A2 != zero_zero(B)) ) ).

% sgn_mult_self_eq
tff(fact_2619_take__bit__minus__one__eq__mask,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),aa(B,B,uminus_uminus(B),one_one(B))) = aa(nat,B,bit_se2239418461657761734s_mask(B),N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_2620_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ! [A2: B] : aa(B,B,sgn_sgn(B),aa(B,B,abs_abs(B),A2)) = aa($o,B,zero_neq_one_of_bool(B),A2 != zero_zero(B)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_2621_sgn__abs,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ! [A2: B] : aa(B,B,abs_abs(B),aa(B,B,sgn_sgn(B),A2)) = aa($o,B,zero_neq_one_of_bool(B),A2 != zero_zero(B)) ) ).

% sgn_abs
tff(fact_2622_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_2623_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_2624_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_2625_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_2626_and__numerals_I5_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,X))),one_one(B)) = zero_zero(B) ) ).

% and_numerals(5)
tff(fact_2627_and__numerals_I1_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Y))) = zero_zero(B) ) ).

% and_numerals(1)
tff(fact_2628_sgn__neg,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,B,sgn_sgn(B),A2) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ) ).

% sgn_neg
tff(fact_2629_sgn__of__nat,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: nat] : aa(B,B,sgn_sgn(B),aa(nat,B,semiring_1_of_nat(B),N)) = aa($o,B,zero_neq_one_of_bool(B),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_2630_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_2631_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_2632_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_2633_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_2634_and__numerals_I7_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num,Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(num,B,numeral_numeral(B),X)),aa(num,B,numeral_numeral(B),Y)))) ) ).

% and_numerals(7)
tff(fact_2635_of__nat__mask__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)) = aa(nat,B,bit_se2239418461657761734s_mask(B),N) ) ).

% of_nat_mask_eq
tff(fact_2636_of__nat__and__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_and_eq
tff(fact_2637_sgn__0__0,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( ( aa(B,B,sgn_sgn(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% sgn_0_0
tff(fact_2638_sgn__eq__0__iff,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ! [A2: B] :
          ( ( aa(B,B,sgn_sgn(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% sgn_eq_0_iff
tff(fact_2639_same__sgn__sgn__add,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [B2: B,A2: B] :
          ( ( aa(B,B,sgn_sgn(B),B2) = aa(B,B,sgn_sgn(B),A2) )
         => ( aa(B,B,sgn_sgn(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,sgn_sgn(B),A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_2640_less__eq__mask,axiom,
    ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)) ).

% less_eq_mask
tff(fact_2641_subset__decode__imp__le,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(Ma)),nat_set_decode(N))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% subset_decode_imp_le
tff(fact_2642_and__eq__minus__1__iff,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),B2) = aa(B,B,uminus_uminus(B),one_one(B)) )
        <=> ( ( A2 = aa(B,B,uminus_uminus(B),one_one(B)) )
            & ( B2 = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_2643_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_2644_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_2645_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y) ) ).

% AND_upper2
tff(fact_2646_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X) ) ).

% AND_upper1
tff(fact_2647_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)) ) ).

% AND_lower
tff(fact_2648_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_sgnE
tff(fact_2649_sgn__not__eq__imp,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [B2: B,A2: B] :
          ( ( aa(B,B,sgn_sgn(B),B2) != aa(B,B,sgn_sgn(B),A2) )
         => ( ( aa(B,B,sgn_sgn(B),A2) != zero_zero(B) )
           => ( ( aa(B,B,sgn_sgn(B),B2) != zero_zero(B) )
             => ( aa(B,B,sgn_sgn(B),A2) = aa(B,B,uminus_uminus(B),aa(B,B,sgn_sgn(B),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_2650_sgn__minus__1,axiom,
    ! [B: $tType] :
      ( idom_abs_sgn(B)
     => ( aa(B,B,sgn_sgn(B),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% sgn_minus_1
tff(fact_2651_same__sgn__abs__add,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [B2: B,A2: B] :
          ( ( aa(B,B,sgn_sgn(B),B2) = aa(B,B,sgn_sgn(B),A2) )
         => ( aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,abs_abs(B),A2)),aa(B,B,abs_abs(B),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_2652_mask__nonnegative__int,axiom,
    ! [N: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(nat,int,bit_se2239418461657761734s_mask(int),N)) ).

% mask_nonnegative_int
tff(fact_2653_not__mask__negative__int,axiom,
    ! [N: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,bit_se2239418461657761734s_mask(int),N)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_2654_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_2655_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_2656_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_2657_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
       => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_2658_sgn__1__pos,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( ( aa(B,B,sgn_sgn(B),A2) = one_one(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% sgn_1_pos
tff(fact_2659_abs__sgn__eq,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          aa(B,B,abs_abs(B),aa(B,B,sgn_sgn(B),A2)) = $ite(A2 = zero_zero(B),zero_zero(B),one_one(B)) ) ).

% abs_sgn_eq
tff(fact_2660_less__mask,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)) ) ).

% less_mask
tff(fact_2661_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_2662_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_2663_sgn__if,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B] :
          aa(B,B,sgn_sgn(B),X) = $ite(
            X = zero_zero(B),
            zero_zero(B),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X),one_one(B),aa(B,B,uminus_uminus(B),one_one(B))) ) ) ).

% sgn_if
tff(fact_2664_sgn__1__neg,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B] :
          ( ( aa(B,B,sgn_sgn(B),A2) = aa(B,B,uminus_uminus(B),one_one(B)) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% sgn_1_neg
tff(fact_2665_one__and__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),one_one(B)),A2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% one_and_eq
tff(fact_2666_and__one__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),one_one(B)) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% and_one_eq
tff(fact_2667_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = aa(nat,int,bit_se2239418461657761734s_mask(int),N) )
    <=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_2668_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_2669_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,B,bit_se2239418461657761734s_mask(B),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_2670_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% and_int_rec
tff(fact_2671_mask__half__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,bit_se2239418461657761734s_mask(int),N)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(nat,int,bit_se2239418461657761734s_mask(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% mask_half_int
tff(fact_2672_mask__nat__def,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2239418461657761734s_mask(nat),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_2673_mask__int__def,axiom,
    ! [N: nat] : aa(nat,int,bit_se2239418461657761734s_mask(int),N) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) ).

% mask_int_def
tff(fact_2674_mask__eq__exp__minus__1,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(nat,B,bit_se2239418461657761734s_mask(B),N) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)),one_one(B)) ) ).

% mask_eq_exp_minus_1
tff(fact_2675_divide__int__unfold,axiom,
    ! [K: int,Ma: nat,L: int,N: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Ma))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( N = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),Ma)))))) ) ).

% divide_int_unfold
tff(fact_2676_modulo__int__unfold,axiom,
    ! [K: int,Ma: nat,L: int,N: nat] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Ma))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( N = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Ma)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),Ma))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),N))))) ) ).

% modulo_int_unfold
tff(fact_2677_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))))) ) ).

% divide_int_def
tff(fact_2678_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,N)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ).

% take_bit_Suc_from_most
tff(fact_2679_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% Suc_0_xor_eq
tff(fact_2680_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% xor_Suc_0_eq
tff(fact_2681_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(list($o),int,aa(int,fun(list($o),int),aa(fun($o,int),fun(int,fun(list($o),int)),groups4207007520872428315er_sum($o,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_2682_push__bit__numeral__minus__1,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: num] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),aa(num,nat,numeral_numeral(nat),N)),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_2683_xor__numerals_I6_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num,Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Y))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),X)),aa(num,B,numeral_numeral(B),Y)))) ) ).

% xor_numerals(6)
tff(fact_2684_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_2685_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_2686_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_2687_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_2688_bit_Oxor__self,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),X),X) = zero_zero(B) ) ).

% bit.xor_self
tff(fact_2689_xor__self__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),A2),A2) = zero_zero(B) ) ).

% xor_self_eq
tff(fact_2690_xor_Oleft__neutral,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),zero_zero(B)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_2691_xor_Oright__neutral,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),A2),zero_zero(B)) = A2 ) ).

% xor.right_neutral
tff(fact_2692_push__bit__of__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),zero_zero(B)) = zero_zero(B) ) ).

% push_bit_of_0
tff(fact_2693_push__bit__eq__0__iff,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat,A2: B] :
          ( ( aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_2694_push__bit__push__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),A2)) = aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),A2) ) ).

% push_bit_push_bit
tff(fact_2695_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : bit_concat_bit(N,zero_zero(int),L) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L) ).

% concat_bit_of_zero_1
tff(fact_2696_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_2697_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% signed_take_bit_negative_iff
tff(fact_2698_xor__numerals_I8_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,X)) ) ).

% xor_numerals(8)
tff(fact_2699_xor__numerals_I5_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,X))),one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_2700_xor__numerals_I2_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_2701_xor__numerals_I1_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Y))) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_2702_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2703_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_2704_bit__0,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),zero_zero(nat))
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2) ) ) ).

% bit_0
tff(fact_2705_even__push__bit__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),A2))
        <=> ( ( N != zero_zero(nat) )
            | aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_2706_push__bit__of__1,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),one_one(B)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) ) ).

% push_bit_of_1
tff(fact_2707_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% push_bit_of_Suc_0
tff(fact_2708_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_2709_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_2710_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_2711_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_2712_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_2713_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_2714_bit__mod__2__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),N)
        <=> ( ( N = zero_zero(nat) )
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_2715_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% and_Suc_0_eq
tff(fact_2716_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Suc_0_and_eq
tff(fact_2717_xor__numerals_I4_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num,Y: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,X))),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),X)),aa(num,B,numeral_numeral(B),Y)))) ) ).

% xor_numerals(4)
tff(fact_2718_flip__bit__nat__def,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),Ma),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Ma),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_2719_of__nat__xor__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_xor_eq
tff(fact_2720_of__nat__push__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Ma),N)) = aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Ma),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_push_bit
tff(fact_2721_push__bit__of__nat,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,Ma: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),aa(nat,B,semiring_1_of_nat(B),Ma)) = aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),Ma)) ) ).

% push_bit_of_nat
tff(fact_2722_bit__of__nat__iff__bit,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(nat,B,semiring_1_of_nat(B),Ma)),N)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Ma),N) ) ) ).

% bit_of_nat_iff_bit
tff(fact_2723_xor__nat__def,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% xor_nat_def
tff(fact_2724_bit__push__bit__iff__int,axiom,
    ! [Ma: nat,K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Ma),K)),N)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) ) ) ).

% bit_push_bit_iff_int
tff(fact_2725_bit__disjunctive__add__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,B2: B,N: nat] :
          ( ! [N2: nat] :
              ( ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N2)
              | ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),B2),N2) )
         => ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),N)
          <=> ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N)
              | aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),B2),N) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_2726_flip__bit__eq__xor,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se8732182000553998342ip_bit(B),N),A2) = aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),A2),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),one_one(B))) ) ).

% flip_bit_eq_xor
tff(fact_2727_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).

% flip_bit_int_def
tff(fact_2728_push__bit__add,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B,B2: B] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),A2)),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),B2)) ) ).

% push_bit_add
tff(fact_2729_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N)
        <=> ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),one_one(B))) != zero_zero(B) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_2730_bit__1__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),one_one(B)),N)
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_2731_not__bit__1__Suc,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),one_one(B)),aa(nat,nat,suc,N)) ) ).

% not_bit_1_Suc
tff(fact_2732_bit__numeral__simps_I1_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: num] : ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),one_one(B)),aa(num,nat,numeral_numeral(nat),N)) ) ).

% bit_numeral_simps(1)
tff(fact_2733_bit__of__bool__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [B2: $o,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa($o,B,zero_neq_one_of_bool(B),(B2))),N)
        <=> ( (B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_2734_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)) ) ) ).

% XOR_lower
tff(fact_2735_push__bit__take__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2)) = aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Ma),A2)) ) ).

% push_bit_take_bit
tff(fact_2736_bit__horner__sum__bit__iff,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Bs: list($o),N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(list($o),B,aa(B,fun(list($o),B),aa(fun($o,B),fun(B,fun(list($o),B)),groups4207007520872428315er_sum($o,B),zero_neq_one_of_bool(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Bs)),N)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),N) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_2737_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int))),N)
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% bit_not_int_iff'
tff(fact_2738_and__nat__def,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% and_nat_def
tff(fact_2739_signed__take__bit__code,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat,A2: B] :
          aa(B,B,bit_ri4674362597316999326ke_bit(B,N),A2) = $let(
            l: B,
            l:= aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,suc,N)),A2),
            $ite(aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),l),N),aa(B,B,aa(B,fun(B,B),plus_plus(B),l),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),aa(nat,nat,suc,N)),aa(B,B,uminus_uminus(B),one_one(B)))),l) ) ) ).

% signed_take_bit_code
tff(fact_2740_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : bit_concat_bit(N,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L)) ).

% concat_bit_eq
tff(fact_2741_bit__imp__take__bit__positive,axiom,
    ! [N: nat,Ma: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Ma),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_2742_bit__concat__bit__iff,axiom,
    ! [Ma: nat,K: int,L: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),bit_concat_bit(Ma,K,L)),N)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_2743_exp__eq__0__imp__not__bit,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [N: nat,A2: B] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) = zero_zero(B) )
         => ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_2744_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M5)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),M5)
              <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N2) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)))
              <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N2) ) ) ) ).

% int_bit_bound
tff(fact_2745_and__exp__eq__0__iff__not__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,N: nat] :
          ( ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = zero_zero(B) )
        <=> ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_2746_push__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% push_bit_minus_one
tff(fact_2747_even__bit__succ__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),A2)),N)
          <=> ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N)
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_2748_odd__bit__iff__bit__pred,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N)
          <=> ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),N)
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_2749_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ) ) ).

% XOR_upper
tff(fact_2750_bit__sum__mult__2__cases,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,B2: B,N: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2))),N)
          <=> $ite(N = zero_zero(nat),~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_2751_bit__rec,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N)
        <=> $ite(N = zero_zero(nat),~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_2752_and__nat__unfold,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),N) = $ite(
        ( ( Ma = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% and_nat_unfold
tff(fact_2753_and__nat__rec,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Ma),N) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% and_nat_rec
tff(fact_2754_xor__nat__unfold,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),N) = $ite(
        Ma = zero_zero(nat),
        N,
        $ite(N = zero_zero(nat),Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% xor_nat_unfold
tff(fact_2755_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K) != ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% xor_int_rec
tff(fact_2756_xor__nat__rec,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Ma),N) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma) != ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% xor_nat_rec
tff(fact_2757_xor__one__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),A2),one_one(B)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa($o,B,zero_neq_one_of_bool(B),aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)))),aa($o,B,zero_neq_one_of_bool(B),~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2))) ) ).

% xor_one_eq
tff(fact_2758_one__xor__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),one_one(B)),A2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa($o,B,zero_neq_one_of_bool(B),aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)))),aa($o,B,zero_neq_one_of_bool(B),~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2))) ) ).

% one_xor_eq
tff(fact_2759_set__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% set_bit_eq
tff(fact_2760_horner__sum__simps_I2_J,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_0(B)
     => ! [F: fun(C,B),A2: B,X: C,Xs: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(C,B,F,X)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_2761_horner__sum__simps_I1_J,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_semiring_0(B)
     => ! [F: fun(C,B),A2: B] : aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),nil(C)) = zero_zero(B) ) ).

% horner_sum_simps(1)
tff(fact_2762_horner__sum__foldr,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_semiring_0(B)
     => ! [F: fun(C,B),A2: B,Xs: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),Xs) = aa(B,B,foldr(C,B,aa(B,fun(C,fun(B,B)),aTP_Lamp_bc(fun(C,B),fun(B,fun(C,fun(B,B))),F),A2),Xs),zero_zero(B)) ) ).

% horner_sum_foldr
tff(fact_2763_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_2764_length__subseqs,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(list(B)),nat,size_size(list(list(B))),subseqs(B,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_subseqs
tff(fact_2765_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_2766_Ints__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ring_1(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(set(C),$o,member(C,aa(B,C,F,X2)),ring_1_Ints(C)) )
         => aa(set(C),$o,member(C,aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),ring_1_Ints(C)) ) ) ).

% Ints_sum
tff(fact_2767_of__int__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ring_1(B)
     => ! [F: fun(C,int),A4: set(C)] : aa(int,B,ring_1_of_int(B),aa(set(C),int,aa(fun(C,int),fun(set(C),int),groups7311177749621191930dd_sum(C,int),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bd(fun(C,int),fun(C,B),F)),A4) ) ).

% of_int_sum
tff(fact_2768_bit_Oconj__cancel__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),X),aa(B,B,bit_ri4277139882892585799ns_not(B),X)) = zero_zero(B) ) ).

% bit.conj_cancel_right
tff(fact_2769_bit_Oconj__cancel__left,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(B,B,bit_ri4277139882892585799ns_not(B),X)),X) = zero_zero(B) ) ).

% bit.conj_cancel_left
tff(fact_2770_of__nat__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_1(B)
     => ! [F: fun(C,nat),A4: set(C)] : aa(nat,B,semiring_1_of_nat(B),aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7311177749621191930dd_sum(C,nat),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_be(fun(C,nat),fun(C,B),F)),A4) ) ).

% of_nat_sum
tff(fact_2771_bit_Ocompl__zero,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ( aa(B,B,bit_ri4277139882892585799ns_not(B),zero_zero(B)) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% bit.compl_zero
tff(fact_2772_bit_Ocompl__one,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ( aa(B,B,bit_ri4277139882892585799ns_not(B),aa(B,B,uminus_uminus(B),one_one(B))) = zero_zero(B) ) ) ).

% bit.compl_one
tff(fact_2773_bit_Oxor__one__left,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(B,B,uminus_uminus(B),one_one(B))),X) = aa(B,B,bit_ri4277139882892585799ns_not(B),X) ) ).

% bit.xor_one_left
tff(fact_2774_bit_Oxor__one__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),X),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,bit_ri4277139882892585799ns_not(B),X) ) ).

% bit.xor_one_right
tff(fact_2775_bit_Oxor__cancel__left,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(B,B,bit_ri4277139882892585799ns_not(B),X)),X) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% bit.xor_cancel_left
tff(fact_2776_bit_Oxor__cancel__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),X),aa(B,B,bit_ri4277139882892585799ns_not(B),X)) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% bit.xor_cancel_right
tff(fact_2777_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_2778_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_2779_push__bit__minus__one__eq__not__mask,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,bit_ri4277139882892585799ns_not(B),aa(nat,B,bit_se2239418461657761734s_mask(B),N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_2780_not__one__eq,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ( aa(B,B,bit_ri4277139882892585799ns_not(B),one_one(B)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ) ).

% not_one_eq
tff(fact_2781_mod__sum__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [F: fun(C,B),A2: B,A4: set(C)] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(B,fun(C,B),aTP_Lamp_bf(fun(C,B),fun(B,fun(C,B)),F),A2)),A4)),A2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)),A2) ) ).

% mod_sum_eq
tff(fact_2782_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),N)
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_2783_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N)) ).

% not_bit_Suc_0_Suc
tff(fact_2784_bit__push__bit__iff__nat,axiom,
    ! [Ma: nat,Q2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Ma),Q2)),N)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
        & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_2785_not__diff__distrib,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B,B2: B] : aa(B,B,bit_ri4277139882892585799ns_not(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,bit_ri4277139882892585799ns_not(B),A2)),B2) ) ).

% not_diff_distrib
tff(fact_2786_not__add__distrib,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B,B2: B] : aa(B,B,bit_ri4277139882892585799ns_not(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,bit_ri4277139882892585799ns_not(B),A2)),B2) ) ).

% not_add_distrib
tff(fact_2787_Misc_Ofoldr__Cons,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),list(B),foldr(B,list(B),cons(B),Xs),nil(B)) = Xs ).

% Misc.foldr_Cons
tff(fact_2788_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N)) ).

% not_bit_Suc_0_numeral
tff(fact_2789_minus__eq__not__plus__1,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B] : aa(B,B,uminus_uminus(B),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,bit_ri4277139882892585799ns_not(B),A2)),one_one(B)) ) ).

% minus_eq_not_plus_1
tff(fact_2790_not__eq__complement,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B] : aa(B,B,bit_ri4277139882892585799ns_not(B),A2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),A2)),one_one(B)) ) ).

% not_eq_complement
tff(fact_2791_minus__eq__not__minus__1,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B] : aa(B,B,uminus_uminus(B),A2) = aa(B,B,bit_ri4277139882892585799ns_not(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))) ) ).

% minus_eq_not_minus_1
tff(fact_2792_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_2793_and__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).

% and_not_numerals(1)
tff(fact_2794_sum__power__add,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,Ma: nat,I5: set(nat)] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_bg(B,fun(nat,fun(nat,B)),X),Ma)),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),I5)) ) ).

% sum_power_add
tff(fact_2795_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int)))) ).

% unset_bit_int_def
tff(fact_2796_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_2797_and__not__numerals_I4_J,axiom,
    ! [Ma: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,Ma)) ).

% and_not_numerals(4)
tff(fact_2798_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(int,nat,nat2,K)),N)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ) ).

% bit_nat_iff
tff(fact_2799_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),K)),N)
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),N) ) ).

% bit_minus_int_iff
tff(fact_2800_take__bit__not__mask__eq__0,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),aa(B,B,bit_ri4277139882892585799ns_not(B),aa(nat,B,bit_se2239418461657761734s_mask(B),N))) = zero_zero(B) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_2801_push__bit__mask__eq,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [Ma: nat,N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Ma),aa(nat,B,bit_se2239418461657761734s_mask(B),N)) = aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(nat,B,bit_se2239418461657761734s_mask(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma))),aa(B,B,bit_ri4277139882892585799ns_not(B),aa(nat,B,bit_se2239418461657761734s_mask(B),Ma))) ) ).

% push_bit_mask_eq
tff(fact_2802_unset__bit__eq__and__not,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2638667681897837118et_bit(B),N),A2) = aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),aa(B,B,bit_ri4277139882892585799ns_not(B),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),one_one(B)))) ) ).

% unset_bit_eq_and_not
tff(fact_2803_and__not__numerals_I7_J,axiom,
    ! [Ma: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,Ma)) ).

% and_not_numerals(7)
tff(fact_2804_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_2805_bit__not__iff__eq,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,bit_ri4277139882892585799ns_not(B),A2)),N)
        <=> ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) != zero_zero(B) )
            & ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N) ) ) ) ).

% bit_not_iff_eq
tff(fact_2806_mask__eq__sum__exp,axiom,
    ! [B: $tType] :
      ( semiring_parity(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)),one_one(B)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N))) ) ).

% mask_eq_sum_exp
tff(fact_2807_and__not__numerals_I8_J,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,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,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_2808_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% not_int_rec
tff(fact_2809_sum__abs__ge__zero,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F: fun(C,B),A4: set(C)] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bh(fun(C,B),fun(C,B),F)),A4)) ) ).

% sum_abs_ge_zero
tff(fact_2810_sum__abs,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F: fun(C,B),A4: set(C)] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bh(fun(C,B),fun(C,B),F)),A4)) ) ).

% sum_abs
tff(fact_2811_convex__sum__bound__le,axiom,
    ! [B: $tType,C: $tType] :
      ( linordered_idom(C)
     => ! [I5: set(B),X: fun(B,C),A2: fun(B,C),B2: C,Delta: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),I5)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,X,I2)) )
         => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),X),I5) = one_one(C) )
           => ( ! [I2: B] :
                  ( aa(set(B),$o,member(B,I2),I5)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(C,C,abs_abs(C),aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(B,C,A2,I2)),B2))),Delta) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(C,C,abs_abs(C),aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aTP_Lamp_bi(fun(B,C),fun(fun(B,C),fun(B,C)),X),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2812_abs__sum__abs,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F: fun(C,B),A4: set(C)] : aa(B,B,abs_abs(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bh(fun(C,B),fun(C,B),F)),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bh(fun(C,B),fun(C,B),F)),A4) ) ).

% abs_sum_abs
tff(fact_2813_sum_Oneutral__const,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bj(C,B)),A4) = zero_zero(B) ) ).

% sum.neutral_const
tff(fact_2814_sum__SucD,axiom,
    ! [B: $tType,F: fun(B,nat),A4: set(B),N: nat] :
      ( ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4) = aa(nat,nat,suc,N) )
     => ? [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,F,X2)) ) ) ).

% sum_SucD
tff(fact_2815_int__sum,axiom,
    ! [B: $tType,F: fun(B,nat),A4: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),aTP_Lamp_bk(fun(B,nat),fun(B,int),F)),A4) ).

% int_sum
tff(fact_2816_sum_Oswap,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,fun(D,B)),B4: set(D),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(set(D),fun(C,B),aTP_Lamp_bl(fun(C,fun(D,B)),fun(set(D),fun(C,B)),G),B4)),A4) = aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7311177749621191930dd_sum(D,B),aa(set(C),fun(D,B),aTP_Lamp_bn(fun(C,fun(D,B)),fun(set(C),fun(D,B)),G),A4)),B4) ) ).

% sum.swap
tff(fact_2817_sum_Oneutral,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => ( aa(B,C,G,X2) = zero_zero(C) ) )
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = zero_zero(C) ) ) ) ).

% sum.neutral
tff(fact_2818_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B),A4: set(C)] :
          ( ( aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4) != zero_zero(B) )
         => ~ ! [A3: C] :
                ( aa(set(C),$o,member(C,A3),A4)
               => ( aa(C,B,G,A3) = zero_zero(B) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_2819_sum__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [K3: set(B),F: fun(B,C),G: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),K3)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),aa(B,C,G,I2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),K3)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),K3)) ) ) ).

% sum_mono
tff(fact_2820_sum_Odistrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B),H: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,B),fun(C,B),aTP_Lamp_bo(fun(C,B),fun(fun(C,B),fun(C,B)),G),H)),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),H),A4)) ) ).

% sum.distrib
tff(fact_2821_sum__distrib__left,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [R: B,F: fun(C,B),A4: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),R),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,B),fun(C,B),aTP_Lamp_bp(B,fun(fun(C,B),fun(C,B)),R),F)),A4) ) ).

% sum_distrib_left
tff(fact_2822_sum__distrib__right,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F: fun(C,B),A4: set(C),R: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)),R) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(B,fun(C,B),aTP_Lamp_bq(fun(C,B),fun(B,fun(C,B)),F),R)),A4) ) ).

% sum_distrib_right
tff(fact_2823_sum__product,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F: fun(C,B),A4: set(C),G: fun(D,B),B4: set(D)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)),aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7311177749621191930dd_sum(D,B),G),B4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(set(D),fun(C,B),aa(fun(D,B),fun(set(D),fun(C,B)),aTP_Lamp_bs(fun(C,B),fun(fun(D,B),fun(set(D),fun(C,B))),F),G),B4)),A4) ) ).

% sum_product
tff(fact_2824_sum__subtractf,axiom,
    ! [B: $tType,C: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(C,B),G: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,B),fun(C,B),aTP_Lamp_bt(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),A4) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4)) ) ).

% sum_subtractf
tff(fact_2825_sum__negf,axiom,
    ! [B: $tType,C: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_bu(fun(C,B),fun(C,B),F)),A4) = aa(B,B,uminus_uminus(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)) ) ).

% sum_negf
tff(fact_2826_sum__divide__distrib,axiom,
    ! [B: $tType,C: $tType] :
      ( field(B)
     => ! [F: fun(C,B),A4: set(C),R: B] : aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4)),R) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(B,fun(C,B),aTP_Lamp_bv(fun(C,B),fun(B,fun(C,B)),F),R)),A4) ) ).

% sum_divide_distrib
tff(fact_2827_sum__nonneg,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,X2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)) ) ) ).

% sum_nonneg
tff(fact_2828_sum__nonpos,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),zero_zero(C)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),zero_zero(C)) ) ) ).

% sum_nonpos
tff(fact_2829_sum__cong__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(nat),F: fun(nat,B),G: fun(nat,B)] :
          ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A4)
         => ( ! [X2: nat] :
                ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,X2)),A4)
               => ( aa(nat,B,F,aa(nat,nat,suc,X2)) = aa(nat,B,G,aa(nat,nat,suc,X2)) ) )
           => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),A4) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),A4) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2830_sum__subtractf__nat,axiom,
    ! [B: $tType,A4: set(B),G: fun(B,nat),F: fun(B,nat)] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,G,X2)),aa(B,nat,F,X2)) )
     => ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(fun(B,nat),fun(B,nat),aTP_Lamp_bw(fun(B,nat),fun(fun(B,nat),fun(B,nat)),G),F)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_2831_choose__odd__sum,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_bx(nat,fun(nat,B),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_2832_choose__even__sum,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_by(nat,fun(nat,B),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_2833_gbinomial__partial__row__sum,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_bz(B,fun(nat,B),A2)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),Ma)),one_one(B))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(nat,B,gbinomial(B,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_2834_sum__gp,axiom,
    ! [B: $tType] :
      ( ( division_ring(B)
        & comm_ring(B) )
     => ! [X: B,Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),set_or1337092689740270186AtMost(nat,Ma,N)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma),
            zero_zero(B),
            $ite(X = one_one(B),aa(nat,B,semiring_1_of_nat(B),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))),Ma)),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,suc,N)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X))) ) ) ).

% sum_gp
tff(fact_2835_gbinomial__r__part__sum,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,B,semiring_1_of_nat(B),Ma))),one_one(B)))),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma)) ) ).

% gbinomial_r_part_sum
tff(fact_2836_gchoose__row__sum__weighted,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [R: B,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_bz(B,fun(nat,B),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Ma)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,Ma))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(nat,B,gbinomial(B,R),aa(nat,nat,suc,Ma))) ) ).

% gchoose_row_sum_weighted
tff(fact_2837_atMost__eq__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,set(B),set_ord_atMost(B),X) = aa(B,set(B),set_ord_atMost(B),Y) )
        <=> ( X = Y ) ) ) ).

% atMost_eq_iff
tff(fact_2838_atLeastAtMost__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,L: B,U: B] :
          ( aa(set(B),$o,member(B,I),set_or1337092689740270186AtMost(B,L,U))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),I)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_2839_Icc__eq__Icc,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,H: B,L3: B,H4: B] :
          ( ( set_or1337092689740270186AtMost(B,L,H) = set_or1337092689740270186AtMost(B,L3,H4) )
        <=> ( ( ( L = L3 )
              & ( H = H4 ) )
            | ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),H)
              & ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L3),H4) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_2840_atMost__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,K: B] :
          ( aa(set(B),$o,member(B,I),aa(B,set(B),set_ord_atMost(B),K))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),K) ) ) ).

% atMost_iff
tff(fact_2841_atLeastatMost__subset__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),set_or1337092689740270186AtMost(B,C2,D2))
        <=> ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_2842_atMost__subset__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atMost(B),X)),aa(B,set(B),set_ord_atMost(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% atMost_subset_iff
tff(fact_2843_Icc__subset__Iic__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [L: B,H: B,H4: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or1337092689740270186AtMost(B,L,H)),aa(B,set(B),set_ord_atMost(B),H4))
        <=> ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),H)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),H),H4) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_2844_sum_OatMost__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,B,G,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_2845_sum_Ocl__ivl__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),Ma),zero_zero(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))),aa(nat,B,G,aa(nat,nat,suc,N)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2846_not__Iic__le__Icc,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [H: B,L3: B,H4: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atMost(B),H)),set_or1337092689740270186AtMost(B,L3,H4)) ) ).

% not_Iic_le_Icc
tff(fact_2847_not__Iic__eq__Icc,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [H4: B,L: B,H: B] : aa(B,set(B),set_ord_atMost(B),H4) != set_or1337092689740270186AtMost(B,L,H) ) ).

% not_Iic_eq_Icc
tff(fact_2848_atMost__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_2849_atMost__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [U: B] : aa(B,set(B),set_ord_atMost(B),U) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ca(B,fun(B,$o),U)) ) ).

% atMost_def
tff(fact_2850_all__nat__less,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N)
         => aa(nat,$o,Pa,M4) )
    <=> ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
         => aa(nat,$o,Pa,X3) ) ) ).

% all_nat_less
tff(fact_2851_ex__nat__less,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N)
          & aa(nat,$o,Pa,M4) )
    <=> ? [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
          & aa(nat,$o,Pa,X3) ) ) ).

% ex_nat_less
tff(fact_2852_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2853_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_cc(fun(nat,B),fun(nat,fun(nat,B)),G),K)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2854_sum__power__shift,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [Ma: nat,N: nat,X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)))) ) ) ) ).

% sum_power_shift
tff(fact_2855_atLeastatMost__psubset__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),set_or1337092689740270186AtMost(B,C2,D2))
        <=> ( ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
              | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2)
                & ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),A2)
                  | aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),D2) ) ) )
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2856_sum_OatLeastAtMost__rev,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,N,Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_cd(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),G),N),Ma)),set_or1337092689740270186AtMost(nat,N,Ma)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2857_sum__choose__upper,axiom,
    ! [Ma: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ce(nat,fun(nat,nat),Ma)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,Ma)) ).

% sum_choose_upper
tff(fact_2858_sum__shift__lb__Suc0__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [F: fun(nat,B),K: nat] :
          ( ( aa(nat,B,F,zero_zero(nat)) = zero_zero(B) )
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2859_sum_OatLeast0__atMost__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,B,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2860_sum_Onat__ivl__Suc_H,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,aa(nat,nat,suc,N))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2861_sum_OatLeast__Suc__atMost,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2862_sum_OatMost__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% sum.atMost_Suc_shift
tff(fact_2863_sum__telescope,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(nat,B),I: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cf(fun(nat,B),fun(nat,B),F)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,zero_zero(nat))),aa(nat,B,F,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_2864_sum_OSuc__reindex__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))),aa(nat,B,G,aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),set_or1337092689740270186AtMost(nat,Ma,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2865_sum__Suc__diff,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Ma: nat,N: nat,F: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cg(fun(nat,B),fun(nat,B),F)),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,aa(nat,nat,suc,N))),aa(nat,B,F,Ma)) ) ) ) ).

% sum_Suc_diff
tff(fact_2866_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ch(nat,fun(nat,nat),R)),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),R),N))),N) ).

% sum_choose_lower
tff(fact_2867_choose__rising__sum_I1_J,axiom,
    ! [N: nat,Ma: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ci(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = 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),Ma)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_2868_choose__rising__sum_I2_J,axiom,
    ! [N: nat,Ma: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ci(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = 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),Ma)),one_one(nat))),Ma) ).

% choose_rising_sum(2)
tff(fact_2869_sum__atLeastAtMost__code,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [F: fun(nat,B),A2: nat,B2: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(B,aTP_Lamp_cj(fun(nat,B),fun(nat,fun(B,B)),F),A2,B2,zero_zero(B)) ) ).

% sum_atLeastAtMost_code
tff(fact_2870_sum_Oub__add__nat,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2871_gbinomial__parallel__sum,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_ck(B,fun(nat,B),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa(nat,B,semiring_1_of_nat(B),N))),one_one(B))),N) ) ).

% gbinomial_parallel_sum
tff(fact_2872_sum__choose__diagonal,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_cl(nat,fun(nat,fun(nat,nat)),Ma),N)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),Ma) ) ) ).

% sum_choose_diagonal
tff(fact_2873_vandermonde,axiom,
    ! [Ma: nat,N: nat,R: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_cm(nat,fun(nat,fun(nat,fun(nat,nat))),Ma),N),R)),aa(nat,set(nat),set_ord_atMost(nat),R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),R) ).

% vandermonde
tff(fact_2874_sum__gp__basic,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_2875_sum__natinterval__diff,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cn(fun(nat,B),fun(nat,B),F)),set_or1337092689740270186AtMost(nat,Ma,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,Ma)),aa(nat,B,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),zero_zero(B)) ) ).

% sum_natinterval_diff
tff(fact_2876_sum__telescope_H_H,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Ma: nat,N: nat,F: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_co(fun(nat,B),fun(nat,B),F)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),N)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,N)),aa(nat,B,F,Ma)) ) ) ) ).

% sum_telescope''
tff(fact_2877_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% choose_row_sum
tff(fact_2878_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),N) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_cp(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ).

% binomial
tff(fact_2879_sum_Oin__pairs__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cq(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.in_pairs_0
tff(fact_2880_gbinomial__sum__lower__neg,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cr(B,fun(nat,B),A2)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),Ma)),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),Ma)) ) ).

% gbinomial_sum_lower_neg
tff(fact_2881_binomial__ring,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,B2: B,N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),N) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aa(B,fun(nat,fun(nat,B)),aTP_Lamp_cs(B,fun(B,fun(nat,fun(nat,B))),A2),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% binomial_ring
tff(fact_2882_sum__gp__multiplied,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [Ma: nat,N: nat,X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),set_or1337092689740270186AtMost(nat,Ma,N))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2883_pochhammer__binomial__sum,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [A2: B,B2: B,N: nat] : comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2),N) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aa(B,fun(nat,fun(nat,B)),aTP_Lamp_ct(B,fun(B,fun(nat,fun(nat,B))),A2),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% pochhammer_binomial_sum
tff(fact_2884_sum_Oin__pairs,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma),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,one2))),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cq(fun(nat,B),fun(nat,B),G)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.in_pairs
tff(fact_2885_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cu(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,one2))),N)),N) ).

% choose_square_sum
tff(fact_2886_sum_Ozero__middle,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [P2: nat,K: nat,G: fun(nat,B),H: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P2)
           => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_cv(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_cw(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2887_gbinomial__partial__sum__poly,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Ma: nat,A2: B,X: B,Y: B] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_cx(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Ma),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_cy(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Ma),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2888_gbinomial__sum__up__index,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cz(nat,fun(nat,B),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2889_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_da(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,one2))) ).

% gauss_sum_nat
tff(fact_2890_sum__gp0,axiom,
    ! [B: $tType] :
      ( ( division_ring(B)
        & comm_ring(B) )
     => ! [X: B,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = $ite(X = one_one(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,suc,N)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X))) ) ).

% sum_gp0
tff(fact_2891_choose__alternating__linear__sum,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_db(nat,fun(nat,B),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(B) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2892_gbinomial__partial__sum__poly__xpos,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Ma: nat,A2: B,X: B,Y: B] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_cx(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Ma),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_dc(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Ma),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2893_gbinomial__sum__nat__pow2,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_dd(nat,fun(nat,B),Ma)),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma) ) ).

% gbinomial_sum_nat_pow2
tff(fact_2894_double__arith__series,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,D2: B,N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aTP_Lamp_de(B,fun(B,fun(nat,B)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),D2))) ) ).

% double_arith_series
tff(fact_2895_double__gauss__sum,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),semiring_1_of_nat(B)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B))) ) ).

% double_gauss_sum
tff(fact_2896_binomial__r__part__sum,axiom,
    ! [Ma: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),Ma)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma)) ).

% binomial_r_part_sum
tff(fact_2897_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_df(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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_2898_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dg(nat,fun(nat,fun(nat,nat)),A2),D2)),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,one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2899_Sum__Icc__nat,axiom,
    ! [Ma: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_da(nat,nat)),set_or1337092689740270186AtMost(nat,Ma,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),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_2900_choose__alternating__sum,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_dh(nat,fun(nat,B),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(B) ) ) ) ).

% choose_alternating_sum
tff(fact_2901_double__gauss__sum__from__Suc__0,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),semiring_1_of_nat(B)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2902_gauss__sum,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),semiring_1_of_nat(B)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B)))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_2903_arith__series,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B,D2: B,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aTP_Lamp_di(B,fun(B,fun(nat,B)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),D2)))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2904_sum__gp__offset,axiom,
    ! [B: $tType] :
      ( ( division_ring(B)
        & comm_ring(B) )
     => ! [X: B,Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N))) = $ite(X = one_one(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B)),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,suc,N))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X))) ) ).

% sum_gp_offset
tff(fact_2905_gauss__sum__from__Suc__0,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),semiring_1_of_nat(B)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),N)),one_one(B)))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2906_pochhammer__times__pochhammer__half,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Z: B,N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),comm_s3205402744901411588hammer(B,Z,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),Z),aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dj(B,fun(nat,B),Z)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_2907_Sum__Icc__int,axiom,
    ! [Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ma),N)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_dk(int,int)),set_or1337092689740270186AtMost(int,Ma,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),Ma),aa(int,int,aa(int,fun(int,int),minus_minus(int),Ma),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_2908_Sum__Ico__nat,axiom,
    ! [Ma: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_da(nat,nat)),set_or7035219750837199246ssThan(nat,Ma,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),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_2909_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_2910_gbinomial__Suc,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),aa(nat,nat,suc,K)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dl(B,fun(nat,B),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(B,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_2911_ivl__diff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [I: B,N: B,Ma: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),N)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),set_or7035219750837199246ssThan(B,I,Ma)),set_or7035219750837199246ssThan(B,I,N)) = set_or7035219750837199246ssThan(B,N,Ma) ) ) ) ).

% ivl_diff
tff(fact_2912_Ints__prod,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comm_monoid_mult(C)
        & ring_1(C) )
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(set(C),$o,member(C,aa(B,C,F,X2)),ring_1_Ints(C)) )
         => aa(set(C),$o,member(C,aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),ring_1_Ints(C)) ) ) ).

% Ints_prod
tff(fact_2913_prod_Oneutral__const,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aTP_Lamp_dm(C,B)),A4) = one_one(B) ) ).

% prod.neutral_const
tff(fact_2914_of__nat__prod,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_1(B)
     => ! [F: fun(C,nat),A4: set(C)] : aa(nat,B,semiring_1_of_nat(B),aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7121269368397514597t_prod(C,nat),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aTP_Lamp_dn(fun(C,nat),fun(C,B),F)),A4) ) ).

% of_nat_prod
tff(fact_2915_of__int__prod,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_ring_1(B)
     => ! [F: fun(C,int),A4: set(C)] : aa(int,B,ring_1_of_int(B),aa(set(C),int,aa(fun(C,int),fun(set(C),int),groups7121269368397514597t_prod(C,int),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aTP_Lamp_do(fun(C,int),fun(C,B),F)),A4) ) ).

% of_int_prod
tff(fact_2916_atLeastLessThan__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,L: B,U: B] :
          ( aa(set(B),$o,member(B,I),set_or7035219750837199246ssThan(B,L,U))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),I)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_2917_ivl__subset,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [I: B,J: B,Ma: B,N: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or7035219750837199246ssThan(B,I,J)),set_or7035219750837199246ssThan(B,Ma,N))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),J),I)
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),I)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),J),N) ) ) ) ) ).

% ivl_subset
tff(fact_2918_prod_OatMost__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,B,G,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_2919_sum_Oop__ivl__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma),zero_zero(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))),aa(nat,B,G,N))) ) ).

% sum.op_ivl_Suc
tff(fact_2920_prod_Ocl__ivl__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),Ma),one_one(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))),aa(nat,B,G,aa(nat,nat,suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_2921_prod_Oop__ivl__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma),one_one(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))),aa(nat,B,G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_2922_prod_Oshift__bounds__nat__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_dp(fun(nat,B),fun(nat,fun(nat,B)),G),K)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_2923_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_2924_prod_Oivl__cong,axiom,
    ! [C: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(C) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,C),H: fun(B,C)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),X2)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),D2)
                   => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_2925_prod_OatLeastLessThan__concat,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,P2: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P2)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_2926_prod_Oswap,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,fun(D,B)),B4: set(D),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(set(D),fun(C,B),aTP_Lamp_dr(fun(C,fun(D,B)),fun(set(D),fun(C,B)),G),B4)),A4) = aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7121269368397514597t_prod(D,B),aa(set(C),fun(D,B),aTP_Lamp_dt(fun(C,fun(D,B)),fun(set(C),fun(D,B)),G),A4)),B4) ) ).

% prod.swap
tff(fact_2927_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,B),A4: set(C)] :
          ( ( aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),G),A4) != one_one(B) )
         => ~ ! [A3: C] :
                ( aa(set(C),$o,member(C,A3),A4)
               => ( aa(C,B,G,A3) = one_one(B) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_2928_prod_Oneutral,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),G: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => ( aa(B,C,G,X2) = one_one(C) ) )
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4) = one_one(C) ) ) ) ).

% prod.neutral
tff(fact_2929_atLeastLessThan__inj_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( set_or7035219750837199246ssThan(B,A2,B2) = set_or7035219750837199246ssThan(B,C2,D2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_2930_atLeastLessThan__inj_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( set_or7035219750837199246ssThan(B,A2,B2) = set_or7035219750837199246ssThan(B,C2,D2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_2931_atLeastLessThan__eq__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),D2)
           => ( ( set_or7035219750837199246ssThan(B,A2,B2) = set_or7035219750837199246ssThan(B,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_2932_prod_Odistrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,B),H: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(C,B),fun(C,B),aTP_Lamp_du(fun(C,B),fun(fun(C,B),fun(C,B)),G),H)),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),G),A4)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),H),A4)) ) ).

% prod.distrib
tff(fact_2933_prod_OatLeast0__lessThan__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,B,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_2934_prod_OatLeast__Suc__lessThan,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_2935_prod_OatLeastLessThan__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: nat,B2: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,B,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_2936_prod__dividef,axiom,
    ! [B: $tType,C: $tType] :
      ( field(B)
     => ! [F: fun(C,B),G: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(C,B),fun(C,B),aTP_Lamp_dv(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),A4) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),F),A4)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),G),A4)) ) ).

% prod_dividef
tff(fact_2937_prod__power__distrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_1(B)
     => ! [F: fun(C,B),A4: set(C),N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),F),A4)),N) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(nat,fun(C,B),aTP_Lamp_dw(fun(C,B),fun(nat,fun(C,B)),F),N)),A4) ) ).

% prod_power_distrib
tff(fact_2938_mod__prod__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [F: fun(C,B),A2: B,A4: set(C)] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(B,fun(C,B),aTP_Lamp_bf(fun(C,B),fun(B,fun(C,B)),F),A2)),A4)),A2) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),F),A4)),A2) ) ).

% mod_prod_eq
tff(fact_2939_prod_Olast__plus,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,N)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))) ) ) ) ).

% prod.last_plus
tff(fact_2940_abs__prod,axiom,
    ! [B: $tType,C: $tType] :
      ( linordered_field(B)
     => ! [F: fun(C,B),A4: set(C)] : aa(B,B,abs_abs(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aTP_Lamp_dx(fun(C,B),fun(C,B),F)),A4) ) ).

% abs_prod
tff(fact_2941_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_2942_prod__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_2943_prod_OatLeastLessThan__rev,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,N,Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_dy(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),G),N),Ma)),set_or7035219750837199246ssThan(nat,N,Ma)) ) ).

% prod.atLeastLessThan_rev
tff(fact_2944_prod_Onested__swap,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: fun(nat,fun(nat,B)),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dz(fun(nat,fun(nat,B)),fun(nat,B),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_eb(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_2945_prod_Ohead__if,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma),one_one(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))),aa(nat,B,G,N))) ) ).

% prod.head_if
tff(fact_2946_prod__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_semidom(C)
     => ! [A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,I2))
                & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),aa(B,C,G,I2)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4)) ) ) ).

% prod_mono
tff(fact_2947_prod__nonneg,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_semidom(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,X2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)) ) ) ).

% prod_nonneg
tff(fact_2948_prod__pos,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_semidom(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(B,C,F,X2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)) ) ) ).

% prod_pos
tff(fact_2949_prod__ge__1,axiom,
    ! [C: $tType,B: $tType] :
      ( linord181362715937106298miring(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),one_one(C)),aa(B,C,F,X2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),one_one(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)) ) ) ).

% prod_ge_1
tff(fact_2950_fact__prod__Suc,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : semiring_char_0_fact(B,N) = aa(nat,B,semiring_1_of_nat(B),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_2951_atLeastLessThan__subset__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or7035219750837199246ssThan(B,A2,B2)),set_or7035219750837199246ssThan(B,C2,D2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_2952_all__nat__less__eq,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
         => aa(nat,$o,Pa,M4) )
    <=> ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
         => aa(nat,$o,Pa,X3) ) ) ).

% all_nat_less_eq
tff(fact_2953_ex__nat__less__eq,axiom,
    ! [N: nat,Pa: fun(nat,$o)] :
      ( ? [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N)
          & aa(nat,$o,Pa,M4) )
    <=> ? [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
          & aa(nat,$o,Pa,X3) ) ) ).

% ex_nat_less_eq
tff(fact_2954_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,N,Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_ec(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),G),N),Ma)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),Ma)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2955_pochhammer__prod,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] : comm_s3205402744901411588hammer(B,A2,N) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_ed(B,fun(nat,B),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_2956_fact__prod__rev,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : semiring_char_0_fact(B,N) = aa(nat,B,semiring_1_of_nat(B),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),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_2957_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_2958_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2959_power__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [C2: B,F: fun(C,nat),A4: set(C)] : aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7311177749621191930dd_sum(C,nat),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(C,nat),fun(C,B),aTP_Lamp_ee(B,fun(fun(C,nat),fun(C,B)),C2),F)),A4) ) ).

% power_sum
tff(fact_2960_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_dp(fun(nat,B),fun(nat,fun(nat,B)),G),K)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2961_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_2962_sum_Oshift__bounds__nat__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_cc(fun(nat,B),fun(nat,fun(nat,B)),G),K)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_2963_prod__le__1,axiom,
    ! [B: $tType,C: $tType] :
      ( linord181362715937106298miring(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,X2))
                & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),one_one(C)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),one_one(C)) ) ) ).

% prod_le_1
tff(fact_2964_fact__split,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( semiring_char_0_fact(B,N) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K),N)))),semiring_char_0_fact(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_split
tff(fact_2965_sum_Oivl__cong,axiom,
    ! [C: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(C) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,C),H: fun(B,C)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),X2)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),D2)
                   => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_2966_gbinomial__altdef__of__nat,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),K) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ef(B,fun(nat,fun(nat,B)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_2967_binomial__altdef__of__nat,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
         => ( aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(N),K)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_eg(nat,fun(nat,fun(nat,B)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_2968_gbinomial__mult__fact_H,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,K: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,A2),K)),semiring_char_0_fact(B,K)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_eh(B,fun(nat,B),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_2969_gbinomial__mult__fact,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [K: nat,A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),semiring_char_0_fact(B,K)),aa(nat,B,gbinomial(B,A2),K)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_eh(B,fun(nat,B),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_2970_gbinomial__prod__rev,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [A2: B,K: nat] : aa(nat,B,gbinomial(B,A2),K) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dl(B,fun(nat,B),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(B,K)) ) ).

% gbinomial_prod_rev
tff(fact_2971_sum_OatLeastLessThan__concat,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,P2: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P2)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_2972_sum__diff__nat__ivl,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Ma: nat,N: nat,P2: nat,F: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P2)
           => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or7035219750837199246ssThan(nat,Ma,P2))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or7035219750837199246ssThan(nat,Ma,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_2973_bset_I1_J,axiom,
    ! [D3: int,B4: set(int),Pa: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,Pa,X2)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_2974_bset_I2_J,axiom,
    ! [D3: int,B4: set(int),Pa: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,Pa,X2)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_2975_aset_I1_J,axiom,
    ! [D3: int,A4: set(int),Pa: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,Pa,X2)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_2976_aset_I2_J,axiom,
    ! [D3: int,A4: set(int),Pa: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,Pa,X2)
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,Pa,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_2977_prod_OatLeastAtMost__rev,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,N,Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_ec(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),G),N),Ma)),set_or1337092689740270186AtMost(nat,N,Ma)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2978_prod_OatLeast0__atMost__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,B,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2979_prod_Onat__ivl__Suc_H,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,aa(nat,nat,suc,N))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2980_prod_OatLeast__Suc__atMost,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2981_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or7035219750837199246ssThan(B,A2,B2)),set_or1337092689740270186AtMost(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_2982_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),set_or7035219750837199246ssThan(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_2983_sum__shift__lb__Suc0__0__upt,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [F: fun(nat,B),K: nat] :
          ( ( aa(nat,B,F,zero_zero(nat)) = zero_zero(B) )
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_2984_sum_OatLeast0__lessThan__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,B,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_2985_sum_OatLeast__Suc__lessThan,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_2986_sum_OatLeastLessThan__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: nat,B2: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,B,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_2987_sum_Olast__plus,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,N)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))) ) ) ) ).

% sum.last_plus
tff(fact_2988_bset_I9_J,axiom,
    ! [D2: int,D3: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)),Ta)) ) ) ) ).

% bset(9)
tff(fact_2989_bset_I10_J,axiom,
    ! [D2: int,D3: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)),Ta)) ) ) ) ).

% bset(10)
tff(fact_2990_aset_I9_J,axiom,
    ! [D2: int,D3: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),Ta)) ) ) ) ).

% aset(9)
tff(fact_2991_aset_I10_J,axiom,
    ! [D2: int,D3: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),Ta)) ) ) ) ).

% aset(10)
tff(fact_2992_prod_OSuc__reindex__ivl,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))),aa(nat,B,G,aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),set_or1337092689740270186AtMost(nat,Ma,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2993_prod_OatMost__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% prod.atMost_Suc_shift
tff(fact_2994_fact__prod,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N: nat] : semiring_char_0_fact(B,N) = aa(nat,B,semiring_1_of_nat(B),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_da(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_2995_sum__Suc__diff_H,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Ma: nat,N: nat,F: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cg(fun(nat,B),fun(nat,B),F)),set_or7035219750837199246ssThan(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,N)),aa(nat,B,F,Ma)) ) ) ) ).

% sum_Suc_diff'
tff(fact_2996_sum_OatLeastLessThan__rev,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,N,Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_ei(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),G),N),Ma)),set_or7035219750837199246ssThan(nat,N,Ma)) ) ).

% sum.atLeastLessThan_rev
tff(fact_2997_sum_Onested__swap,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: fun(nat,fun(nat,B)),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_ej(fun(nat,fun(nat,B)),fun(nat,B),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_el(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_2998_prod__atLeastAtMost__code,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [F: fun(nat,B),A2: nat,B2: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),F),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(B,aTP_Lamp_em(fun(nat,B),fun(nat,fun(B,B)),F),A2,B2,one_one(B)) ) ).

% prod_atLeastAtMost_code
tff(fact_2999_prod_Oub__add__nat,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3000_sum_Ohead__if,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma),zero_zero(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N))),aa(nat,B,G,N))) ) ).

% sum.head_if
tff(fact_3001_periodic__finite__ex,axiom,
    ! [D2: int,Pa: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K4: int] :
            ( aa(int,$o,Pa,X2)
          <=> aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K4),D2))) )
       => ( ? [X_12: int] : aa(int,$o,Pa,X_12)
        <=> ? [X3: int] :
              ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,Pa,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3002_bset_I3_J,axiom,
    ! [D3: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_3003_bset_I4_J,axiom,
    ! [D3: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,Ta),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_3004_bset_I5_J,axiom,
    ! [D3: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)),Ta) ) ) ) ).

% bset(5)
tff(fact_3005_bset_I7_J,axiom,
    ! [D3: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,Ta),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)) ) ) ) ) ).

% bset(7)
tff(fact_3006_aset_I3_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X4 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_3007_aset_I4_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,Ta),A4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X4 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_3008_aset_I5_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,Ta),A4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_3009_aset_I7_J,axiom,
    ! [D3: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ).

% aset(7)
tff(fact_3010_fact__eq__fact__times,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( semiring_char_0_fact(nat,Ma) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_da(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),Ma))) ) ) ).

% fact_eq_fact_times
tff(fact_3011_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat,Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,N,Ma)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_cd(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),G),N),Ma)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),Ma)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3012_bset_I6_J,axiom,
    ! [D3: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)),Ta) ) ) ) ).

% bset(6)
tff(fact_3013_bset_I8_J,axiom,
    ! [D3: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)) ) ) ) ) ).

% bset(8)
tff(fact_3014_aset_I6_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_3015_aset_I8_J,axiom,
    ! [D3: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ).

% aset(8)
tff(fact_3016_cppi,axiom,
    ! [D3: int,Pa: fun(int,$o),P: fun(int,$o),A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X2)
           => ( aa(int,$o,Pa,X2)
            <=> aa(int,$o,P,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),A4)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,Pa,X2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
         => ( ! [X2: int,K4: int] :
                ( aa(int,$o,P,X2)
              <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K4),D3))) )
           => ( ? [X_12: int] : aa(int,$o,Pa,X_12)
            <=> ( ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P,X3) )
                | ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa4: int] :
                        ( aa(set(int),$o,member(int,Xa4),A4)
                        & aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa4),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3017_cpmi,axiom,
    ! [D3: int,Pa: fun(int,$o),P: fun(int,$o),B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z3)
           => ( aa(int,$o,Pa,X2)
            <=> aa(int,$o,P,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),B4)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,Pa,X2)
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D3)) ) )
         => ( ! [X2: int,K4: int] :
                ( aa(int,$o,P,X2)
              <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K4),D3))) )
           => ( ? [X_12: int] : aa(int,$o,Pa,X_12)
            <=> ( ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P,X3) )
                | ? [X3: int] :
                    ( aa(set(int),$o,member(int,X3),set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa4: int] :
                        ( aa(set(int),$o,member(int,Xa4),B4)
                        & aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa4),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3018_pochhammer__Suc__prod,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] : comm_s3205402744901411588hammer(B,A2,aa(nat,nat,suc,N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_ed(B,fun(nat,B),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_3019_pochhammer__prod__rev,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] : comm_s3205402744901411588hammer(B,A2,N) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_en(B,fun(nat,fun(nat,B)),A2),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_3020_fact__div__fact,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Ma)),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_da(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),Ma)) ) ) ).

% fact_div_fact
tff(fact_3021_take__bit__sum,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_eo(B,fun(nat,B),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_3022_prod_Oin__pairs,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma),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,one2))),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_ep(fun(nat,B),fun(nat,B),G)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.in_pairs
tff(fact_3023_prod_Oin__pairs__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_ep(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.in_pairs_0
tff(fact_3024_pochhammer__Suc__prod__rev,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: B,N: nat] : comm_s3205402744901411588hammer(B,A2,aa(nat,nat,suc,N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_en(B,fun(nat,fun(nat,B)),A2),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3025_horner__sum__eq__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_1(B)
     => ! [F: fun(C,B),A2: B,Xs: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(list(C),fun(nat,B),aa(B,fun(list(C),fun(nat,B)),aTP_Lamp_eq(fun(C,B),fun(B,fun(list(C),fun(nat,B))),F),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(C),nat,size_size(list(C)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3026_prod_Ozero__middle,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [P2: nat,K: nat,G: fun(nat,B),H: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P2)
           => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_er(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_es(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3027_case__nat__add__eq__if,axiom,
    ! [B: $tType,A2: B,F: fun(nat,B),V2: num,N: nat] : case_nat(B,A2,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(nat,B,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N)) ).

% case_nat_add_eq_if
tff(fact_3028_integer__of__int__code,axiom,
    ! [K: int] :
      aa(int,code_integer,code_integer_of_int,K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
            $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_3029_sum__zero__power_H,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [C2: fun(nat,B),D2: fun(nat,B),A4: set(nat)] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,B),fun(nat,B),aTP_Lamp_et(fun(nat,B),fun(fun(nat,B),fun(nat,B)),C2),D2)),A4) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A4)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A4) ),
            aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,C2,zero_zero(nat))),aa(nat,B,D2,zero_zero(nat))),
            zero_zero(B) ) ) ).

% sum_zero_power'
tff(fact_3030_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            | ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% or_int_rec
tff(fact_3031_or__numerals_I4_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num,Y: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,X))),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),X)),aa(num,B,numeral_numeral(B),Y)))) ) ).

% or_numerals(4)
tff(fact_3032_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_1_of_int(code_integer) ).

% integer_of_int_eq_of_int
tff(fact_3033_or_Oright__neutral,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),zero_zero(B)) = A2 ) ).

% or.right_neutral
tff(fact_3034_or_Oleft__neutral,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),zero_zero(B)),A2) = A2 ) ).

% or.left_neutral
tff(fact_3035_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U)) ).

% finite_atLeastAtMost
tff(fact_3036_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U)) ).

% finite_atLeastLessThan
tff(fact_3037_finite__atMost,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% finite_atMost
tff(fact_3038_sum_Oinfinite,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = zero_zero(C) ) ) ) ).

% sum.infinite
tff(fact_3039_sum__eq__0__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( canoni5634975068530333245id_add(C)
     => ! [F3: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),F3)
         => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),F3) = zero_zero(C) )
          <=> ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),F3)
               => ( aa(B,C,F,X3) = zero_zero(C) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_3040_prod__zero__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( semidom(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4) = zero_zero(C) )
          <=> ? [X3: B] :
                ( aa(set(B),$o,member(B,X3),A4)
                & ( aa(B,C,F,X3) = zero_zero(C) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_3041_infinite__Icc__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(set(B),$o,finite_finite2(B),set_or1337092689740270186AtMost(B,A2,B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_3042_infinite__Ico__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(set(B),$o,finite_finite2(B),set_or7035219750837199246ssThan(B,A2,B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_3043_prod_Oinfinite,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),G: fun(B,C)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4) = one_one(C) ) ) ) ).

% prod.infinite
tff(fact_3044_bit_Odisj__one__left,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(B,B,uminus_uminus(B),one_one(B))),X) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% bit.disj_one_left
tff(fact_3045_bit_Odisj__one__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),X),aa(B,B,uminus_uminus(B),one_one(B))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% bit.disj_one_right
tff(fact_3046_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_3047_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_3048_sum_Odelta,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),A2: B,B2: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aTP_Lamp_eu(B,fun(fun(B,C),fun(B,C)),A2),B2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(B,C,B2,A2),zero_zero(C)) ) ) ) ).

% sum.delta
tff(fact_3049_sum_Odelta_H,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),A2: B,B2: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aTP_Lamp_ev(B,fun(fun(B,C),fun(B,C)),A2),B2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(B,C,B2,A2),zero_zero(C)) ) ) ) ).

% sum.delta'
tff(fact_3050_prod__eq__1__iff,axiom,
    ! [B: $tType,A4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A4) = one_one(nat) )
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
           => ( aa(B,nat,F,X3) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3051_prod_Odelta_H,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),A2: B,B2: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,C),fun(B,C),aTP_Lamp_ew(B,fun(fun(B,C),fun(B,C)),A2),B2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(B,C,B2,A2),one_one(C)) ) ) ) ).

% prod.delta'
tff(fact_3052_prod_Odelta,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),A2: B,B2: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,C),fun(B,C),aTP_Lamp_ex(B,fun(fun(B,C),fun(B,C)),A2),B2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(B,C,B2,A2),one_one(C)) ) ) ) ).

% prod.delta
tff(fact_3053_case__nat__numeral,axiom,
    ! [B: $tType,A2: B,F: fun(nat,B),V2: num] : case_nat(B,A2,F,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,B,F,pred_numeral(V2)) ).

% case_nat_numeral
tff(fact_3054_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_3055_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_3056_or__numerals_I2_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Y: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y)) ) ).

% or_numerals(2)
tff(fact_3057_or__numerals_I8_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_3058_bit_Odisj__cancel__left,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(B,B,bit_ri4277139882892585799ns_not(B),X)),X) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% bit.disj_cancel_left
tff(fact_3059_bit_Odisj__cancel__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),X),aa(B,B,bit_ri4277139882892585799ns_not(B),X)) = aa(B,B,uminus_uminus(B),one_one(B)) ) ).

% bit.disj_cancel_right
tff(fact_3060_prod__pos__nat__iff,axiom,
    ! [B: $tType,A4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A4))
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3061_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_3062_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_3063_or__numerals_I1_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Y: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),one_one(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Y))) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_3064_or__numerals_I5_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,X))),one_one(B)) = aa(num,B,numeral_numeral(B),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_3065_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(2)
tff(fact_3066_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(6)
tff(fact_3067_sum__zero__power,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [C2: fun(nat,B),A4: set(nat)] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_ey(fun(nat,B),fun(nat,B),C2)),A4) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A4)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A4) ),
            aa(nat,B,C2,zero_zero(nat)),
            zero_zero(B) ) ) ).

% sum_zero_power
tff(fact_3068_and__minus__minus__numerals,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Ma)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_3069_or__minus__minus__numerals,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ma))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Ma)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_3070_or__numerals_I7_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num,Y: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Y))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),X)),aa(num,B,numeral_numeral(B),Y)))) ) ).

% or_numerals(7)
tff(fact_3071_or__numerals_I6_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [X: num,Y: num] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,X))),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Y))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(num,B,numeral_numeral(B),X)),aa(num,B,numeral_numeral(B),Y)))) ) ).

% or_numerals(6)
tff(fact_3072_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_3073_of__nat__or__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),N)) = aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(nat,B,semiring_1_of_nat(B),Ma)),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_or_eq
tff(fact_3074_int__prod,axiom,
    ! [B: $tType,F: fun(B,nat),A4: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A4)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),aTP_Lamp_bk(fun(B,nat),fun(B,int),F)),A4) ).

% int_prod
tff(fact_3075_or__nat__def,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),Ma)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% or_nat_def
tff(fact_3076_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa2),X) ) ).

% less_eq_integer.abs_eq
tff(fact_3077_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa2),X) ) ).

% less_integer.abs_eq
tff(fact_3078_one__integer__def,axiom,
    one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).

% one_integer_def
tff(fact_3079_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(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),times_times(int),Xa2),X)) ).

% times_integer.abs_eq
tff(fact_3080_uminus__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),X)) ).

% uminus_integer.abs_eq
tff(fact_3081_or__eq__0__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),B2) = zero_zero(B) )
        <=> ( ( A2 = zero_zero(B) )
            & ( B2 = zero_zero(B) ) ) ) ) ).

% or_eq_0_iff
tff(fact_3082_bit_Odisj__zero__right,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),X),zero_zero(B)) = X ) ).

% bit.disj_zero_right
tff(fact_3083_zero__integer__def,axiom,
    zero_zero(code_integer) = aa(int,code_integer,code_integer_of_int,zero_zero(int)) ).

% zero_integer_def
tff(fact_3084_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,sgn_sgn(code_integer),K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_3085_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_3086_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_3087_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).

% minus_integer_code(2)
tff(fact_3088_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_3089_finite__if__eq__beyond__finite,axiom,
    ! [B: $tType,S: set(B),S4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),S)
     => aa(set(set(B)),$o,finite_finite2(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aa(set(B),fun(set(B),$o),aTP_Lamp_ez(set(B),fun(set(B),fun(set(B),$o)),S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3090_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_3091_or__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(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),bit_se1065995026697491101ons_or(int),Xa2),X)) ).

% or_integer.abs_eq
tff(fact_3092_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_3093_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_3094_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_3095_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).

% times_integer_code(2)
tff(fact_3096_nat_Ocase__distrib,axiom,
    ! [C: $tType,B: $tType,H: fun(C,B),F1: C,F22: fun(nat,C),Nat: nat] : aa(C,B,H,case_nat(C,F1,F22,Nat)) = case_nat(B,aa(C,B,H,F1),aa(fun(nat,C),fun(nat,B),aTP_Lamp_fa(fun(C,B),fun(fun(nat,C),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_3097_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_3098_minus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(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),minus_minus(int),Xa2),X)) ).

% minus_integer.abs_eq
tff(fact_3099_abs__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,abs_abs(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,abs_abs(int),X)) ).

% abs_integer.abs_eq
tff(fact_3100_divide__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(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),divide_divide(int),Xa2),X)) ).

% divide_integer.abs_eq
tff(fact_3101_modulo__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(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),modulo_modulo(int),Xa2),X)) ).

% modulo_integer.abs_eq
tff(fact_3102_take__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),Xa2),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Xa2),X)) ).

% take_bit_integer.abs_eq
tff(fact_3103_not__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,bit_ri4277139882892585799ns_not(int),X)) ).

% not_integer.abs_eq
tff(fact_3104_and__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(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),bit_se5824344872417868541ns_and(int),Xa2),X)) ).

% and_integer.abs_eq
tff(fact_3105_bounded__nat__set__is__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( ! [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),N) )
     => aa(set(nat),$o,finite_finite2(nat),N5) ) ).

% bounded_nat_set_is_finite
tff(fact_3106_finite__nat__set__iff__bounded,axiom,
    ! [N5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N5)
    <=> ? [M4: nat] :
        ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),M4) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3107_sgn__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,sgn_sgn(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,sgn_sgn(int),X)) ).

% sgn_integer.abs_eq
tff(fact_3108_finite__nat__set__iff__bounded__le,axiom,
    ! [N5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N5)
    <=> ? [M4: nat] :
        ! [X3: nat] :
          ( aa(set(nat),$o,member(nat,X3),N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M4) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_3109_bit__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),X) ).

% bit_integer.abs_eq
tff(fact_3110_infinite__Iic,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [A2: B] : ~ aa(set(B),$o,finite_finite2(B),aa(B,set(B),set_ord_atMost(B),A2)) ) ).

% infinite_Iic
tff(fact_3111_xor__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(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),bit_se5824344971392196577ns_xor(int),Xa2),X)) ).

% xor_integer.abs_eq
tff(fact_3112_push__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Xa2),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Xa2),X)) ).

% push_bit_integer.abs_eq
tff(fact_3113_mask__integer_Oabs__eq,axiom,
    ! [X: nat] : aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),X) = aa(int,code_integer,code_integer_of_int,aa(nat,int,bit_se2239418461657761734s_mask(int),X)) ).

% mask_integer.abs_eq
tff(fact_3114_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),Xa2),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Xa2),X)) ).

% unset_bit_integer.abs_eq
tff(fact_3115_set__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xa2),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xa2),X)) ).

% set_bit_integer.abs_eq
tff(fact_3116_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),Xa2),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Xa2),X)) ).

% flip_bit_integer.abs_eq
tff(fact_3117_finite__M__bounded__by__nat,axiom,
    ! [Pa: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_fb(fun(nat,$o),fun(nat,fun(nat,$o)),Pa),I))) ).

% finite_M_bounded_by_nat
tff(fact_3118_finite__less__ub,axiom,
    ! [F: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),aa(nat,nat,F,N2))
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_fc(fun(nat,nat),fun(nat,fun(nat,$o)),F),U))) ) ).

% finite_less_ub
tff(fact_3119_sum_Oswap__restrict,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [A4: set(B),B4: set(C),G: fun(B,fun(C,D)),R2: fun(B,fun(C,$o))] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(C),$o,finite_finite2(C),B4)
           => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(B,fun(C,$o)),fun(B,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D)),aTP_Lamp_fe(set(C),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D))),B4),G),R2)),A4) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),aa(fun(B,fun(C,$o)),fun(C,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D)),aTP_Lamp_fh(set(B),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D))),A4),G),R2)),B4) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3120_prod_Oswap__restrict,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [A4: set(B),B4: set(C),G: fun(B,fun(C,D)),R2: fun(B,fun(C,$o))] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(C),$o,finite_finite2(C),B4)
           => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(fun(B,fun(C,$o)),fun(B,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D)),aTP_Lamp_fi(set(C),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D))),B4),G),R2)),A4) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),aa(fun(B,fun(C,$o)),fun(C,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D)),aTP_Lamp_fk(set(B),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D))),A4),G),R2)),B4) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3121_old_Onat_Osimps_I4_J,axiom,
    ! [B: $tType,F1: B,F22: fun(nat,B)] : case_nat(B,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_3122_old_Onat_Osimps_I5_J,axiom,
    ! [B: $tType,F1: B,F22: fun(nat,B),X22: nat] : case_nat(B,F1,F22,aa(nat,nat,suc,X22)) = aa(nat,B,F22,X22) ).

% old.nat.simps(5)
tff(fact_3123_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_3124_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) ) ) ).

% OR_lower
tff(fact_3125_disjunctive__add,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,B2: B] :
          ( ! [N2: nat] :
              ( ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N2)
              | ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),B2),N2) )
         => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),B2) ) ) ) ).

% disjunctive_add
tff(fact_3126_plus__and__or,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ).

% plus_and_or
tff(fact_3127_sum__mono__inv,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [F: fun(C,B),I5: set(C),G: fun(C,B),I: C] :
          ( ( aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),I5) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),I5) )
         => ( ! [I2: C] :
                ( aa(set(C),$o,member(C,I2),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,I2)),aa(C,B,G,I2)) )
           => ( aa(set(C),$o,member(C,I),I5)
             => ( aa(set(C),$o,finite_finite2(C),I5)
               => ( aa(C,B,F,I) = aa(C,B,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_3128_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [S5: set(B),T3: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T4: set(C),G: fun(B,D),H: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),S5)
         => ( aa(set(C),$o,finite_finite2(C),T3)
           => ( ! [A3: B] :
                  ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5))
                 => ( aa(C,B,I,aa(B,C,J,A3)) = A3 ) )
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5))
                   => aa(set(C),$o,member(C,aa(B,C,J,A3)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3)) )
               => ( ! [B3: C] :
                      ( aa(set(C),$o,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( aa(set(C),$o,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
                       => aa(set(B),$o,member(B,aa(C,B,I,B3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5)) )
                   => ( ! [A3: B] :
                          ( aa(set(B),$o,member(B,A3),S5)
                         => ( aa(B,D,G,A3) = zero_zero(D) ) )
                     => ( ! [B3: C] :
                            ( aa(set(C),$o,member(C,B3),T3)
                           => ( aa(C,D,H,B3) = zero_zero(D) ) )
                       => ( ! [A3: B] :
                              ( aa(set(B),$o,member(B,A3),S)
                             => ( aa(C,D,H,aa(B,C,J,A3)) = aa(B,D,G,A3) ) )
                         => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),G),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_3129_prod__zero,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_1(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ? [X4: B] :
                ( aa(set(B),$o,member(B,X4),A4)
                & ( aa(B,C,F,X4) = zero_zero(C) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4) = zero_zero(C) ) ) ) ) ).

% prod_zero
tff(fact_3130_infinite__Icc,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ aa(set(B),$o,finite_finite2(B),set_or1337092689740270186AtMost(B,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_3131_infinite__Ico,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ aa(set(B),$o,finite_finite2(B),set_or7035219750837199246ssThan(B,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_3132_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_fl(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_3133_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_fm(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_3134_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [S5: set(B),T3: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T4: set(C),G: fun(B,D),H: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),S5)
         => ( aa(set(C),$o,finite_finite2(C),T3)
           => ( ! [A3: B] :
                  ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5))
                 => ( aa(C,B,I,aa(B,C,J,A3)) = A3 ) )
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5))
                   => aa(set(C),$o,member(C,aa(B,C,J,A3)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3)) )
               => ( ! [B3: C] :
                      ( aa(set(C),$o,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( aa(set(C),$o,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
                       => aa(set(B),$o,member(B,aa(C,B,I,B3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5)) )
                   => ( ! [A3: B] :
                          ( aa(set(B),$o,member(B,A3),S5)
                         => ( aa(B,D,G,A3) = one_one(D) ) )
                     => ( ! [B3: C] :
                            ( aa(set(C),$o,member(C,B3),T3)
                           => ( aa(C,D,H,B3) = one_one(D) ) )
                       => ( ! [A3: B] :
                              ( aa(set(B),$o,member(B,A3),S)
                             => ( aa(C,D,H,aa(B,C,J,A3)) = aa(B,D,G,A3) ) )
                         => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),G),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3135_set__bit__nat__def,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),Ma),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Ma),one_one(nat))) ).

% set_bit_nat_def
tff(fact_3136_sum_Ofinite__Collect__op,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(B),X: fun(B,C),Y: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fn(set(B),fun(fun(B,C),fun(B,$o)),I5),X)))
         => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fn(set(B),fun(fun(B,C),fun(B,$o)),I5),Y)))
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aa(fun(B,C),fun(fun(B,C),fun(B,$o)),aTP_Lamp_fo(set(B),fun(fun(B,C),fun(fun(B,C),fun(B,$o))),I5),X),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3137_prod_Ofinite__Collect__op,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(B),X: fun(B,C),Y: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fp(set(B),fun(fun(B,C),fun(B,$o)),I5),X)))
         => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fp(set(B),fun(fun(B,C),fun(B,$o)),I5),Y)))
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aa(fun(B,C),fun(fun(B,C),fun(B,$o)),aTP_Lamp_fq(set(B),fun(fun(B,C),fun(fun(B,C),fun(B,$o))),I5),X),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3138_sum_Osetdiff__irrelevant,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_fr(fun(B,C),fun(B,$o),G)))) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3139_sum_Ointer__filter,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C),Pa: fun(B,$o)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa))) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,$o),fun(B,C),aTP_Lamp_fs(fun(B,C),fun(fun(B,$o),fun(B,C)),G),Pa)),A4) ) ) ) ).

% sum.inter_filter
tff(fact_3140_filter__preserves__multiset,axiom,
    ! [B: $tType,M: fun(B,nat),Pa: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ft(fun(B,nat),fun(B,$o),M)))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_fu(fun(B,nat),fun(fun(B,$o),fun(B,$o)),M),Pa))) ) ).

% filter_preserves_multiset
tff(fact_3141_prod_Ointer__filter,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),G: fun(B,C),Pa: fun(B,$o)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa))) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,$o),fun(B,C),aTP_Lamp_fv(fun(B,C),fun(fun(B,$o),fun(B,C)),G),Pa)),A4) ) ) ) ).

% prod.inter_filter
tff(fact_3142_prod_Osetdiff__irrelevant,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_fw(fun(B,C),fun(B,$o),G)))) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3143_finite__int__segment,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [A2: B,B2: B] : aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_fx(B,fun(B,fun(B,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_3144_or__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(1)
tff(fact_3145_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_3146_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,X2)) )
           => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4) = zero_zero(C) )
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => ( aa(B,C,F,X3) = zero_zero(C) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_3147_sum__le__included,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(D)
     => ! [S2: set(B),Ta: set(C),G: fun(C,D),I: fun(C,B),F: fun(B,D)] :
          ( aa(set(B),$o,finite_finite2(B),S2)
         => ( aa(set(C),$o,finite_finite2(C),Ta)
           => ( ! [X2: C] :
                  ( aa(set(C),$o,member(C,X2),Ta)
                 => aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),zero_zero(D)),aa(C,D,G,X2)) )
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S2)
                   => ? [Xa: C] :
                        ( aa(set(C),$o,member(C,Xa),Ta)
                        & ( aa(C,B,I,Xa) = X2 )
                        & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(B,D,F,X2)),aa(C,D,G,Xa)) ) )
               => aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),F),S2)),aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_3148_sum__strict__mono__ex1,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(C)
     => ! [A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2)) )
           => ( ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),A4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X4)),aa(B,C,G,X4)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3149_sum_Orelated,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [R2: fun(B,fun(B,$o)),S: set(C),H: fun(C,B),G: fun(C,B)] :
          ( aa(B,$o,aa(B,fun(B,$o),R2,zero_zero(B)),zero_zero(B))
         => ( ! [X1: B,Y1: B,X23: B,Y23: B] :
                ( ( aa(B,$o,aa(B,fun(B,$o),R2,X1),X23)
                  & aa(B,$o,aa(B,fun(B,$o),R2,Y1),Y23) )
               => aa(B,$o,aa(B,fun(B,$o),R2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X1),Y1)),aa(B,B,aa(B,fun(B,B),plus_plus(B),X23),Y23)) )
           => ( aa(set(C),$o,finite_finite2(C),S)
             => ( ! [X2: C] :
                    ( aa(set(C),$o,member(C,X2),S)
                   => aa(B,$o,aa(B,fun(B,$o),R2,aa(C,B,H,X2)),aa(C,B,G,X2)) )
               => aa(B,$o,aa(B,fun(B,$o),R2,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),H),S)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),S)) ) ) ) ) ) ).

% sum.related
tff(fact_3150_sum_Osame__carrier,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [C6: set(B),A4: set(B),B4: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),C6)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),C6)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),C6)
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A4))
                   => ( aa(B,C,G,A3) = zero_zero(C) ) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4))
                     => ( aa(B,C,H,B3) = zero_zero(C) ) )
                 => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),B4) )
                  <=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),C6) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),C6) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_3151_sum_Osame__carrierI,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [C6: set(B),A4: set(B),B4: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),C6)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),C6)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),C6)
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A4))
                   => ( aa(B,C,G,A3) = zero_zero(C) ) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4))
                     => ( aa(B,C,H,B3) = zero_zero(C) ) )
                 => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),C6) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),C6) )
                   => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3152_sum_Omono__neutral__left,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [T4: set(B),S: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,G,X2) = zero_zero(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T4) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3153_sum_Omono__neutral__right,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [T4: set(B),S: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,G,X2) = zero_zero(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3154_sum_Omono__neutral__cong__left,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [T4: set(B),S: set(B),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,H,X2) = zero_zero(C) ) )
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
               => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3155_sum_Omono__neutral__cong__right,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [T4: set(B),S: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,G,X2) = zero_zero(C) ) )
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
               => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3156_prod_Orelated,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [R2: fun(B,fun(B,$o)),S: set(C),H: fun(C,B),G: fun(C,B)] :
          ( aa(B,$o,aa(B,fun(B,$o),R2,one_one(B)),one_one(B))
         => ( ! [X1: B,Y1: B,X23: B,Y23: B] :
                ( ( aa(B,$o,aa(B,fun(B,$o),R2,X1),X23)
                  & aa(B,$o,aa(B,fun(B,$o),R2,Y1),Y23) )
               => aa(B,$o,aa(B,fun(B,$o),R2,aa(B,B,aa(B,fun(B,B),times_times(B),X1),Y1)),aa(B,B,aa(B,fun(B,B),times_times(B),X23),Y23)) )
           => ( aa(set(C),$o,finite_finite2(C),S)
             => ( ! [X2: C] :
                    ( aa(set(C),$o,member(C,X2),S)
                   => aa(B,$o,aa(B,fun(B,$o),R2,aa(C,B,H,X2)),aa(C,B,G,X2)) )
               => aa(B,$o,aa(B,fun(B,$o),R2,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),H),S)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),G),S)) ) ) ) ) ) ).

% prod.related
tff(fact_3157_sum_Osubset__diff,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [B4: set(B),A4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
         => ( aa(set(B),$o,finite_finite2(B),A4)
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3158_prod_Omono__neutral__cong__right,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [T4: set(B),S: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,G,X2) = one_one(C) ) )
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
               => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3159_prod_Omono__neutral__cong__left,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [T4: set(B),S: set(B),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,H,X2) = one_one(C) ) )
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
               => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T4) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3160_prod_Omono__neutral__right,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [T4: set(B),S: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,G,X2) = one_one(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3161_prod_Omono__neutral__left,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [T4: set(B),S: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,G,X2) = one_one(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T4) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3162_prod_Osame__carrierI,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [C6: set(B),A4: set(B),B4: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),C6)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),C6)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),C6)
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A4))
                   => ( aa(B,C,G,A3) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4))
                     => ( aa(B,C,H,B3) = one_one(C) ) )
                 => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),C6) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),C6) )
                   => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3163_prod_Osame__carrier,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [C6: set(B),A4: set(B),B4: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),C6)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),C6)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),C6)
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A4))
                   => ( aa(B,C,G,A3) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4))
                     => ( aa(B,C,H,B3) = one_one(C) ) )
                 => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),B4) )
                  <=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),C6) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),C6) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3164_set__bit__eq__or,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se5668285175392031749et_bit(B),N),A2) = aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),one_one(B))) ) ).

% set_bit_eq_or
tff(fact_3165_sum__eq__Suc0__iff,axiom,
    ! [B: $tType,A4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
            & ( aa(B,nat,F,X3) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa4: B] :
                ( aa(set(B),$o,member(B,Xa4),A4)
               => ( ( X3 != Xa4 )
                 => ( aa(B,nat,F,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_3166_sum__eq__1__iff,axiom,
    ! [B: $tType,A4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4) = one_one(nat) )
      <=> ? [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
            & ( aa(B,nat,F,X3) = one_one(nat) )
            & ! [Xa4: B] :
                ( aa(set(B),$o,member(B,Xa4),A4)
               => ( ( X3 != Xa4 )
                 => ( aa(B,nat,F,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_3167_prod__int__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_dk(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_3168_sum__nonneg__0,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S2: set(B),F: fun(B,C),I: B] :
          ( aa(set(B),$o,finite_finite2(B),S2)
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),S2)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,I2)) )
           => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),S2) = zero_zero(C) )
             => ( aa(set(B),$o,member(B,I),S2)
               => ( aa(B,C,F,I) = zero_zero(C) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3169_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S2: set(B),F: fun(B,C),B4: C,I: B] :
          ( aa(set(B),$o,finite_finite2(B),S2)
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),S2)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,I2)) )
           => ( ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),S2) = B4 )
             => ( aa(set(B),$o,member(B,I),S2)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I)),B4) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3170_set__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).

% set_bit_int_def
tff(fact_3171_add__mset__in__multiset,axiom,
    ! [B: $tType,M: fun(B,nat),A2: B] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ft(fun(B,nat),fun(B,$o),M)))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_fy(fun(B,nat),fun(B,fun(B,$o)),M),A2))) ) ).

% add_mset_in_multiset
tff(fact_3172_less__eq__nat_Osimps_I2_J,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ma)),N)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% less_eq_nat.simps(2)
tff(fact_3173_diff__preserves__multiset,axiom,
    ! [B: $tType,M: fun(B,nat),N5: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ft(fun(B,nat),fun(B,$o),M)))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,nat),fun(B,$o),aTP_Lamp_fz(fun(B,nat),fun(fun(B,nat),fun(B,$o)),M),N5))) ) ).

% diff_preserves_multiset
tff(fact_3174_finite__divisors__nat,axiom,
    ! [Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),Ma))) ) ).

% finite_divisors_nat
tff(fact_3175_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N5),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
     => aa(set(nat),$o,finite_finite2(nat),N5) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3176_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
     => aa(set(nat),$o,finite_finite2(nat),N5) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_3177_finite__abs__int__segment,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [A2: B] : aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_gb(B,fun(B,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_3178_bit_Ocomplement__unique,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B,X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),X) = zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),X) = aa(B,B,uminus_uminus(B),one_one(B)) )
           => ( ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),Y) = zero_zero(B) )
             => ( ( aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),Y) = aa(B,B,uminus_uminus(B),one_one(B)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3179_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(2)
tff(fact_3180_or__not__numerals_I4_J,axiom,
    ! [Ma: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_3181_sum__pos2,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [I5: set(B),I: B,F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( aa(set(B),$o,member(B,I),I5)
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(B,C,F,I))
             => ( ! [I2: B] :
                    ( aa(set(B),$o,member(B,I2),I5)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,I2)) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_3182_nth__Cons,axiom,
    ! [B: $tType,X: B,Xs: list(B),N: nat] : aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),N) = case_nat(B,X,nth(B,Xs),N) ).

% nth_Cons
tff(fact_3183_sum__mono2,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [B4: set(B),A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),B4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
           => ( ! [B3: B] :
                  ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4))
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,B3)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),B4)) ) ) ) ) ).

% sum_mono2
tff(fact_3184_less__1__prod2,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_idom(C)
     => ! [I5: set(B),I: B,F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( aa(set(B),$o,member(B,I),I5)
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),one_one(C)),aa(B,C,F,I))
             => ( ! [I2: B] :
                    ( aa(set(B),$o,member(B,I2),I5)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),one_one(C)),aa(B,C,F,I2)) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less(C),one_one(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3185_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_dk(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_3186_diff__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_da(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)) ).

% diff_Suc
tff(fact_3187_bit_Ocompl__unique,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),X),Y) = zero_zero(B) )
         => ( ( aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),X),Y) = aa(B,B,uminus_uminus(B),one_one(B)) )
           => ( aa(B,B,bit_ri4277139882892585799ns_not(B),X) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_3188_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(3)
tff(fact_3189_or__not__numerals_I7_J,axiom,
    ! [Ma: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_3190_bit__numeral__rec_I1_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [W: num,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,W))),N)
        <=> case_nat($o,$false,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(num,B,numeral_numeral(B),W)),N) ) ) ).

% bit_numeral_rec(1)
tff(fact_3191_bit__numeral__rec_I2_J,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [W: num,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(num,B,numeral_numeral(B),aa(num,num,bit1,W))),N)
        <=> case_nat($o,$true,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(num,B,numeral_numeral(B),W)),N) ) ) ).

% bit_numeral_rec(2)
tff(fact_3192_Nitpick_Ocase__nat__unfold,axiom,
    ! [B: $tType,X: B,F: fun(nat,B),N: nat] :
      case_nat(B,X,F,N) = $ite(N = zero_zero(nat),X,aa(nat,B,F,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_3193_sum__strict__mono2,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(C)
     => ! [B4: set(B),A4: set(B),B2: B,F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),B4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
           => ( aa(set(B),$o,member(B,B2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4))
             => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(B,C,F,B2))
               => ( ! [X2: B] :
                      ( aa(set(B),$o,member(B,X2),B4)
                     => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,X2)) )
                 => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),B4)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3194_prod__mono2,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_idom(C)
     => ! [B4: set(B),A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),B4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
           => ( ! [B3: B] :
                  ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4))
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),one_one(C)),aa(B,C,F,B3)) )
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),A4)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,A3)) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),B4)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3195_one__or__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),one_one(B)),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa($o,B,zero_neq_one_of_bool(B),aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2))) ) ).

% one_or_eq
tff(fact_3196_or__one__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),A2),one_one(B)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),aa($o,B,zero_neq_one_of_bool(B),aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2))) ) ).

% or_one_eq
tff(fact_3197_mask__Suc__double,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(nat,B,bit_se2239418461657761734s_mask(B),aa(nat,nat,suc,N)) = aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),one_one(B)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,B,bit_se2239418461657761734s_mask(B),N))) ) ).

% mask_Suc_double
tff(fact_3198_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ) ) ) ).

% OR_upper
tff(fact_3199_or__not__numerals_I5_J,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Ma))),aa(int,int,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,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_3200_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% or_Suc_0_eq
tff(fact_3201_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% Suc_0_or_eq
tff(fact_3202_or__not__numerals_I9_J,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_3203_or__not__numerals_I8_J,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Ma))),aa(int,int,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,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_3204_or__nat__rec,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),N) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Ma)
            | ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% or_nat_rec
tff(fact_3205_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ap(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_3206_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_3207_finite__Collect__subsets,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(set(set(B)),$o,finite_finite2(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_gc(set(B),fun(set(B),$o),A4))) ) ).

% finite_Collect_subsets
tff(fact_3208_finite__Collect__conjI,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] :
      ( ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
        | aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Q)) )
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_gd(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q))) ) ).

% finite_Collect_conjI
tff(fact_3209_finite__Collect__disjI,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ge(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)))
    <=> ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
        & aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3210_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or1337092689740270186AtMost(int,L,U)) ).

% finite_atLeastAtMost_int
tff(fact_3211_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or7035219750837199246ssThan(int,L,U)) ).

% finite_atLeastLessThan_int
tff(fact_3212_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_gf(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_3213_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_gg(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_3214_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_3215_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or1337092689740270186AtMost(code_integer,L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_3216_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_gh(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_3217_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_gi(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_3218_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),$o,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_3219_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_3220_finite__maxlen,axiom,
    ! [B: $tType,M: set(list(B))] :
      ( aa(set(list(B)),$o,finite_finite2(list(B)),M)
     => ? [N2: nat] :
        ! [X4: list(B)] :
          ( aa(set(list(B)),$o,member(list(B),X4),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),X4)),N2) ) ) ).

% finite_maxlen
tff(fact_3221_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_gj(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_3222_pigeonhole__infinite__rel,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C),R2: fun(B,fun(C,$o))] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => ? [Xa: C] :
                  ( aa(set(C),$o,member(C,Xa),B4)
                  & aa(C,$o,aa(B,fun(C,$o),R2,X2),Xa) ) )
         => ? [X2: C] :
              ( aa(set(C),$o,member(C,X2),B4)
              & ~ aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,fun(C,$o)),fun(C,fun(B,$o)),aTP_Lamp_fg(set(B),fun(fun(B,fun(C,$o)),fun(C,fun(B,$o))),A4),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3223_not__finite__existsD,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ~ aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
     => ? [X_13: B] : aa(B,$o,Pa,X_13) ) ).

% not_finite_existsD
tff(fact_3224_finite__has__maximal2,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => ? [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                & ! [Xa: B] :
                    ( aa(set(B),$o,member(B,Xa),A4)
                   => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3225_finite__has__minimal2,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => ? [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),A2)
                & ! [Xa: B] :
                    ( aa(set(B),$o,member(B,Xa),A4)
                   => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3226_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),ord_max(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% or_int_unfold
tff(fact_3227_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => boolea3799213064322606851m_diff(B,bit_se5824344872417868541ns_and(B),bit_se1065995026697491101ons_or(B),bit_ri4277139882892585799ns_not(B),zero_zero(B),aa(B,B,uminus_uminus(B),one_one(B)),bit_se5824344971392196577ns_xor(B)) ) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_3228_even__set__encode__iff,axiom,
    ! [A4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),nat_set_encode(A4))
      <=> ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A4) ) ) ).

% even_set_encode_iff
tff(fact_3229_Code__Numeral_Opositive__def,axiom,
    code_positive = numeral_numeral(code_integer) ).

% Code_Numeral.positive_def
tff(fact_3230_or__nat__unfold,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Ma),N) = $ite(
        Ma = zero_zero(nat),
        N,
        $ite(N = zero_zero(nat),Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% or_nat_unfold
tff(fact_3231_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_3232_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_3233_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_3234_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_3235_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_3236_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_3237_max__Suc__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),N)) ).

% max_Suc_Suc
tff(fact_3238_max__number__of_I1_J,axiom,
    ! [B: $tType] :
      ( ( numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_max(B),aa(num,B,numeral_numeral(B),U)),aa(num,B,numeral_numeral(B),V2)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),U)),aa(num,B,numeral_numeral(B),V2)),aa(num,B,numeral_numeral(B),V2),aa(num,B,numeral_numeral(B),U)) ) ).

% max_number_of(1)
tff(fact_3239_max__0__1_I4_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_max(B),aa(num,B,numeral_numeral(B),X)),zero_zero(B)) = aa(num,B,numeral_numeral(B),X) ) ).

% max_0_1(4)
tff(fact_3240_max__0__1_I3_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_max(B),zero_zero(B)),aa(num,B,numeral_numeral(B),X)) = aa(num,B,numeral_numeral(B),X) ) ).

% max_0_1(3)
tff(fact_3241_max__0__1_I1_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ( aa(B,B,aa(B,fun(B,B),ord_max(B),zero_zero(B)),one_one(B)) = one_one(B) ) ) ).

% max_0_1(1)
tff(fact_3242_max__0__1_I2_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ( aa(B,B,aa(B,fun(B,B),ord_max(B),one_one(B)),zero_zero(B)) = one_one(B) ) ) ).

% max_0_1(2)
tff(fact_3243_max__0__1_I6_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_max(B),aa(num,B,numeral_numeral(B),X)),one_one(B)) = aa(num,B,numeral_numeral(B),X) ) ).

% max_0_1(6)
tff(fact_3244_max__0__1_I5_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_max(B),one_one(B)),aa(num,B,numeral_numeral(B),X)) = aa(num,B,numeral_numeral(B),X) ) ).

% max_0_1(5)
tff(fact_3245_max__number__of_I4_J,axiom,
    ! [B: $tType] :
      ( ( uminus(B)
        & numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))) ) ).

% max_number_of(4)
tff(fact_3246_max__number__of_I3_J,axiom,
    ! [B: $tType] :
      ( ( uminus(B)
        & numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(num,B,numeral_numeral(B),V2)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(num,B,numeral_numeral(B),V2)),aa(num,B,numeral_numeral(B),V2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))) ) ).

% max_number_of(3)
tff(fact_3247_max__number__of_I2_J,axiom,
    ! [B: $tType] :
      ( ( uminus(B)
        & numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_max(B),aa(num,B,numeral_numeral(B),U)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),U)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2)),aa(num,B,numeral_numeral(B),U)) ) ).

% max_number_of(2)
tff(fact_3248_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),N)) ).

% max_numeral_Suc
tff(fact_3249_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_3250_max__diff__distrib__left,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B,Y: B,Z: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),Z) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Z)),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_3251_max__absorb2,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_3252_max__absorb1,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_3253_max__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2),B2,A2) ) ).

% max_def
tff(fact_3254_of__int__max,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: int,Y: int] : aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y)) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(int,B,ring_1_of_int(B),X)),aa(int,B,ring_1_of_int(B),Y)) ) ).

% of_int_max
tff(fact_3255_of__nat__max,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: nat,Y: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(nat,B,semiring_1_of_nat(B),X)),aa(nat,B,semiring_1_of_nat(B),Y)) ) ).

% of_nat_max
tff(fact_3256_nat__mult__max__right,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q2)) ).

% nat_mult_max_right
tff(fact_3257_nat__mult__max__left,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_max_left
tff(fact_3258_nat__add__max__left,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q2)) ).

% nat_add_max_left
tff(fact_3259_nat__add__max__right,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),Q2)) ).

% nat_add_max_right
tff(fact_3260_max__add__distrib__right,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [X: B,Y: B,Z: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),ord_max(B),Y),Z)) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Z)) ) ).

% max_add_distrib_right
tff(fact_3261_max__add__distrib__left,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [X: B,Y: B,Z: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),Z) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Z)),aa(B,B,aa(B,fun(B,B),plus_plus(B),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_3262_max__def__raw,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X4: B,Xa: B] :
          aa(B,B,aa(B,fun(B,B),ord_max(B),X4),Xa) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Xa),Xa,X4) ) ).

% max_def_raw
tff(fact_3263_nat__minus__add__max,axiom,
    ! [N: nat,Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)),Ma) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Ma) ).

% nat_minus_add_max
tff(fact_3264_max__Suc1,axiom,
    ! [N: nat,Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),Ma) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_gk(nat,fun(nat,nat),N),Ma) ).

% max_Suc1
tff(fact_3265_max__Suc2,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Ma),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_gl(nat,fun(nat,nat),N),Ma) ).

% max_Suc2
tff(fact_3266_set__encode__inf,axiom,
    ! [A4: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A4)
     => ( nat_set_encode(A4) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_3267_max_Oabsorb1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_3268_max_Oabsorb2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_3269_max_Obounded__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_max(B),B2),C2)),A2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2) ) ) ) ).

% max.bounded_iff
tff(fact_3270_infinite__int__iff__unbounded__le,axiom,
    ! [S: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S)
    <=> ! [M4: int] :
        ? [N4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M4),aa(int,int,abs_abs(int),N4))
          & aa(set(int),$o,member(int,N4),S) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_3271_set__encode__insert,axiom,
    ! [A4: set(nat),N: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A4)
     => ( ~ aa(set(nat),$o,member(nat,N),A4)
       => ( nat_set_encode(aa(set(nat),set(nat),insert(nat,N),A4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),nat_set_encode(A4)) ) ) ) ).

% set_encode_insert
tff(fact_3272_sum_Oinsert,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),X: B,G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),insert(B,X),A4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,G,X)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4)) ) ) ) ) ).

% sum.insert
tff(fact_3273_atMost__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_3274_sum_Oinsert__if,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),insert(B,X),A4)) = $ite(aa(set(B),$o,member(B,X),A4),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,G,X)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4))) ) ) ) ).

% sum.insert_if
tff(fact_3275_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_3276_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_3277_Icc__eq__insert__lb__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( set_or1337092689740270186AtMost(nat,Ma,N) = aa(set(nat),set(nat),insert(nat,Ma),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3278_atLeastAtMostSuc__conv,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),aa(nat,nat,suc,N))
     => ( set_or1337092689740270186AtMost(nat,Ma,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3279_atLeastAtMost__insertL,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(set(nat),set(nat),insert(nat,Ma),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),N)) = set_or1337092689740270186AtMost(nat,Ma,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_3280_atMost__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_3281_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ aa(set(nat),$o,member(nat,N),nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),Z)) = aa(set(nat),set(nat),insert(nat,N),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_3282_max_OcoboundedI2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_3283_max_OcoboundedI1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_3284_max_Oabsorb__iff2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_3285_max_Oabsorb__iff1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_3286_le__max__iff__disj,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Z: B,X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_3287_max_Ocobounded2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2)) ) ).

% max.cobounded2
tff(fact_3288_max_Ocobounded1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2)) ) ).

% max.cobounded1
tff(fact_3289_max_Oorder__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( A2 = aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_3290_max_OboundedI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_max(B),B2),C2)),A2) ) ) ) ).

% max.boundedI
tff(fact_3291_max_OboundedE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_max(B),B2),C2)),A2)
         => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2) ) ) ) ).

% max.boundedE
tff(fact_3292_max_OorderI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% max.orderI
tff(fact_3293_max_OorderE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( A2 = aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2) ) ) ) ).

% max.orderE
tff(fact_3294_max_Omono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B,A2: B,D2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_max(B),C2),D2)),aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2)) ) ) ) ).

% max.mono
tff(fact_3295_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
    <=> ! [M4: nat] :
        ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N4)
          & aa(set(nat),$o,member(nat,N4),S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_3296_integer__of__num_I3_J,axiom,
    ! [N: num] :
      aa(num,code_integer,code_integer_of_num,aa(num,num,bit1,N)) = $let(
        k: code_integer,
        k:= aa(num,code_integer,code_integer_of_num,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),k),k)),one_one(code_integer)) ) ).

% integer_of_num(3)
tff(fact_3297_even__sum__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_parity(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(C,$o,aa(C,fun(C,$o),dvd_dvd(C),aa(num,C,numeral_numeral(C),aa(num,num,bit0,one2))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_gm(set(B),fun(fun(B,C),fun(B,$o)),A4),F)))) ) ) ) ).

% even_sum_iff
tff(fact_3298_nat_Osplit__sels_I2_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),F1: B,F22: fun(nat,B),Nat: nat] :
      ( aa(B,$o,Pa,case_nat(B,F1,F22,Nat))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ aa(B,$o,Pa,F1) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ aa(B,$o,Pa,aa(nat,B,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_3299_nat_Osplit__sels_I1_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),F1: B,F22: fun(nat,B),Nat: nat] :
      ( aa(B,$o,Pa,case_nat(B,F1,F22,Nat))
    <=> ( ( ( Nat = zero_zero(nat) )
         => aa(B,$o,Pa,F1) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => aa(B,$o,Pa,aa(nat,B,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_3300_drop__bit__rec,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] :
          aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2) = $ite(N = zero_zero(nat),A2,aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))))) ) ).

% drop_bit_rec
tff(fact_3301_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N))) = N ).

% card_Collect_less_nat
tff(fact_3302_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_3303_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_3304_drop__bit__of__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),zero_zero(B)) = zero_zero(B) ) ).

% drop_bit_of_0
tff(fact_3305_drop__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_3306_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_atLeastLessThan
tff(fact_3307_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_3308_drop__bit__drop__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2)) = aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),A2) ) ).

% drop_bit_drop_bit
tff(fact_3309_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ap(nat,fun(nat,$o)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_3310_card_Oinfinite,axiom,
    ! [B: $tType,A4: set(B)] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),nat,finite_card(B),A4) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_3311_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_3312_drop__bit__of__bool,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,B2: $o] :
          aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),aa($o,B,zero_neq_one_of_bool(B),(B2))) = aa($o,B,zero_neq_one_of_bool(B),
            ( ( N = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_3313_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),N = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_3314_prod__constant,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Y: B,A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aTP_Lamp_gn(B,fun(C,B),Y)),A4) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(set(C),nat,finite_card(C),A4)) ) ).

% prod_constant
tff(fact_3315_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_atLeastLessThan_int
tff(fact_3316_drop__bit__of__1,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),one_one(B)) = aa($o,B,zero_neq_one_of_bool(B),N = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_3317_sum__constant,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Y: B,A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_go(B,fun(C,B),Y)),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(C),nat,finite_card(C),A4))),Y) ) ).

% sum_constant
tff(fact_3318_card__Diff__insert,axiom,
    ! [B: $tType,A2: B,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,member(B,A2),A4)
     => ( ~ aa(set(B),$o,member(B,A2),B4)
       => ( aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_3319_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_3320_drop__bit__of__nat,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat,Ma: nat] : aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),aa(nat,B,semiring_1_of_nat(B),Ma)) = aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),Ma)) ) ).

% drop_bit_of_nat
tff(fact_3321_of__nat__drop__bit,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Ma: nat,N: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Ma),N)) = aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),Ma),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% of_nat_drop_bit
tff(fact_3322_drop__bit__integer_Oabs__eq,axiom,
    ! [Xa2: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Xa2),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa2),X)) ).

% drop_bit_integer.abs_eq
tff(fact_3323_n__subsets,axiom,
    ! [B: $tType,A4: set(B),K: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(set(B)),nat,finite_card(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aa(nat,fun(set(B),$o),aTP_Lamp_gp(set(B),fun(nat,fun(set(B),$o)),A4),K))) = aa(nat,nat,binomial(aa(set(B),nat,finite_card(B),A4)),K) ) ) ).

% n_subsets
tff(fact_3324_card__le__if__inj__on__rel,axiom,
    ! [C: $tType,B: $tType,B4: set(B),A4: set(C),R: fun(C,fun(B,$o))] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => ( ! [A3: C] :
            ( aa(set(C),$o,member(C,A3),A4)
           => ? [B7: B] :
                ( aa(set(B),$o,member(B,B7),B4)
                & aa(B,$o,aa(C,fun(B,$o),R,A3),B7) ) )
       => ( ! [A1: C,A22: C,B3: B] :
              ( aa(set(C),$o,member(C,A1),A4)
             => ( aa(set(C),$o,member(C,A22),A4)
               => ( aa(set(B),$o,member(B,B3),B4)
                 => ( aa(B,$o,aa(C,fun(B,$o),R,A1),B3)
                   => ( aa(B,$o,aa(C,fun(B,$o),R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(C),nat,finite_card(C),A4)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3325_card__insert__le,axiom,
    ! [B: $tType,A4: set(B),X: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),aa(set(B),set(B),insert(B,X),A4))) ).

% card_insert_le
tff(fact_3326_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] :
          ( ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) = A2 )
        <=> ( aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2) = zero_zero(B) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_3327_take__bit__drop__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2)) = aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),A2)) ) ).

% take_bit_drop_bit
tff(fact_3328_sum__multicount__gen,axiom,
    ! [B: $tType,C: $tType,S2: set(B),Ta: set(C),R2: fun(B,fun(C,$o)),K: fun(C,nat)] :
      ( aa(set(B),$o,finite_finite2(B),S2)
     => ( aa(set(C),$o,finite_finite2(C),Ta)
       => ( ! [X2: C] :
              ( aa(set(C),$o,member(C,X2),Ta)
             => ( aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,fun(C,$o)),fun(C,fun(B,$o)),aTP_Lamp_fg(set(B),fun(fun(B,fun(C,$o)),fun(C,fun(B,$o))),S2),R2),X2))) = aa(C,nat,K,X2) ) )
         => ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(fun(B,fun(C,$o)),fun(B,nat),aTP_Lamp_gq(set(C),fun(fun(B,fun(C,$o)),fun(B,nat)),Ta),R2)),S2) = aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7311177749621191930dd_sum(C,nat),K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3329_card__eq__sum,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(B),nat,finite_card(B),A4) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aTP_Lamp_gr(B,nat)),A4) ).

% card_eq_sum
tff(fact_3330_card__ge__0__finite,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A4))
     => aa(set(B),$o,finite_finite2(B),A4) ) ).

% card_ge_0_finite
tff(fact_3331_integer__of__num__def,axiom,
    code_integer_of_num = numeral_numeral(code_integer) ).

% integer_of_num_def
tff(fact_3332_card__2__iff_H,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( aa(set(B),nat,finite_card(B),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),S)
          & ? [Xa4: B] :
              ( aa(set(B),$o,member(B,Xa4),S)
              & ( X3 != Xa4 )
              & ! [Xb4: B] :
                  ( aa(set(B),$o,member(B,Xb4),S)
                 => ( ( Xb4 = X3 )
                    | ( Xb4 = Xa4 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3333_obtain__subset__with__card__n,axiom,
    ! [B: $tType,N: nat,S: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(set(B),nat,finite_card(B),S))
     => ~ ! [T5: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T5),S)
           => ( ( aa(set(B),nat,finite_card(B),T5) = N )
             => ~ aa(set(B),$o,finite_finite2(B),T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3334_card__mono,axiom,
    ! [B: $tType,B4: set(B),A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),B4)) ) ) ).

% card_mono
tff(fact_3335_card__seteq,axiom,
    ! [B: $tType,B4: set(B),A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B4)),aa(set(B),nat,finite_card(B),A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_3336_finite__if__finite__subsets__card__bdd,axiom,
    ! [B: $tType,F3: set(B),C6: nat] :
      ( ! [G2: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),G2),F3)
         => ( aa(set(B),$o,finite_finite2(B),G2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),G2)),C6) ) )
     => ( aa(set(B),$o,finite_finite2(B),F3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),F3)),C6) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3337_card__le__sym__Diff,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3338_card__less__Suc2,axiom,
    ! [M: set(nat),I: nat] :
      ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gs(set(nat),fun(nat,fun(nat,$o)),M),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gt(set(nat),fun(nat,fun(nat,$o)),M),I))) ) ) ).

% card_less_Suc2
tff(fact_3339_card__less__Suc,axiom,
    ! [M: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gs(set(nat),fun(nat,fun(nat,$o)),M),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gt(set(nat),fun(nat,fun(nat,$o)),M),I))) ) ) ).

% card_less_Suc
tff(fact_3340_card__less,axiom,
    ! [M: set(nat),I: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_gt(set(nat),fun(nat,fun(nat,$o)),M),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3341_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_3342_subset__card__intvl__is__intvl,axiom,
    ! [A4: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A4),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A4))))
     => ( A4 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_3343_sum__Suc,axiom,
    ! [B: $tType,F: fun(B,nat),A4: set(B)] : aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aTP_Lamp_gu(fun(B,nat),fun(B,nat),F)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)),aa(set(B),nat,finite_card(B),A4)) ).

% sum_Suc
tff(fact_3344_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,N: nat] : aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),one_one(B))) = aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_3345_sum__multicount,axiom,
    ! [B: $tType,C: $tType,S: set(B),T4: set(C),R2: fun(B,fun(C,$o)),K: nat] :
      ( aa(set(B),$o,finite_finite2(B),S)
     => ( aa(set(C),$o,finite_finite2(C),T4)
       => ( ! [X2: C] :
              ( aa(set(C),$o,member(C,X2),T4)
             => ( aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,fun(C,$o)),fun(C,fun(B,$o)),aTP_Lamp_fg(set(B),fun(fun(B,fun(C,$o)),fun(C,fun(B,$o))),S),R2),X2))) = K ) )
         => ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(fun(B,fun(C,$o)),fun(B,nat),aTP_Lamp_gq(set(C),fun(fun(B,fun(C,$o)),fun(B,nat)),T4),R2)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(C),nat,finite_card(C),T4)) ) ) ) ) ).

% sum_multicount
tff(fact_3346_bit__iff__and__drop__bit__eq__1,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),N)
        <=> ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2)),one_one(B)) = one_one(B) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_3347_bits__ident,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2))),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2)) = A2 ) ).

% bits_ident
tff(fact_3348_sum__bounded__below,axiom,
    ! [C: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(C)
        & semiring_1(C) )
     => ! [A4: set(B),K3: C,F: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),K3),aa(B,C,F,I2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(B),nat,finite_card(B),A4))),K3)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)) ) ) ).

% sum_bounded_below
tff(fact_3349_sum__bounded__above,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(C)
        & semiring_1(C) )
     => ! [A4: set(B),F: fun(B,C),K3: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),K3) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(B),nat,finite_card(B),A4))),K3)) ) ) ).

% sum_bounded_above
tff(fact_3350_card__le__Suc0__iff__eq,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
           => ! [Xa4: B] :
                ( aa(set(B),$o,member(B,Xa4),A4)
               => ( X3 = Xa4 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3351_card__le__Suc__iff,axiom,
    ! [B: $tType,N: nat,A4: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(B),nat,finite_card(B),A4))
    <=> ? [A6: B,B8: set(B)] :
          ( ( A4 = aa(set(B),set(B),insert(B,A6),B8) )
          & ~ aa(set(B),$o,member(B,A6),B8)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(set(B),nat,finite_card(B),B8))
          & aa(set(B),$o,finite_finite2(B),B8) ) ) ).

% card_le_Suc_iff
tff(fact_3352_diff__card__le__card__Diff,axiom,
    ! [B: $tType,B4: set(B),A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),B4))),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))) ) ).

% diff_card_le_card_Diff
tff(fact_3353_atLeastAtMostPlus1__int__conv,axiom,
    ! [Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ma),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N))
     => ( set_or1337092689740270186AtMost(int,Ma,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,Ma,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3354_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set(nat),N: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N5)),N) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_3355_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_da(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_da(nat,nat)),S)) ).

% card_sum_le_nat_sum
tff(fact_3356_card__insert__disjoint_H,axiom,
    ! [B: $tType,A4: set(B),X: B] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ~ aa(set(B),$o,member(B,X),A4)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),insert(B,X),A4))),aa(nat,nat,suc,zero_zero(nat))) = aa(set(B),nat,finite_card(B),A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_3357_integer__of__num__triv_I1_J,axiom,
    aa(num,code_integer,code_integer_of_num,one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_3358_integer__of__num_I2_J,axiom,
    ! [N: num] :
      aa(num,code_integer,code_integer_of_num,aa(num,num,bit0,N)) = $let(
        k: code_integer,
        k:= aa(num,code_integer,code_integer_of_num,N),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k) ) ).

% integer_of_num(2)
tff(fact_3359_prod__le__power,axiom,
    ! [B: $tType,C: $tType] :
      ( linordered_semidom(C)
     => ! [A4: set(B),F: fun(B,C),N: C,K: nat] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,I2))
                & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),N) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),K)
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),one_one(C)),N)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),aa(nat,C,aa(C,fun(nat,C),power_power(C),N),K)) ) ) ) ) ).

% prod_le_power
tff(fact_3360_sum__bounded__above__strict,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere8940638589300402666id_add(C)
        & semiring_1(C) )
     => ! [A4: set(B),F: fun(B,C),K3: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,I2)),K3) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A4))
           => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(B),nat,finite_card(B),A4))),K3)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3361_card__insert__le__m1,axiom,
    ! [B: $tType,N: nat,Y: set(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),insert(B,X),Y))),N) ) ) ).

% card_insert_le_m1
tff(fact_3362_prod__gen__delta,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),A2: B,B2: fun(B,C),C2: C] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(C,fun(B,C),aa(fun(B,C),fun(C,fun(B,C)),aTP_Lamp_gv(B,fun(fun(B,C),fun(C,fun(B,C))),A2),B2),C2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,B2,A2)),aa(nat,C,aa(C,fun(nat,C),power_power(C),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),S)),one_one(nat)))),aa(nat,C,aa(C,fun(nat,C),power_power(C),C2),aa(set(B),nat,finite_card(B),S))) ) ) ) ).

% prod_gen_delta
tff(fact_3363_integer__of__num__triv_I2_J,axiom,
    aa(num,code_integer,code_integer_of_num,aa(num,num,bit0,one2)) = aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)) ).

% integer_of_num_triv(2)
tff(fact_3364_slice__eq__mask,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat,Ma: nat,A2: B] : aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),N),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2))) = aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),A2),aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(nat,B,bit_se2239418461657761734s_mask(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N))),aa(B,B,bit_ri4277139882892585799ns_not(B),aa(nat,B,bit_se2239418461657761734s_mask(B),N)))) ) ).

% slice_eq_mask
tff(fact_3365_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_da(nat,nat),Nat) ).

% pred_def
tff(fact_3366_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( aa(set(int),$o,member(int,K),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & aa(set(int),$o,member(int,L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,
          aa(int,fun(int,int),plus_plus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
              & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ).

% and_int.simps
tff(fact_3367_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( Y = $ite(
            ( aa(set(int),$o,member(int,X),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,Xa2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.elims
tff(fact_3368_card__lists__length__le,axiom,
    ! [B: $tType,A4: set(B),N: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(list(B)),nat,finite_card(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_gw(set(B),fun(nat,fun(list(B),$o)),A4),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(B),nat,finite_card(B),A4))),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_3369_horner__sum__eq__sum__funpow,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_0(B)
     => ! [F: fun(C,B),A2: B,Xs: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(list(C),fun(nat,B),aa(B,fun(list(C),fun(nat,B)),aTP_Lamp_gx(fun(C,B),fun(B,fun(list(C),fun(nat,B))),F),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(C),nat,size_size(list(C)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_3370_bot__apply,axiom,
    ! [C: $tType,B: $tType] :
      ( bot(B)
     => ! [X: C] : aa(C,B,bot_bot(fun(C,B)),X) = bot_bot(B) ) ).

% bot_apply
tff(fact_3371_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_3372_max__bot,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),bot_bot(B)),X) = X ) ).

% max_bot
tff(fact_3373_max__bot2,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),X),bot_bot(B)) = X ) ).

% max_bot2
tff(fact_3374_funpow__0,axiom,
    ! [B: $tType,F: fun(B,B),X: B] : aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),zero_zero(nat)),F),X) = X ).

% funpow_0
tff(fact_3375_sum_Oempty,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),bot_bot(set(C))) = zero_zero(B) ) ).

% sum.empty
tff(fact_3376_atLeastatMost__empty__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( ( set_or1337092689740270186AtMost(B,A2,B2) = bot_bot(set(B)) )
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_3377_atLeastatMost__empty__iff2,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( ( bot_bot(set(B)) = set_or1337092689740270186AtMost(B,A2,B2) )
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_3378_atLeastatMost__empty,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( set_or1337092689740270186AtMost(B,A2,B2) = bot_bot(set(B)) ) ) ) ).

% atLeastatMost_empty
tff(fact_3379_atLeastLessThan__empty,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( set_or7035219750837199246ssThan(B,A2,B2) = bot_bot(set(B)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_3380_atLeastLessThan__empty__iff2,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( ( bot_bot(set(B)) = set_or7035219750837199246ssThan(B,A2,B2) )
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_3381_atLeastLessThan__empty__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B,B2: B] :
          ( ( set_or7035219750837199246ssThan(B,A2,B2) = bot_bot(set(B)) )
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_3382_prod_Oempty,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,B)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),G),bot_bot(set(C))) = one_one(B) ) ).

% prod.empty
tff(fact_3383_card_Oempty,axiom,
    ! [B: $tType] : aa(set(B),nat,finite_card(B),bot_bot(set(B))) = zero_zero(nat) ).

% card.empty
tff(fact_3384_atLeastAtMost__singleton__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( set_or1337092689740270186AtMost(B,A2,B2) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3385_atLeastAtMost__singleton,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B] : set_or1337092689740270186AtMost(B,A2,A2) = aa(set(B),set(B),insert(B,A2),bot_bot(set(B))) ) ).

% atLeastAtMost_singleton
tff(fact_3386_some__insert__self,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( S != bot_bot(set(B)) )
     => ( aa(set(B),set(B),insert(B,fChoice(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S))),S) = S ) ) ).

% some_insert_self
tff(fact_3387_card__0__eq,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ( aa(set(B),nat,finite_card(B),A4) = zero_zero(nat) )
      <=> ( A4 = bot_bot(set(B)) ) ) ) ).

% card_0_eq
tff(fact_3388_card__doubleton__eq__2__iff,axiom,
    ! [B: $tType,A2: B,B2: B] :
      ( ( aa(set(B),nat,finite_card(B),aa(set(B),set(B),insert(B,A2),aa(set(B),set(B),insert(B,B2),bot_bot(set(B))))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_3389_funpow__mult,axiom,
    ! [B: $tType,N: nat,Ma: nat,F: fun(B,B)] : aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Ma),F)) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)),F) ).

% funpow_mult
tff(fact_3390_funpow__swap1,axiom,
    ! [B: $tType,F: fun(B,B),N: nat,X: B] : aa(B,B,F,aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),X)) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),aa(B,B,F,X)) ).

% funpow_swap1
tff(fact_3391_bot__fun__def,axiom,
    ! [B: $tType,C: $tType] :
      ( bot(C)
     => ! [X4: B] : aa(B,C,bot_bot(fun(B,C)),X4) = bot_bot(C) ) ).

% bot_fun_def
tff(fact_3392_set__notEmptyE,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( S != bot_bot(set(B)) )
     => ~ ! [X2: B] : ~ aa(set(B),$o,member(B,X2),S) ) ).

% set_notEmptyE
tff(fact_3393_memb__imp__not__empty,axiom,
    ! [B: $tType,X: B,S: set(B)] :
      ( aa(set(B),$o,member(B,X),S)
     => ( S != bot_bot(set(B)) ) ) ).

% memb_imp_not_empty
tff(fact_3394_bot_Oextremum__strict,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),bot_bot(B)) ) ).

% bot.extremum_strict
tff(fact_3395_bot_Onot__eq__extremum,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [A2: B] :
          ( ( A2 != bot_bot(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),bot_bot(B)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_3396_bot_Oextremum,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),bot_bot(B)),A2) ) ).

% bot.extremum
tff(fact_3397_bot_Oextremum__unique,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),bot_bot(B))
        <=> ( A2 = bot_bot(B) ) ) ) ).

% bot.extremum_unique
tff(fact_3398_bot_Oextremum__uniqueI,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),bot_bot(B))
         => ( A2 = bot_bot(B) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_3399_not__empty__eq__Iic__eq__empty,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [H: B] : bot_bot(set(B)) != aa(B,set(B),set_ord_atMost(B),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_3400_some__elem,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( S != bot_bot(set(B)) )
     => aa(set(B),$o,member(B,fChoice(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S))),S) ) ).

% some_elem
tff(fact_3401_some__in__eq,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,member(B,fChoice(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4))),A4)
    <=> ( A4 != bot_bot(set(B)) ) ) ).

% some_in_eq
tff(fact_3402_diff__shunt__var,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y) = bot_bot(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% diff_shunt_var
tff(fact_3403_finite__has__minimal,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ? [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
                & ! [Xa: B] :
                    ( aa(set(B),$o,member(B,Xa),A4)
                   => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3404_finite__has__maximal,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ? [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
                & ! [Xa: B] :
                    ( aa(set(B),$o,member(B,Xa),A4)
                   => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3405_ex__min__if__finite,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [S: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ? [X2: B] :
                ( aa(set(B),$o,member(B,X2),S)
                & ~ ? [Xa: B] :
                      ( aa(set(B),$o,member(B,Xa),S)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xa),X2) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3406_infinite__growing,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X5: set(B)] :
          ( ( X5 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => ? [Xa: B] :
                    ( aa(set(B),$o,member(B,Xa),X5)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Xa) ) )
           => ~ aa(set(B),$o,finite_finite2(B),X5) ) ) ) ).

% infinite_growing
tff(fact_3407_subset__minus__empty,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4) = bot_bot(set(B)) ) ) ).

% subset_minus_empty
tff(fact_3408_insert__minus__eq,axiom,
    ! [B: $tType,X: B,Y: B,A4: set(B)] :
      ( ( X != Y )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),insert(B,X),A4)),aa(set(B),set(B),insert(B,Y),bot_bot(set(B)))) = aa(set(B),set(B),insert(B,X),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,Y),bot_bot(set(B))))) ) ) ).

% insert_minus_eq
tff(fact_3409_set__minus__singleton__eq,axiom,
    ! [B: $tType,X: B,X5: set(B)] :
      ( ~ aa(set(B),$o,member(B,X),X5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),X5),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_3410_atLeastAtMost__singleton_H,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(B,A2,B2) = aa(set(B),set(B),insert(B,A2),bot_bot(set(B))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3411_arg__min__if__finite_I1_J,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [S: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => aa(set(B),$o,member(B,lattic7623131987881927897min_on(B,C,F,S)),S) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_3412_length__pos__if__in__set,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_3413_sorted__simps_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Ys: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Ys))
        <=> ( ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Ys))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),X3) )
            & sorted_wrt(B,ord_less_eq(B),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_3414_all__set__conv__all__nth,axiom,
    ! [B: $tType,Xs: list(B),Pa: fun(B,$o)] :
      ( ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Xs))
         => aa(B,$o,Pa,X3) )
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
         => aa(B,$o,Pa,aa(nat,B,nth(B,Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_3415_all__nth__imp__all__set,axiom,
    ! [B: $tType,Xs: list(B),Pa: fun(B,$o),X: B] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Xs))
         => aa(B,$o,Pa,aa(nat,B,nth(B,Xs),I2)) )
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
       => aa(B,$o,Pa,X) ) ) ).

% all_nth_imp_all_set
tff(fact_3416_in__set__conv__nth,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
          & ( aa(nat,B,nth(B,Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3417_list__ball__nth,axiom,
    ! [B: $tType,N: nat,Xs: list(B),Pa: fun(B,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
           => aa(B,$o,Pa,X2) )
       => aa(B,$o,Pa,aa(nat,B,nth(B,Xs),N)) ) ) ).

% list_ball_nth
tff(fact_3418_nth__mem,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => aa(set(B),$o,member(B,aa(nat,B,nth(B,Xs),N)),aa(list(B),set(B),set2(B),Xs)) ) ).

% nth_mem
tff(fact_3419_all__set__conv__nth,axiom,
    ! [B: $tType,L: list(B),Pa: fun(B,$o)] :
      ( ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),L))
         => aa(B,$o,Pa,X3) )
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),L))
         => aa(B,$o,Pa,aa(nat,B,nth(B,L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_3420_card__length,axiom,
    ! [B: $tType,Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(list(B),set(B),set2(B),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% card_length
tff(fact_3421_ex__is__arg__min__if__finite,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [S: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ? [X_13: B] : aa(B,$o,lattic501386751177426532rg_min(B,C,F,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S)),X_13) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_3422_finite__ranking__induct,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [S: set(B),Pa: fun(set(B),$o),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),$o,Pa,bot_bot(set(B)))
           => ( ! [X2: B,S6: set(B)] :
                  ( aa(set(B),$o,finite_finite2(B),S6)
                 => ( ! [Y3: B] :
                        ( aa(set(B),$o,member(B,Y3),S6)
                       => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y3)),aa(B,C,F,X2)) )
                   => ( aa(set(B),$o,Pa,S6)
                     => aa(set(B),$o,Pa,aa(set(B),set(B),insert(B,X2),S6)) ) ) )
             => aa(set(B),$o,Pa,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3423_finite__linorder__min__induct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),Pa: fun(set(B),$o)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,Pa,bot_bot(set(B)))
           => ( ! [B3: B,A7: set(B)] :
                  ( aa(set(B),$o,finite_finite2(B),A7)
                 => ( ! [X4: B] :
                        ( aa(set(B),$o,member(B,X4),A7)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),X4) )
                   => ( aa(set(B),$o,Pa,A7)
                     => aa(set(B),$o,Pa,aa(set(B),set(B),insert(B,B3),A7)) ) ) )
             => aa(set(B),$o,Pa,A4) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3424_finite__linorder__max__induct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),Pa: fun(set(B),$o)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,Pa,bot_bot(set(B)))
           => ( ! [B3: B,A7: set(B)] :
                  ( aa(set(B),$o,finite_finite2(B),A7)
                 => ( ! [X4: B] :
                        ( aa(set(B),$o,member(B,X4),A7)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),B3) )
                   => ( aa(set(B),$o,Pa,A7)
                     => aa(set(B),$o,Pa,aa(set(B),set(B),insert(B,B3),A7)) ) ) )
             => aa(set(B),$o,Pa,A4) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3425_card__eq__0__iff,axiom,
    ! [B: $tType,A4: set(B)] :
      ( ( aa(set(B),nat,finite_card(B),A4) = zero_zero(nat) )
    <=> ( ( A4 = bot_bot(set(B)) )
        | ~ aa(set(B),$o,finite_finite2(B),A4) ) ) ).

% card_eq_0_iff
tff(fact_3426_card__1__singletonE,axiom,
    ! [B: $tType,A4: set(B)] :
      ( ( aa(set(B),nat,finite_card(B),A4) = one_one(nat) )
     => ~ ! [X2: B] : A4 != aa(set(B),set(B),insert(B,X2),bot_bot(set(B))) ) ).

% card_1_singletonE
tff(fact_3427_remove__subset,axiom,
    ! [B: $tType,X: B,S: set(B)] :
      ( aa(set(B),$o,member(B,X),S)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))),S) ) ).

% remove_subset
tff(fact_3428_bot_Oordering__top__axioms,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ordering_top(B,aTP_Lamp_gy(B,fun(B,$o)),aTP_Lamp_gz(B,fun(B,$o)),bot_bot(B)) ) ).

% bot.ordering_top_axioms
tff(fact_3429_of__nat__def,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] : aa(nat,B,semiring_1_of_nat(B),N) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),aa(B,fun(B,B),plus_plus(B),one_one(B))),zero_zero(B)) ) ).

% of_nat_def
tff(fact_3430_numeral__add__unfold__funpow,axiom,
    ! [B: $tType] :
      ( semiring_numeral(B)
     => ! [K: num,A2: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),K)),A2) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(num,nat,numeral_numeral(nat),K)),aa(B,fun(B,B),plus_plus(B),one_one(B))),A2) ) ).

% numeral_add_unfold_funpow
tff(fact_3431_arg__min__least,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [S: set(B),Y: B,F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(B),$o,member(B,Y),S)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,lattic7623131987881927897min_on(B,C,F,S))),aa(B,C,F,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_3432_arg__min__if__finite_I2_J,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [S: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ~ ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),S)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X4)),aa(B,C,F,lattic7623131987881927897min_on(B,C,F,S))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_3433_sum__pos,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [I5: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I2: B] :
                  ( aa(set(B),$o,member(B,I2),I5)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(B,C,F,I2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),zero_zero(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),I5)) ) ) ) ) ).

% sum_pos
tff(fact_3434_less__1__prod,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_idom(C)
     => ! [I5: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I2: B] :
                  ( aa(set(B),$o,member(B,I2),I5)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less(C),one_one(C)),aa(B,C,F,I2)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),one_one(C)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_3435_card__gt__0__iff,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A4))
    <=> ( ( A4 != bot_bot(set(B)) )
        & aa(set(B),$o,finite_finite2(B),A4) ) ) ).

% card_gt_0_iff
tff(fact_3436_card__Suc__eq,axiom,
    ! [B: $tType,A4: set(B),K: nat] :
      ( ( aa(set(B),nat,finite_card(B),A4) = aa(nat,nat,suc,K) )
    <=> ? [B9: B,B8: set(B)] :
          ( ( A4 = aa(set(B),set(B),insert(B,B9),B8) )
          & ~ aa(set(B),$o,member(B,B9),B8)
          & ( aa(set(B),nat,finite_card(B),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(B)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3437_card__eq__SucD,axiom,
    ! [B: $tType,A4: set(B),K: nat] :
      ( ( aa(set(B),nat,finite_card(B),A4) = aa(nat,nat,suc,K) )
     => ? [B3: B,B10: set(B)] :
          ( ( A4 = aa(set(B),set(B),insert(B,B3),B10) )
          & ~ aa(set(B),$o,member(B,B3),B10)
          & ( aa(set(B),nat,finite_card(B),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(B)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3438_card__1__singleton__iff,axiom,
    ! [B: $tType,A4: set(B)] :
      ( ( aa(set(B),nat,finite_card(B),A4) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X3: B] : A4 = aa(set(B),set(B),insert(B,X3),bot_bot(set(B))) ) ).

% card_1_singleton_iff
tff(fact_3439_card__1__singletonI,axiom,
    ! [B: $tType,S: set(B),X: B] :
      ( aa(set(B),$o,finite_finite2(B),S)
     => ( ( aa(set(B),nat,finite_card(B),S) = one_one(nat) )
       => ( aa(set(B),$o,member(B,X),S)
         => ( S = aa(set(B),set(B),insert(B,X),bot_bot(set(B))) ) ) ) ) ).

% card_1_singletonI
tff(fact_3440_card__Diff1__le,axiom,
    ! [B: $tType,A4: set(B),X: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))),aa(set(B),nat,finite_card(B),A4)) ).

% card_Diff1_le
tff(fact_3441_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] : set_or7035219750837199246ssThan(B,A2,B2) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),aa(set(B),set(B),insert(B,B2),bot_bot(set(B)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3442_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_3443_numeral__unfold__funpow,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [K: num] : aa(num,B,numeral_numeral(B),K) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(num,nat,numeral_numeral(nat),K)),aa(B,fun(B,B),plus_plus(B),one_one(B))),zero_zero(B)) ) ).

% numeral_unfold_funpow
tff(fact_3444_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),insert(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_3445_finite__lists__length__eq,axiom,
    ! [B: $tType,A4: set(B),N: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(set(list(B)),$o,finite_finite2(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_ha(set(B),fun(nat,fun(list(B),$o)),A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3446_nth__equal__first__eq,axiom,
    ! [B: $tType,X: B,Xs: list(B),N: nat] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
       => ( ( aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_3447_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(B,ord_less(B),L2)
               => ( ( aa(list(B),set(B),set2(B),L2) = A4 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != aa(set(B),nat,finite_card(B),A4) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_3448_prod__mono__strict,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_semidom(C)
     => ! [A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),A4)
               => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,I2))
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,I2)),aa(B,C,G,I2)) ) )
           => ( ( A4 != bot_bot(set(B)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3449_sum_Oremove,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),X: B,G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,G,X)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_3450_sum_Oinsert__remove,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),insert(B,X),A4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,G,X)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3451_card__2__iff,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( aa(set(B),nat,finite_card(B),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X3: B,Y5: B] :
          ( ( S = aa(set(B),set(B),insert(B,X3),aa(set(B),set(B),insert(B,Y5),bot_bot(set(B)))) )
          & ( X3 != Y5 ) ) ) ).

% card_2_iff
tff(fact_3452_card__3__iff,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( aa(set(B),nat,finite_card(B),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X3: B,Y5: B,Z5: B] :
          ( ( S = aa(set(B),set(B),insert(B,X3),aa(set(B),set(B),insert(B,Y5),aa(set(B),set(B),insert(B,Z5),bot_bot(set(B))))) )
          & ( X3 != Y5 )
          & ( Y5 != Z5 )
          & ( X3 != Z5 ) ) ) ).

% card_3_iff
tff(fact_3453_card__lists__length__eq,axiom,
    ! [B: $tType,A4: set(B),N: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(list(B)),nat,finite_card(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_ha(set(B),fun(nat,fun(list(B),$o)),A4),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(B),nat,finite_card(B),A4)),N) ) ) ).

% card_lists_length_eq
tff(fact_3454_card__Diff__singleton__if,axiom,
    ! [B: $tType,A4: set(B),X: B] :
      aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) = $ite(aa(set(B),$o,member(B,X),A4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),A4)),one_one(nat)),aa(set(B),nat,finite_card(B),A4)) ).

% card_Diff_singleton_if
tff(fact_3455_card__Diff__singleton,axiom,
    ! [B: $tType,X: B,A4: set(B)] :
      ( aa(set(B),$o,member(B,X),A4)
     => ( aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),A4)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3456_finite__lists__length__le,axiom,
    ! [B: $tType,A4: set(B),N: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(set(list(B)),$o,finite_finite2(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_gw(set(B),fun(nat,fun(list(B),$o)),A4),N))) ) ).

% finite_lists_length_le
tff(fact_3457_sum_Odelta__remove,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),A2: B,B2: fun(B,C),C2: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_hb(B,fun(fun(B,C),fun(fun(B,C),fun(B,C))),A2),B2),C2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,B2,A2)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3458_prod_Odelta__remove,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),A2: B,B2: fun(B,C),C2: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_hc(B,fun(fun(B,C),fun(fun(B,C),fun(B,C))),A2),B2),C2)),S) = $ite(aa(set(B),$o,member(B,A2),S),aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,B2,A2)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3459_member__le__sum,axiom,
    ! [C: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(C)
        & semiring_1(C) )
     => ! [I: B,A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,member(B,I),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,I),bot_bot(set(B)))))
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,F,X2)) )
           => ( aa(set(B),$o,finite_finite2(B),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)) ) ) ) ) ).

% member_le_sum
tff(fact_3460_prod__diff1,axiom,
    ! [C: $tType,B: $tType] :
      ( semidom_divide(C)
     => ! [A4: set(B),F: fun(B,C),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( aa(B,C,F,A2) != zero_zero(C) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) = $ite(aa(set(B),$o,member(B,A2),A4),aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),aa(B,C,F,A2)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)) ) ) ) ) ).

% prod_diff1
tff(fact_3461_sum__bounded__above__divide,axiom,
    ! [B: $tType,C: $tType] :
      ( linordered_field(C)
     => ! [A4: set(B),F: fun(B,C),K3: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),aa(C,C,aa(C,fun(C,C),divide_divide(C),K3),aa(nat,C,semiring_1_of_nat(C),aa(set(B),nat,finite_card(B),A4)))) )
         => ( aa(set(B),$o,finite_finite2(B),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),K3) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3462_singleton__conv2,axiom,
    ! [B: $tType,A2: B] : aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),fequal(B),A2)) = aa(set(B),set(B),insert(B,A2),bot_bot(set(B))) ).

% singleton_conv2
tff(fact_3463_singleton__conv,axiom,
    ! [B: $tType,A2: B] : aa(fun(B,$o),set(B),collect(B),aTP_Lamp_af(B,fun(B,$o),A2)) = aa(set(B),set(B),insert(B,A2),bot_bot(set(B))) ).

% singleton_conv
tff(fact_3464_card__lists__distinct__length__eq,axiom,
    ! [B: $tType,A4: set(B),K: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(B),nat,finite_card(B),A4))
       => ( aa(set(list(B)),nat,finite_card(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_hd(set(B),fun(nat,fun(list(B),$o)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_da(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(B),nat,finite_card(B),A4)),K)),one_one(nat)),aa(set(B),nat,finite_card(B),A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3465_card__lists__distinct__length__eq_H,axiom,
    ! [B: $tType,K: nat,A4: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(B),nat,finite_card(B),A4))
     => ( aa(set(list(B)),nat,finite_card(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(set(B),fun(list(B),$o),aTP_Lamp_he(nat,fun(set(B),fun(list(B),$o)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_da(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(B),nat,finite_card(B),A4)),K)),one_one(nat)),aa(set(B),nat,finite_card(B),A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3466_relpowp__fun__conv,axiom,
    ! [B: $tType,N: nat,Pa: fun(B,fun(B,$o)),X: B,Y: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),N),Pa),X),Y)
    <=> ? [F5: fun(nat,B)] :
          ( ( aa(nat,B,F5,zero_zero(nat)) = X )
          & ( aa(nat,B,F5,N) = Y )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
             => aa(B,$o,aa(B,fun(B,$o),Pa,aa(nat,B,F5,I3)),aa(nat,B,F5,aa(nat,nat,suc,I3))) ) ) ) ).

% relpowp_fun_conv
tff(fact_3467_bit__0__eq,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ( aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),zero_zero(B)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_3468_relpowp__1,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o))] : aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),one_one(nat)),Pa) = Pa ).

% relpowp_1
tff(fact_3469_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3470_set__encode__empty,axiom,
    nat_set_encode(bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_3471_atLeastLessThan__singleton,axiom,
    ! [Ma: nat] : set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,Ma)) = aa(set(nat),set(nat),insert(nat,Ma),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_3472_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3473_finite__lists__distinct__length__eq,axiom,
    ! [B: $tType,A4: set(B),N: nat] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(set(list(B)),$o,finite_finite2(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_hd(set(B),fun(nat,fun(list(B),$o)),A4),N))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3474_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_3475_relpowp__bot,axiom,
    ! [B: $tType,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),N),bot_bot(fun(B,fun(B,$o)))) = bot_bot(fun(B,fun(B,$o))) ) ) ).

% relpowp_bot
tff(fact_3476_atLeastLessThan0,axiom,
    ! [Ma: nat] : set_or7035219750837199246ssThan(nat,Ma,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3477_distinct__finite__set,axiom,
    ! [B: $tType,X: set(B)] : aa(set(list(B)),$o,finite_finite2(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aTP_Lamp_hf(set(B),fun(list(B),$o),X))) ).

% distinct_finite_set
tff(fact_3478_strict__sorted__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B)] :
          ( sorted_wrt(B,ord_less(B),L)
        <=> ( sorted_wrt(B,ord_less_eq(B),L)
            & distinct(B,L) ) ) ) ).

% strict_sorted_iff
tff(fact_3479_distinct__length__le,axiom,
    ! [B: $tType,Ys: list(B),Xs: list(B)] :
      ( distinct(B,Ys)
     => ( ( aa(list(B),set(B),set2(B),Ys) = aa(list(B),set(B),set2(B),Xs) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Ys)),aa(list(B),nat,size_size(list(B)),Xs)) ) ) ).

% distinct_length_le
tff(fact_3480_sorted__distinct__set__unique,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Ys: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( distinct(B,Xs)
           => ( sorted_wrt(B,ord_less_eq(B),Ys)
             => ( distinct(B,Ys)
               => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_3481_distinct__conv__nth,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( distinct(B,Xs)
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(B),nat,size_size(list(B)),Xs))
             => ( ( I3 != J3 )
               => ( aa(nat,B,nth(B,Xs),I3) != aa(nat,B,nth(B,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_3482_nth__eq__iff__index__eq,axiom,
    ! [B: $tType,Xs: list(B),I: nat,J: nat] :
      ( distinct(B,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
         => ( ( aa(nat,B,nth(B,Xs),I) = aa(nat,B,nth(B,Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_3483_card__distinct,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( aa(set(B),nat,finite_card(B),aa(list(B),set(B),set2(B),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) )
     => distinct(B,Xs) ) ).

% card_distinct
tff(fact_3484_distinct__card,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( distinct(B,Xs)
     => ( aa(set(B),nat,finite_card(B),aa(list(B),set(B),set2(B),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ) ).

% distinct_card
tff(fact_3485_relpowp_Osimps_I1_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o))] : aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),zero_zero(nat)),R2) = fequal(B) ).

% relpowp.simps(1)
tff(fact_3486_relpowp__0__E,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),X: B,Y: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),zero_zero(nat)),Pa),X),Y)
     => ( X = Y ) ) ).

% relpowp_0_E
tff(fact_3487_relpowp__0__I,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),X: B] : aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),zero_zero(nat)),Pa),X),X) ).

% relpowp_0_I
tff(fact_3488_finite__sorted__distinct__unique,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ? [X2: list(B)] :
              ( ( aa(list(B),set(B),set2(B),X2) = A4 )
              & sorted_wrt(B,ord_less_eq(B),X2)
              & distinct(B,X2)
              & ! [Y3: list(B)] :
                  ( ( ( aa(list(B),set(B),set2(B),Y3) = A4 )
                    & sorted_wrt(B,ord_less_eq(B),Y3)
                    & distinct(B,Y3) )
                 => ( Y3 = X2 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_3489_distinct__Ex1,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( distinct(B,Xs)
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
       => ? [X2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(list(B),nat,size_size(list(B)),Xs))
            & ( aa(nat,B,nth(B,Xs),X2) = X )
            & ! [Y3: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y3),aa(list(B),nat,size_size(list(B)),Xs))
                  & ( aa(nat,B,nth(B,Xs),Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_3490_distinct__finite__subset,axiom,
    ! [B: $tType,X: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),X)
     => aa(set(list(B)),$o,finite_finite2(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aTP_Lamp_hg(set(B),fun(list(B),$o),X))) ) ).

% distinct_finite_subset
tff(fact_3491_atLeastLessThanSuc,axiom,
    ! [Ma: nat,N: nat] :
      set_or7035219750837199246ssThan(nat,Ma,aa(nat,nat,suc,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N),aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,Ma,N)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_3492_Set_Oempty__def,axiom,
    ! [B: $tType] : bot_bot(set(B)) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_hh(B,$o)) ).

% Set.empty_def
tff(fact_3493_insert__compr,axiom,
    ! [B: $tType,A2: B,B4: set(B)] : aa(set(B),set(B),insert(B,A2),B4) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_hi(B,fun(set(B),fun(B,$o)),A2),B4)) ).

% insert_compr
tff(fact_3494_insert__Collect,axiom,
    ! [B: $tType,A2: B,Pa: fun(B,$o)] : aa(set(B),set(B),insert(B,A2),aa(fun(B,$o),set(B),collect(B),Pa)) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_hj(B,fun(fun(B,$o),fun(B,$o)),A2),Pa)) ).

% insert_Collect
tff(fact_3495_set__diff__eq,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_hk(set(B),fun(set(B),fun(B,$o)),A4),B4)) ).

% set_diff_eq
tff(fact_3496_minus__set__def,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),minus_minus(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),B4))) ).

% minus_set_def
tff(fact_3497_less__set__def,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),A4),B4)
    <=> aa(fun(B,$o),$o,aa(fun(B,$o),fun(fun(B,$o),$o),ord_less(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),B4)) ) ).

% less_set_def
tff(fact_3498_distinct__sorted__strict__mono__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B),I: nat,J: nat] :
          ( distinct(B,L)
         => ( sorted_wrt(B,ord_less_eq(B),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,nth(B,L),I)),aa(nat,B,nth(B,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_3499_distinct__sorted__mono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B),I: nat,J: nat] :
          ( sorted_wrt(B,ord_less_eq(B),L)
         => ( distinct(B,L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,nth(B,L),I)),aa(nat,B,nth(B,L),J)) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_3500_distinct__sorted__mono__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B),I: nat,J: nat] :
          ( distinct(B,L)
         => ( sorted_wrt(B,ord_less_eq(B),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,L),I)),aa(nat,B,nth(B,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_3501_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atMost(nat),N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3502_atLeastLessThan__nat__numeral,axiom,
    ! [Ma: nat,K: num] :
      set_or7035219750837199246ssThan(nat,Ma,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),pred_numeral(K)),aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,Ma,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_3503_relpowp__E2,axiom,
    ! [B: $tType,N: nat,Pa: fun(B,fun(B,$o)),X: B,Z: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),N),Pa),X),Z)
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: B,M2: nat] :
              ( ( N = aa(nat,nat,suc,M2) )
             => ( aa(B,$o,aa(B,fun(B,$o),Pa,X),Y2)
               => ~ aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),M2),Pa),Y2),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_3504_relpowp__E,axiom,
    ! [B: $tType,N: nat,Pa: fun(B,fun(B,$o)),X: B,Z: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),N),Pa),X),Z)
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: B,M2: nat] :
              ( ( N = aa(nat,nat,suc,M2) )
             => ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),M2),Pa),X),Y2)
               => ~ aa(B,$o,aa(B,fun(B,$o),Pa,Y2),Z) ) ) ) ) ).

% relpowp_E
tff(fact_3505_Collect__conv__if,axiom,
    ! [B: $tType,A2: B,Pa: fun(B,$o)] :
      aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_hl(B,fun(fun(B,$o),fun(B,$o)),A2),Pa)) = $ite(aa(B,$o,Pa,A2),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))),bot_bot(set(B))) ).

% Collect_conv_if
tff(fact_3506_Collect__conv__if2,axiom,
    ! [B: $tType,A2: B,Pa: fun(B,$o)] :
      aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_hm(B,fun(fun(B,$o),fun(B,$o)),A2),Pa)) = $ite(aa(B,$o,Pa,A2),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))),bot_bot(set(B))) ).

% Collect_conv_if2
tff(fact_3507_mergesort__remdups__correct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B)] :
          ( distinct(B,aa(list(B),list(B),mergesort_remdups(B),L))
          & sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),mergesort_remdups(B),L))
          & ( aa(list(B),set(B),set2(B),aa(list(B),list(B),mergesort_remdups(B),L)) = aa(list(B),set(B),set2(B),L) ) ) ) ).

% mergesort_remdups_correct
tff(fact_3508_sum__count__set,axiom,
    ! [B: $tType,Xs: list(B),X5: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),X5)
     => ( aa(set(B),$o,finite_finite2(B),X5)
       => ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),count_list(B,Xs)),X5) = aa(list(B),nat,size_size(list(B)),Xs) ) ) ) ).

% sum_count_set
tff(fact_3509_sum__diff1_H__aux,axiom,
    ! [C: $tType,B: $tType] :
      ( ab_group_add(C)
     => ! [F3: set(B),I5: set(B),F: fun(B,C),I: B] :
          ( aa(set(B),$o,finite_finite2(B),F3)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_hn(set(B),fun(fun(B,C),fun(B,$o)),I5),F))),F3)
           => ( groups1027152243600224163dd_sum(B,C,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),I5),aa(set(B),set(B),insert(B,I),bot_bot(set(B))))) = $ite(aa(set(B),$o,member(B,I),I5),aa(C,C,aa(C,fun(C,C),minus_minus(C),groups1027152243600224163dd_sum(B,C,F,I5)),aa(B,C,F,I)),groups1027152243600224163dd_sum(B,C,F,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3510_Nat_Ofunpow__code__def,axiom,
    ! [B: $tType] : funpow(B) = compow(fun(B,B)) ).

% Nat.funpow_code_def
tff(fact_3511_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_3512_greaterThanLessThan__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,L: B,U: B] :
          ( aa(set(B),$o,member(B,I),set_or5935395276787703475ssThan(B,L,U))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),I)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_3513_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or5935395276787703475ssThan(int,L,U)) ).

% finite_greaterThanLessThan_int
tff(fact_3514_greaterThanLessThan__empty__iff2,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( ( bot_bot(set(B)) = set_or5935395276787703475ssThan(B,A2,B2) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_3515_greaterThanLessThan__empty__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( ( set_or5935395276787703475ssThan(B,A2,B2) = bot_bot(set(B)) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_3516_greaterThanLessThan__empty,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,K: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),K)
         => ( set_or5935395276787703475ssThan(B,K,L) = bot_bot(set(B)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_3517_infinite__Ioo__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(set(B),$o,finite_finite2(B),set_or5935395276787703475ssThan(B,A2,B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_3518_sum_Oempty_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [P2: fun(C,B)] : groups1027152243600224163dd_sum(C,B,P2,bot_bot(set(C))) = zero_zero(B) ) ).

% sum.empty'
tff(fact_3519_count__notin,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
     => ( aa(B,nat,count_list(B,Xs),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_3520_sum_Oinsert_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(B),P2: fun(B,C),I: B] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fn(set(B),fun(fun(B,C),fun(B,$o)),I5),P2)))
         => ( groups1027152243600224163dd_sum(B,C,P2,aa(set(B),set(B),insert(B,I),I5)) = $ite(aa(set(B),$o,member(B,I),I5),groups1027152243600224163dd_sum(B,C,P2,I5),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,P2,I)),groups1027152243600224163dd_sum(B,C,P2,I5))) ) ) ) ).

% sum.insert'
tff(fact_3521_bot2E,axiom,
    ! [B: $tType,C: $tType,X: B,Y: C] : ~ aa(C,$o,aa(B,fun(C,$o),bot_bot(fun(B,fun(C,$o))),X),Y) ).

% bot2E
tff(fact_3522_sum_Onon__neutral_H,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B),I5: set(C)] : groups1027152243600224163dd_sum(C,B,G,aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aTP_Lamp_ho(fun(C,B),fun(set(C),fun(C,$o)),G),I5))) = groups1027152243600224163dd_sum(C,B,G,I5) ) ).

% sum.non_neutral'
tff(fact_3523_infinite__Ioo,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ aa(set(B),$o,finite_finite2(B),set_or5935395276787703475ssThan(B,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_3524_sum_Odistrib__triv_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( groups1027152243600224163dd_sum(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_hp(fun(B,C),fun(fun(B,C),fun(B,C)),G),H),I5) = aa(C,C,aa(C,fun(C,C),plus_plus(C),groups1027152243600224163dd_sum(B,C,G,I5)),groups1027152243600224163dd_sum(B,C,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3525_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or5935395276787703475ssThan(B,A2,B2)),set_or5935395276787703475ssThan(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_3526_sum_Omono__neutral__left_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),T4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,G,X2) = zero_zero(C) ) )
           => ( groups1027152243600224163dd_sum(B,C,G,S) = groups1027152243600224163dd_sum(B,C,G,T4) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3527_sum_Omono__neutral__right_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),T4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,G,X2) = zero_zero(C) ) )
           => ( groups1027152243600224163dd_sum(B,C,G,T4) = groups1027152243600224163dd_sum(B,C,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3528_sum_Omono__neutral__cong__left_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),T4: set(B),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,H,I2) = zero_zero(C) ) )
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),S)
                 => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
             => ( groups1027152243600224163dd_sum(B,C,G,S) = groups1027152243600224163dd_sum(B,C,H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3529_sum_Omono__neutral__cong__right_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),T4: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,G,X2) = zero_zero(C) ) )
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),S)
                 => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
             => ( groups1027152243600224163dd_sum(B,C,G,T4) = groups1027152243600224163dd_sum(B,C,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3530_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_3531_sum_Odistrib_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fn(set(B),fun(fun(B,C),fun(B,$o)),I5),G)))
         => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fn(set(B),fun(fun(B,C),fun(B,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_hp(fun(B,C),fun(fun(B,C),fun(B,C)),G),H),I5) = aa(C,C,aa(C,fun(C,C),plus_plus(C),groups1027152243600224163dd_sum(B,C,G,I5)),groups1027152243600224163dd_sum(B,C,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3532_sum_OG__def,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [P2: fun(C,B),I5: set(C)] :
          groups1027152243600224163dd_sum(C,B,P2,I5) = $ite(aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aTP_Lamp_ho(fun(C,B),fun(set(C),fun(C,$o)),P2),I5))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),P2),aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aTP_Lamp_ho(fun(C,B),fun(set(C),fun(C,$o)),P2),I5))),zero_zero(B)) ) ).

% sum.G_def
tff(fact_3533_count__list_Osimps_I1_J,axiom,
    ! [B: $tType,Y: B] : aa(B,nat,count_list(B,nil(B)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_3534_count__le__length,axiom,
    ! [B: $tType,Xs: list(B),X: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,count_list(B,Xs),X)),aa(list(B),nat,size_size(list(B)),Xs)) ).

% count_le_length
tff(fact_3535_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or5935395276787703475ssThan(B,A2,B2)),set_or1337092689740270186AtMost(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_3536_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or5935395276787703475ssThan(B,A2,B2)),set_or7035219750837199246ssThan(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_3537_atLeastAtMost__diff__ends,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),aa(set(B),set(B),insert(B,A2),aa(set(B),set(B),insert(B,B2),bot_bot(set(B))))) = set_or5935395276787703475ssThan(B,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_3538_count__list_Osimps_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),Y: B] :
      aa(B,nat,count_list(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),Y) = $ite(X = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,count_list(B,Xs),Y)),one_one(nat)),aa(B,nat,count_list(B,Xs),Y)) ).

% count_list.simps(2)
tff(fact_3539_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_3540_sum__diff1_H,axiom,
    ! [C: $tType,B: $tType] :
      ( ab_group_add(C)
     => ! [I5: set(B),F: fun(B,C),I: B] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_hn(set(B),fun(fun(B,C),fun(B,$o)),I5),F)))
         => ( groups1027152243600224163dd_sum(B,C,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),I5),aa(set(B),set(B),insert(B,I),bot_bot(set(B))))) = $ite(aa(set(B),$o,member(B,I),I5),aa(C,C,aa(C,fun(C,C),minus_minus(C),groups1027152243600224163dd_sum(B,C,F,I5)),aa(B,C,F,I)),groups1027152243600224163dd_sum(B,C,F,I5)) ) ) ) ).

% sum_diff1'
tff(fact_3541_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),L: list(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( sorted_wrt(B,ord_less(B),L)
              & ( aa(list(B),set(B),set2(B),L) = A4 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A4) ) )
          <=> ( aa(set(B),list(B),linord4507533701916653071of_set(B),A4) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_3542_set__update__distinct,axiom,
    ! [B: $tType,Xs: list(B),N: nat,X: B] :
      ( distinct(B,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(list(B),set(B),set2(B),list_update(B,Xs,N,X)) = aa(set(B),set(B),insert(B,X),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(set(B),set(B),insert(B,aa(nat,B,nth(B,Xs),N)),bot_bot(set(B))))) ) ) ) ).

% set_update_distinct
tff(fact_3543_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ap(nat,fun(nat,$o)),aTP_Lamp_aq(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_3544_bit_Oabstract__boolean__algebra__axioms,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => boolea2506097494486148201lgebra(B,bit_se5824344872417868541ns_and(B),bit_se1065995026697491101ons_or(B),bit_ri4277139882892585799ns_not(B),zero_zero(B),aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% bit.abstract_boolean_algebra_axioms
tff(fact_3545_sub__num__simps_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [L: num] : neg_numeral_sub(B,one2,aa(num,num,bit1,L)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_3546_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U)) ).

% finite_greaterThanLessThan
tff(fact_3547_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_3548_length__list__update,axiom,
    ! [B: $tType,Xs: list(B),I: nat,X: B] : aa(list(B),nat,size_size(list(B)),list_update(B,Xs,I,X)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_list_update
tff(fact_3549_sub__num__simps_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ( neg_numeral_sub(B,one2,one2) = zero_zero(B) ) ) ).

% sub_num_simps(1)
tff(fact_3550_diff__numeral__simps_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = neg_numeral_sub(B,Ma,N) ) ).

% diff_numeral_simps(1)
tff(fact_3551_list__update__beyond,axiom,
    ! [B: $tType,Xs: list(B),I: nat,X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),I)
     => ( list_update(B,Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_3552_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] : aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),linord4507533701916653071of_set(B),A4)) = aa(set(B),nat,finite_card(B),A4) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_3553_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_3554_sub__num__simps_I6_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num,L: num] : neg_numeral_sub(B,aa(num,num,bit0,K),aa(num,num,bit0,L)) = neg_numeral_dbl(B,neg_numeral_sub(B,K,L)) ) ).

% sub_num_simps(6)
tff(fact_3555_sub__num__simps_I9_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num,L: num] : neg_numeral_sub(B,aa(num,num,bit1,K),aa(num,num,bit1,L)) = neg_numeral_dbl(B,neg_numeral_sub(B,K,L)) ) ).

% sub_num_simps(9)
tff(fact_3556_add__neg__numeral__simps_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(num,B,numeral_numeral(B),N)) = neg_numeral_sub(B,N,Ma) ) ).

% add_neg_numeral_simps(2)
tff(fact_3557_add__neg__numeral__simps_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = neg_numeral_sub(B,Ma,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_3558_semiring__norm_I166_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [V2: num,W: num,Y: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),V2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))),Y)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),neg_numeral_sub(B,V2,W)),Y) ) ).

% semiring_norm(166)
tff(fact_3559_semiring__norm_I167_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [V2: num,W: num,Y: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),W)),Y)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),neg_numeral_sub(B,W,V2)),Y) ) ).

% semiring_norm(167)
tff(fact_3560_diff__numeral__simps_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = neg_numeral_sub(B,N,Ma) ) ).

% diff_numeral_simps(4)
tff(fact_3561_nth__update__invalid,axiom,
    ! [B: $tType,I: nat,L: list(B),J: nat,X: B] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
     => ( aa(nat,B,nth(B,list_update(B,L,J,X)),I) = aa(nat,B,nth(B,L),I) ) ) ).

% nth_update_invalid
tff(fact_3562_nth__list__update__eq,axiom,
    ! [B: $tType,I: nat,Xs: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,B,nth(B,list_update(B,Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_3563_sub__num__simps_I7_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num,L: num] : neg_numeral_sub(B,aa(num,num,bit0,K),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(B,neg_numeral_sub(B,K,L)) ) ).

% sub_num_simps(7)
tff(fact_3564_sub__num__simps_I8_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num,L: num] : neg_numeral_sub(B,aa(num,num,bit1,K),aa(num,num,bit0,L)) = neg_numeral_dbl_inc(B,neg_numeral_sub(B,K,L)) ) ).

% sub_num_simps(8)
tff(fact_3565_diff__numeral__special_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),Ma)),one_one(B)) = neg_numeral_sub(B,Ma,one2) ) ).

% diff_numeral_special(2)
tff(fact_3566_diff__numeral__special_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(num,B,numeral_numeral(B),N)) = neg_numeral_sub(B,one2,N) ) ).

% diff_numeral_special(1)
tff(fact_3567_sub__num__simps_I5_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_sub(B,aa(num,num,bit1,K),one2) = aa(num,B,numeral_numeral(B),aa(num,num,bit0,K)) ) ).

% sub_num_simps(5)
tff(fact_3568_set__swap,axiom,
    ! [B: $tType,I: nat,Xs: list(B),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(list(B),set(B),set2(B),list_update(B,list_update(B,Xs,I,aa(nat,B,nth(B,Xs),J)),J,aa(nat,B,nth(B,Xs),I))) = aa(list(B),set(B),set2(B),Xs) ) ) ) ).

% set_swap
tff(fact_3569_distinct__swap,axiom,
    ! [B: $tType,I: nat,Xs: list(B),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
       => ( distinct(B,list_update(B,list_update(B,Xs,I,aa(nat,B,nth(B,Xs),J)),J,aa(nat,B,nth(B,Xs),I)))
        <=> distinct(B,Xs) ) ) ) ).

% distinct_swap
tff(fact_3570_add__neg__numeral__special_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(num,B,numeral_numeral(B),N)) = neg_numeral_sub(B,N,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_3571_add__neg__numeral__special_I3_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(num,B,numeral_numeral(B),Ma)),aa(B,B,uminus_uminus(B),one_one(B))) = neg_numeral_sub(B,Ma,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_3572_add__neg__numeral__special_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),one_one(B)) = neg_numeral_sub(B,one2,Ma) ) ).

% add_neg_numeral_special(2)
tff(fact_3573_add__neg__numeral__special_I1_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),plus_plus(B),one_one(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))) = neg_numeral_sub(B,one2,Ma) ) ).

% add_neg_numeral_special(1)
tff(fact_3574_minus__sub__one__diff__one,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),neg_numeral_sub(B,Ma,one2))),one_one(B)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma)) ) ).

% minus_sub_one_diff_one
tff(fact_3575_diff__numeral__special_I7_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),one_one(B))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = neg_numeral_sub(B,N,one2) ) ).

% diff_numeral_special(7)
tff(fact_3576_diff__numeral__special_I8_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [Ma: num] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Ma))),aa(B,B,uminus_uminus(B),one_one(B))) = neg_numeral_sub(B,one2,Ma) ) ).

% diff_numeral_special(8)
tff(fact_3577_list__update_Osimps_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),I: nat,V2: B] : list_update(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),I,V2) = case_nat(list(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Xs),aa(B,fun(nat,list(B)),aa(list(B),fun(B,fun(nat,list(B))),aTP_Lamp_hq(B,fun(list(B),fun(B,fun(nat,list(B)))),X),Xs),V2),I) ).

% list_update.simps(2)
tff(fact_3578_list__update__code_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),Y: B] : list_update(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),zero_zero(nat),Y) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Xs) ).

% list_update_code(2)
tff(fact_3579_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_3580_neg__numeral__class_Osub__def,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num,L: num] : neg_numeral_sub(B,K,L) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),K)),aa(num,B,numeral_numeral(B),L)) ) ).

% neg_numeral_class.sub_def
tff(fact_3581_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] : sorted_wrt(B,ord_less_eq(B),aa(set(B),list(B),linord4507533701916653071of_set(B),A4)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_3582_in__set__upd__eq,axiom,
    ! [B: $tType,I: nat,L: list(B),X: B,Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),list_update(B,L,I,Y)))
      <=> ( ( X = Y )
          | ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),L))
            & ! [Y5: B] : aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),list_update(B,L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3583_in__set__upd__cases,axiom,
    ! [B: $tType,X: B,L: list(B),I: nat,Y: B] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),list_update(B,L,I,Y)))
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
         => ( X != Y ) )
       => aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),L)) ) ) ).

% in_set_upd_cases
tff(fact_3584_in__set__upd__eq__aux,axiom,
    ! [B: $tType,I: nat,L: list(B),X: B,Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),list_update(B,L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y5: B] : aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),list_update(B,L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3585_set__update__memI,axiom,
    ! [B: $tType,N: nat,Xs: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),list_update(B,Xs,N,X))) ) ).

% set_update_memI
tff(fact_3586_nth__list__update_H,axiom,
    ! [B: $tType,L: list(B),I: nat,X: B,J: nat] :
      aa(nat,B,nth(B,list_update(B,L,I,X)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L)) ),
        X,
        aa(nat,B,nth(B,L),J) ) ).

% nth_list_update'
tff(fact_3587_nth__list__update,axiom,
    ! [B: $tType,I: nat,Xs: list(B),X: B,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,B,nth(B,list_update(B,Xs,I,X)),J) = $ite(I = J,X,aa(nat,B,nth(B,Xs),J)) ) ) ).

% nth_list_update
tff(fact_3588_list__update__same__conv,axiom,
    ! [B: $tType,I: nat,Xs: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( ( list_update(B,Xs,I,X) = Xs )
      <=> ( aa(nat,B,nth(B,Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_3589_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_3590_sub__non__positive,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num,Ma: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),neg_numeral_sub(B,N,Ma)),zero_zero(B))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N),Ma) ) ) ).

% sub_non_positive
tff(fact_3591_sub__non__negative,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num,Ma: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),neg_numeral_sub(B,N,Ma))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),N) ) ) ).

% sub_non_negative
tff(fact_3592_sub__negative,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num,Ma: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),neg_numeral_sub(B,N,Ma)),zero_zero(B))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),N),Ma) ) ) ).

% sub_negative
tff(fact_3593_sub__positive,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [N: num,Ma: num] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),neg_numeral_sub(B,N,Ma))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),N) ) ) ).

% sub_positive
tff(fact_3594_sub__inc__One__eq,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : neg_numeral_sub(B,inc(N),one2) = aa(num,B,numeral_numeral(B),N) ) ).

% sub_inc_One_eq
tff(fact_3595_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_3596_insert__swap__set__eq,axiom,
    ! [B: $tType,I: nat,L: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
     => ( aa(set(B),set(B),insert(B,aa(nat,B,nth(B,L),I)),aa(list(B),set(B),set2(B),list_update(B,L,I,X))) = aa(set(B),set(B),insert(B,X),aa(list(B),set(B),set2(B),L)) ) ) ).

% insert_swap_set_eq
tff(fact_3597_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( distinct(B,Xs)
           => ( aa(set(B),list(B),linord4507533701916653071of_set(B),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_3598_sub__num__simps_I2_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [L: num] : neg_numeral_sub(B,one2,aa(num,num,bit0,L)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_3599_subset__mset_Osum__list__update,axiom,
    ! [B: $tType,K: nat,Xs: list(multiset(B)),X: multiset(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(multiset(B)),nat,size_size(list(multiset(B))),Xs))
     => ( groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),list_update(multiset(B),Xs,K,X)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Xs)),X)),aa(nat,multiset(B),nth(multiset(B),Xs),K)) ) ) ).

% subset_mset.sum_list_update
tff(fact_3600_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      aa(nat,num,num_of_nat,aa(nat,nat,suc,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N),inc(aa(nat,num,num_of_nat,N)),one2) ).

% num_of_nat.simps(2)
tff(fact_3601_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),list(B),linord4507533701916653071of_set(B),aa(set(B),set(B),insert(B,X),A4)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),aa(set(B),list(B),linord4507533701916653071of_set(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_3602_eq__numeral__iff__iszero_I7_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X)) = one_one(B) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_3603_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa(nat,num,num_of_nat,aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_3604_length__insort,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,C,F),X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_insort
tff(fact_3605_iszero__neg__numeral,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [W: num] :
          ( ring_1_iszero(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W)))
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),W)) ) ) ).

% iszero_neg_numeral
tff(fact_3606_subset__mset_Osum__list__eq__0__iff,axiom,
    ! [B: $tType,Ns: list(multiset(B))] :
      ( ( groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Ns) = zero_zero(multiset(B)) )
    <=> ! [X3: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X3),aa(list(multiset(B)),set(multiset(B)),set2(multiset(B)),Ns))
         => ( X3 = zero_zero(multiset(B)) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
tff(fact_3607_dbl__dec__simps_I5_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_dbl_dec(B,aa(num,B,numeral_numeral(B),K)) = aa(num,B,numeral_numeral(B),bitM(K)) ) ).

% dbl_dec_simps(5)
tff(fact_3608_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit0,K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_3609_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),list(B),linord4507533701916653071of_set(B),aa(set(B),set(B),insert(B,X),A4)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),aa(set(B),list(B),linord4507533701916653071of_set(B),A4)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_3610_sub__num__simps_I4_J,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [K: num] : neg_numeral_sub(B,aa(num,num,bit0,K),one2) = aa(num,B,numeral_numeral(B),bitM(K)) ) ).

% sub_num_simps(4)
tff(fact_3611_insort__left__comm,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B,Xs: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),Y),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),Xs)) ) ).

% insort_left_comm
tff(fact_3612_iszero__def,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Z: B] :
          ( ring_1_iszero(B,Z)
        <=> ( Z = zero_zero(B) ) ) ) ).

% iszero_def
tff(fact_3613_iszero__0,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ring_1_iszero(B,zero_zero(B)) ) ).

% iszero_0
tff(fact_3614_not__iszero__numeral,axiom,
    ! [B: $tType] :
      ( ring_char_0(B)
     => ! [W: num] : ~ ring_1_iszero(B,aa(num,B,numeral_numeral(B),W)) ) ).

% not_iszero_numeral
tff(fact_3615_not__iszero__1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ~ ring_1_iszero(B,one_one(B)) ) ).

% not_iszero_1
tff(fact_3616_eq__iff__iszero__diff,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: B,Y: B] :
          ( ( X = Y )
        <=> ring_1_iszero(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_3617_insort__key_Osimps_I2_J,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),X: B,Y: B,Ys: list(B)] :
          aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,C,F),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = $ite(aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,Y)),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),Ys)),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,C,F),X),Ys))) ) ).

% insort_key.simps(2)
tff(fact_3618_sorted__insort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),Xs))
        <=> sorted_wrt(B,ord_less_eq(B),Xs) ) ) ).

% sorted_insort
tff(fact_3619_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = aa(num,num,bit0,N) ).

% inc_BitM_eq
tff(fact_3620_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_3621_eq__numeral__iff__iszero_I9_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num] :
          ( ( aa(num,B,numeral_numeral(B),X) = zero_zero(B) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),X)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_3622_eq__numeral__iff__iszero_I10_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Y: num] :
          ( ( zero_zero(B) = aa(num,B,numeral_numeral(B),Y) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_3623_num__of__nat_Osimps_I1_J,axiom,
    aa(nat,num,num_of_nat,zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_3624_not__iszero__Numeral1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ~ ring_1_iszero(B,aa(num,B,numeral_numeral(B),one2)) ) ).

% not_iszero_Numeral1
tff(fact_3625_not__iszero__neg__1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ~ ring_1_iszero(B,aa(B,B,uminus_uminus(B),one_one(B))) ) ).

% not_iszero_neg_1
tff(fact_3626_eq__numeral__iff__iszero_I1_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num,Y: num] :
          ( ( aa(num,B,numeral_numeral(B),X) = aa(num,B,numeral_numeral(B),Y) )
        <=> ring_1_iszero(B,neg_numeral_sub(B,X,Y)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_3627_insort__is__Cons,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Xs: list(B),F: fun(B,C),A2: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,A2)),aa(B,C,F,X2)) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,C,F),A2),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_3628_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_3629_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = aa(num,num,bit0,N) ).

% BitM_plus_one
tff(fact_3630_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = aa(num,num,bit0,N) ).

% one_plus_BitM
tff(fact_3631_eq__numeral__iff__iszero_I11_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X)) = zero_zero(B) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),X)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_3632_eq__numeral__iff__iszero_I12_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Y: num] :
          ( ( zero_zero(B) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Y)) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_3633_not__iszero__neg__Numeral1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ~ ring_1_iszero(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_3634_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(num,nat,numeral_numeral(nat),aa(nat,num,num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_3635_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),one_one(nat))
     => ( aa(nat,num,num_of_nat,N) = one2 ) ) ).

% num_of_nat_One
tff(fact_3636_eq__numeral__iff__iszero_I3_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num,Y: num] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X)) = aa(num,B,numeral_numeral(B),Y) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_3637_eq__numeral__iff__iszero_I2_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num,Y: num] :
          ( ( aa(num,B,numeral_numeral(B),X) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Y)) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_3638_eq__numeral__iff__iszero_I4_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num,Y: num] :
          ( ( aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),X)) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Y)) )
        <=> ring_1_iszero(B,neg_numeral_sub(B,Y,X)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_3639_numeral__BitM,axiom,
    ! [B: $tType] :
      ( neg_numeral(B)
     => ! [N: num] : aa(num,B,numeral_numeral(B),bitM(N)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,N))),one_one(B)) ) ).

% numeral_BitM
tff(fact_3640_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub(int,bitM(N),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),neg_numeral_sub(int,N,one2)) ).

% sub_BitM_One_eq
tff(fact_3641_numeral__num__of__nat__unfold,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] :
          aa(num,B,numeral_numeral(B),aa(nat,num,num_of_nat,N)) = $ite(N = zero_zero(nat),one_one(B),aa(nat,B,semiring_1_of_nat(B),N)) ) ).

% numeral_num_of_nat_unfold
tff(fact_3642_num__of__nat__double,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = aa(num,num,bit0,aa(nat,num,num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_3643_num__of__nat__plus__distrib,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ma)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(nat,num,num_of_nat,Ma)),aa(nat,num,num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_3644_eq__numeral__iff__iszero_I6_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Y: num] :
          ( ( one_one(B) = aa(num,B,numeral_numeral(B),Y) )
        <=> ring_1_iszero(B,neg_numeral_sub(B,one2,Y)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_3645_eq__numeral__iff__iszero_I5_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: num] :
          ( ( aa(num,B,numeral_numeral(B),X) = one_one(B) )
        <=> ring_1_iszero(B,neg_numeral_sub(B,X,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_3646_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),list(B),linord4507533701916653071of_set(B),A4) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),aa(set(B),list(B),linord4507533701916653071of_set(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_3647_eq__numeral__iff__iszero_I8_J,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Y: num] :
          ( ( one_one(B) = aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),Y)) )
        <=> ring_1_iszero(B,aa(num,B,numeral_numeral(B),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_3648_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_3649_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_3650_subset__mset_Oelem__le__sum__list,axiom,
    ! [B: $tType,K: nat,Ns: list(multiset(B))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(multiset(B)),nat,size_size(list(multiset(B))),Ns))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(nat,multiset(B),nth(multiset(B),Ns),K)),groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Ns)) ) ).

% subset_mset.elem_le_sum_list
tff(fact_3651_num__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,num,code_num_of_integer,aa(int,code_integer,code_integer_of_int,X)) = aa(nat,num,num_of_nat,aa(int,nat,nat2,X)) ).

% num_of_integer.abs_eq
tff(fact_3652_subset__mset_Oadd__le__cancel__left,axiom,
    ! [B: $tType,C2: multiset(B),A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% subset_mset.add_le_cancel_left
tff(fact_3653_subset__mset_Oadd__le__cancel__right,axiom,
    ! [B: $tType,A2: multiset(B),C2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% subset_mset.add_le_cancel_right
tff(fact_3654_mset__subset__eq__mono__add__left__cancel,axiom,
    ! [B: $tType,C6: multiset(B),A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),A4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),B4))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4) ) ).

% mset_subset_eq_mono_add_left_cancel
tff(fact_3655_mset__subset__eq__mono__add__right__cancel,axiom,
    ! [B: $tType,A4: multiset(B),C6: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),C6)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4) ) ).

% mset_subset_eq_mono_add_right_cancel
tff(fact_3656_subset__mset_Oadd__le__same__cancel1,axiom,
    ! [B: $tType,B2: multiset(B),A2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),A2)),B2)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),zero_zero(multiset(B))) ) ).

% subset_mset.add_le_same_cancel1
tff(fact_3657_subset__mset_Oadd__le__same__cancel2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)),B2)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),zero_zero(multiset(B))) ) ).

% subset_mset.add_le_same_cancel2
tff(fact_3658_subset__mset_Ole__add__same__cancel1,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),B2) ) ).

% subset_mset.le_add_same_cancel1
tff(fact_3659_subset__mset_Ole__add__same__cancel2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),A2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),B2) ) ).

% subset_mset.le_add_same_cancel2
tff(fact_3660_subset__mset_Odiff__add,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)),A2) = B2 ) ) ).

% subset_mset.diff_add
tff(fact_3661_subset__mset_Oadd__diff__assoc,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2)),A2) ) ) ).

% subset_mset.add_diff_assoc
tff(fact_3662_subset__mset_Oadd__diff__assoc2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)),C2) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2)),A2) ) ) ).

% subset_mset.add_diff_assoc2
tff(fact_3663_mset__subset__eq__multiset__union__diff__commute,axiom,
    ! [B: $tType,B4: multiset(B),A4: multiset(B),C6: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),A4)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),B4)),C6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),C6)),B4) ) ) ).

% mset_subset_eq_multiset_union_diff_commute
tff(fact_3664_mset__le__subtract,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),C6)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),C6)) ) ).

% mset_le_subtract
tff(fact_3665_mset__subset__eq__exists__conv,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4)
    <=> ? [C8: multiset(B)] : B4 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),C8) ) ).

% mset_subset_eq_exists_conv
tff(fact_3666_mset__subset__eq__add__right,axiom,
    ! [B: $tType,B4: multiset(B),A4: multiset(B)] : aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) ).

% mset_subset_eq_add_right
tff(fact_3667_mset__subset__eq__mono__add,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B),D3: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C6),D3)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),C6)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),D3)) ) ) ).

% mset_subset_eq_mono_add
tff(fact_3668_mset__subset__eq__add__left,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) ).

% mset_subset_eq_add_left
tff(fact_3669_subset__mset_Oadd__le__imp__le__right,axiom,
    ! [B: $tType,A2: multiset(B),C2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% subset_mset.add_le_imp_le_right
tff(fact_3670_subset__mset_Oadd__le__imp__le__left,axiom,
    ! [B: $tType,C2: multiset(B),A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% subset_mset.add_le_imp_le_left
tff(fact_3671_subset__mset_Oadd__right__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2)) ) ).

% subset_mset.add_right_mono
tff(fact_3672_subset__mset_Oadd__left__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2)) ) ).

% subset_mset.add_left_mono
tff(fact_3673_subset__mset_Ole__iff__add,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
    <=> ? [C5: multiset(B)] : B2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C5) ) ).

% subset_mset.le_iff_add
tff(fact_3674_subset__mset_Oless__eqE,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ~ ! [C4: multiset(B)] : B2 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C4) ) ).

% subset_mset.less_eqE
tff(fact_3675_subset__mset_Oadd__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B),D2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C2),D2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),D2)) ) ) ).

% subset_mset.add_mono
tff(fact_3676_mset__le__addE,axiom,
    ! [B: $tType,Xs: multiset(B),Ys: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Xs),Ys)
     => ~ ! [Zs2: multiset(B)] : Ys != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Xs),Zs2) ) ).

% mset_le_addE
tff(fact_3677_mset__le__distrib,axiom,
    ! [B: $tType,X5: multiset(B),A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X5),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4))
     => ~ ! [Xa5: multiset(B),Xb5: multiset(B)] :
            ( ( X5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Xa5),Xb5) )
           => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Xa5),A4)
             => ~ aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Xb5),B4) ) ) ) ).

% mset_le_distrib
tff(fact_3678_mset__union__subset,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),C6)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),C6)
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),C6) ) ) ).

% mset_union_subset
tff(fact_3679_mset__le__decr__left1,axiom,
    ! [B: $tType,A2: multiset(B),C2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% mset_le_decr_left1
tff(fact_3680_mset__le__decr__left2,axiom,
    ! [B: $tType,C2: multiset(B),A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% mset_le_decr_left2
tff(fact_3681_mset__le__incr__right1,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2)) ) ).

% mset_le_incr_right1
tff(fact_3682_mset__le__incr__right2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2)) ) ).

% mset_le_incr_right2
tff(fact_3683_subset__mset_Olift__Suc__antimono__le,axiom,
    ! [B: $tType,F: fun(nat,multiset(B)),N: nat,N3: nat] :
      ( ! [N2: nat] : aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(nat,multiset(B),F,aa(nat,nat,suc,N2))),aa(nat,multiset(B),F,N2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(nat,multiset(B),F,N3)),aa(nat,multiset(B),F,N)) ) ) ).

% subset_mset.lift_Suc_antimono_le
tff(fact_3684_subset__mset_Olift__Suc__mono__le,axiom,
    ! [B: $tType,F: fun(nat,multiset(B)),N: nat,N3: nat] :
      ( ! [N2: nat] : aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(nat,multiset(B),F,N2)),aa(nat,multiset(B),F,aa(nat,nat,suc,N2)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(nat,multiset(B),F,N)),aa(nat,multiset(B),F,N3)) ) ) ).

% subset_mset.lift_Suc_mono_le
tff(fact_3685_subset__mset_Oadd__decreasing,axiom,
    ! [B: $tType,A2: multiset(B),C2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C2),B2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),B2) ) ) ).

% subset_mset.add_decreasing
tff(fact_3686_subset__mset_Oadd__increasing,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),C2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)) ) ) ).

% subset_mset.add_increasing
tff(fact_3687_subset__mset_Oadd__decreasing2,axiom,
    ! [B: $tType,C2: multiset(B),A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C2),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),B2) ) ) ).

% subset_mset.add_decreasing2
tff(fact_3688_subset__mset_Oadd__increasing2,axiom,
    ! [B: $tType,C2: multiset(B),B2: multiset(B),A2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),C2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),A2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)) ) ) ).

% subset_mset.add_increasing2
tff(fact_3689_subset__mset_Oadd__nonneg__nonneg,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),B2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)) ) ) ).

% subset_mset.add_nonneg_nonneg
tff(fact_3690_subset__mset_Oadd__nonpos__nonpos,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),zero_zero(multiset(B)))
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)),zero_zero(multiset(B))) ) ) ).

% subset_mset.add_nonpos_nonpos
tff(fact_3691_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),X)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),Y)
       => ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Y) = zero_zero(multiset(B)) )
        <=> ( ( X = zero_zero(multiset(B)) )
            & ( Y = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.add_nonneg_eq_0_iff
tff(fact_3692_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Y),zero_zero(multiset(B)))
       => ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Y) = zero_zero(multiset(B)) )
        <=> ( ( X = zero_zero(multiset(B)) )
            & ( Y = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.add_nonpos_eq_0_iff
tff(fact_3693_multiset__diff__union__assoc,axiom,
    ! [B: $tType,C6: multiset(B),B4: multiset(B),A4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C6),B4)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),C6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),C6)) ) ) ).

% multiset_diff_union_assoc
tff(fact_3694_subset__eq__diff__conv,axiom,
    ! [B: $tType,A4: multiset(B),C6: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),C6)),B4)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6)) ) ).

% subset_eq_diff_conv
tff(fact_3695_subset__mset_Ole__imp__diff__is__add,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
       => ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2) = C2 )
        <=> ( B2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2) ) ) ) ) ).

% subset_mset.le_imp_diff_is_add
tff(fact_3696_subset__mset_Oadd__diff__inverse,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)) = B2 ) ) ).

% subset_mset.add_diff_inverse
tff(fact_3697_subset__mset_Odiff__diff__right,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),B2) ) ) ).

% subset_mset.diff_diff_right
tff(fact_3698_subset__mset_Odiff__add__assoc2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2)),A2) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)),C2) ) ) ).

% subset_mset.diff_add_assoc2
tff(fact_3699_subset__mset_Odiff__add__assoc,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2)),A2) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2)) ) ) ).

% subset_mset.diff_add_assoc
tff(fact_3700_subset__mset_Ole__diff__conv2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B2),A2))
      <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),B2) ) ) ).

% subset_mset.le_diff_conv2
tff(fact_3701_subset__mset_Ole__add__diff,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2)),A2)) ) ).

% subset_mset.le_add_diff
tff(fact_3702_mset__le__subtract__left,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),X5: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),X5)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),X5),A4))
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),X5) ) ) ).

% mset_le_subtract_left
tff(fact_3703_mset__le__subtract__right,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),X5: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),X5)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),X5),B4))
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),X5) ) ) ).

% mset_le_subtract_right
tff(fact_3704_size__mset__mono,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(B),nat,size_size(multiset(B)),A4)),aa(multiset(B),nat,size_size(multiset(B)),B4)) ) ).

% size_mset_mono
tff(fact_3705_size__Diff__submset,axiom,
    ! [B: $tType,M: multiset(B),M7: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),M7)
     => ( aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M7),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(B),nat,size_size(multiset(B)),M7)),aa(multiset(B),nat,size_size(multiset(B)),M)) ) ) ).

% size_Diff_submset
tff(fact_3706_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [B: $tType,Xs: list(multiset(B))] :
      ( ! [X2: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X2),aa(list(multiset(B)),set(multiset(B)),set2(multiset(B)),Xs))
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),X2) )
     => ( ( groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Xs) = zero_zero(multiset(B)) )
      <=> ! [X3: multiset(B)] :
            ( aa(set(multiset(B)),$o,member(multiset(B),X3),aa(list(multiset(B)),set(multiset(B)),set2(multiset(B)),Xs))
           => ( X3 = zero_zero(multiset(B)) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
tff(fact_3707_subset__mset_Omember__le__sum__list,axiom,
    ! [B: $tType,X: multiset(B),Xs: list(multiset(B))] :
      ( aa(set(multiset(B)),$o,member(multiset(B),X),aa(list(multiset(B)),set(multiset(B)),set2(multiset(B)),Xs))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X),groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Xs)) ) ).

% subset_mset.member_le_sum_list
tff(fact_3708_subset__mset_Osum__list__nonpos,axiom,
    ! [B: $tType,Xs: list(multiset(B))] :
      ( ! [X2: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X2),aa(list(multiset(B)),set(multiset(B)),set2(multiset(B)),Xs))
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X2),zero_zero(multiset(B))) )
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Xs)),zero_zero(multiset(B))) ) ).

% subset_mset.sum_list_nonpos
tff(fact_3709_subset__mset_Osum__list__nonneg,axiom,
    ! [B: $tType,Xs: list(multiset(B))] :
      ( ! [X2: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X2),aa(list(multiset(B)),set(multiset(B)),set2(multiset(B)),Xs))
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),X2) )
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),groups4543113879258116180m_list(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),Xs)) ) ).

% subset_mset.sum_list_nonneg
tff(fact_3710_subset__mset_Osum__mono2,axiom,
    ! [C: $tType,B: $tType,B4: set(B),A4: set(B),F: fun(B,multiset(C))] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
       => ( ! [B3: B] :
              ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4))
             => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,B3)) )
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,A4)),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,B4)) ) ) ) ).

% subset_mset.sum_mono2
tff(fact_3711_mset__size2elem,axiom,
    ! [B: $tType,Pa: multiset(B),Q2: B,Q5: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(B),nat,size_size(multiset(B)),Pa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),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),Q2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Q5),zero_zero(multiset(B))))),Pa)
       => ( Pa = 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),Q2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Q5),zero_zero(multiset(B)))) ) ) ) ).

% mset_size2elem
tff(fact_3712_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_3713_take__bit__numeral__minus__numeral__int,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),Ma)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_hs(num,fun(num,int),Ma),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Ma),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_3714_union__mset__add__mset__right,axiom,
    ! [B: $tType,A4: multiset(B),A2: B,B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),B4)) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) ).

% union_mset_add_mset_right
tff(fact_3715_union__mset__add__mset__left,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B4: multiset(B)] : 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),A2),A4)),B4) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) ).

% union_mset_add_mset_left
tff(fact_3716_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U)) ).

% finite_greaterThanAtMost
tff(fact_3717_greaterThanAtMost__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,L: B,U: B] :
          ( aa(set(B),$o,member(B,I),set_or3652927894154168847AtMost(B,L,U))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),I)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_3718_greaterThanAtMost__empty,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,K: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),K)
         => ( set_or3652927894154168847AtMost(B,K,L) = bot_bot(set(B)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_3719_greaterThanAtMost__empty__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [K: B,L: B] :
          ( ( set_or3652927894154168847AtMost(B,K,L) = bot_bot(set(B)) )
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_3720_greaterThanAtMost__empty__iff2,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [K: B,L: B] :
          ( ( bot_bot(set(B)) = set_or3652927894154168847AtMost(B,K,L) )
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_3721_infinite__Ioc__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( ~ aa(set(B),$o,finite_finite2(B),set_or3652927894154168847AtMost(B,A2,B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_3722_size__add__mset,axiom,
    ! [B: $tType,A2: B,A4: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),A4)) = aa(nat,nat,suc,aa(multiset(B),nat,size_size(multiset(B)),A4)) ).

% size_add_mset
tff(fact_3723_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_greaterThanAtMost
tff(fact_3724_subset__mset_Osum__eq__0__iff,axiom,
    ! [B: $tType,C: $tType,F3: set(B),F: fun(B,multiset(C))] :
      ( aa(set(B),$o,finite_finite2(B),F3)
     => ( ( groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,F3) = zero_zero(multiset(C)) )
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),F3)
           => ( aa(B,multiset(C),F,X3) = zero_zero(multiset(C)) ) ) ) ) ).

% subset_mset.sum_eq_0_iff
tff(fact_3725_size__multiset__add__mset,axiom,
    ! [B: $tType,F: fun(B,nat),A2: B,M: multiset(B)] : aa(multiset(B),nat,size_multiset(B,F),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,aa(B,nat,F,A2))),aa(multiset(B),nat,size_multiset(B,F),M)) ).

% size_multiset_add_mset
tff(fact_3726_take__bit__numeral__numeral,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),aa(num,nat,numeral_numeral(nat),Ma)),aa(num,B,numeral_numeral(B),N)) = case_option(B,num,zero_zero(B),numeral_numeral(B),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Ma),N)) ) ).

% take_bit_numeral_numeral
tff(fact_3727_mset__le__add__mset__decr__left1,axiom,
    ! [B: $tType,C2: B,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),C2),A2)),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2) ) ).

% mset_le_add_mset_decr_left1
tff(fact_3728_mset__union__2__elem,axiom,
    ! [B: $tType,A2: B,B2: B,C2: B,M: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B)))) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),C2),M) )
     => ( ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) = M )
          & ( B2 = C2 ) )
        | ( ( A2 = C2 )
          & ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B))) = M ) ) ) ) ).

% mset_union_2_elem
tff(fact_3729_Ioc__inj,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( set_or3652927894154168847AtMost(B,A2,B2) = set_or3652927894154168847AtMost(B,C2,D2) )
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D2),C2) )
            | ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_3730_mset__le__add__mset,axiom,
    ! [B: $tType,X: B,B4: multiset(B),C6: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),B4)),C6)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))),C6)
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),C6) ) ) ).

% mset_le_add_mset
tff(fact_3731_mset__le__single__cases,axiom,
    ! [B: $tType,M: multiset(B),A2: B] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))))
     => ( ( M != zero_zero(multiset(B)) )
       => ( M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) ) ) ) ).

% mset_le_single_cases
tff(fact_3732_mset__le__add__mset__decr__left2,axiom,
    ! [B: $tType,C2: B,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),C2),A2)),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),C2),zero_zero(multiset(B)))),B2) ) ).

% mset_le_add_mset_decr_left2
tff(fact_3733_add__mset__add__single,axiom,
    ! [B: $tType,A2: B,A4: multiset(B)] : aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),A4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))) ).

% add_mset_add_single
tff(fact_3734_union__is__single,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B),A2: B] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) )
    <=> ( ( ( M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) )
          & ( N5 = zero_zero(multiset(B)) ) )
        | ( ( M = zero_zero(multiset(B)) )
          & ( N5 = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) ) ) ) ) ).

% union_is_single
tff(fact_3735_single__is__union,axiom,
    ! [B: $tType,A2: B,M: multiset(B),N5: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5) )
    <=> ( ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) = M )
          & ( N5 = zero_zero(multiset(B)) ) )
        | ( ( M = zero_zero(multiset(B)) )
          & ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) = N5 ) ) ) ) ).

% single_is_union
tff(fact_3736_add__mset__replicate__mset__safe,axiom,
    ! [B: $tType,C: $tType,M: multiset(C),A2: C] :
      ( nO_MATCH(multiset(B),multiset(C),zero_zero(multiset(B)),M)
     => ( aa(multiset(C),multiset(C),aa(C,fun(multiset(C),multiset(C)),add_mset(C),A2),M) = aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),plus_plus(multiset(C)),aa(multiset(C),multiset(C),aa(C,fun(multiset(C),multiset(C)),add_mset(C),A2),zero_zero(multiset(C)))),M) ) ) ).

% add_mset_replicate_mset_safe
tff(fact_3737_size__eq__Suc__imp__eq__union,axiom,
    ! [B: $tType,M: multiset(B),N: nat] :
      ( ( aa(multiset(B),nat,size_size(multiset(B)),M) = aa(nat,nat,suc,N) )
     => ? [A3: B,N9: multiset(B)] : M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),N9) ) ).

% size_eq_Suc_imp_eq_union
tff(fact_3738_infinite__Ioc,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ~ aa(set(B),$o,finite_finite2(B),set_or3652927894154168847AtMost(B,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_3739_multiset__induct2__size,axiom,
    ! [E: $tType,D: $tType,B: $tType,C: $tType,M: multiset(B),N5: multiset(C),Pa: fun(multiset(B),fun(multiset(C),$o))] :
      ( ( aa(multiset(B),nat,size_size(multiset(B)),M) = aa(multiset(C),nat,size_size(multiset(C)),N5) )
     => ( aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),Pa,zero_zero(multiset(B))),zero_zero(multiset(C)))
       => ( ! [A3: D,B3: E,M8: multiset(B),N9: multiset(C),Aa2: B,Ba: C] :
              ( aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),Pa,M8),N9)
             => aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),Pa,aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Aa2),M8)),aa(multiset(C),multiset(C),aa(C,fun(multiset(C),multiset(C)),add_mset(C),Ba),N9)) )
         => aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),Pa,M),N5) ) ) ) ).

% multiset_induct2_size
tff(fact_3740_Ioc__subset__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or3652927894154168847AtMost(B,A2,B2)),set_or3652927894154168847AtMost(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            | ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_3741_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_3742_mset__le__subtract__add__mset__left,axiom,
    ! [B: $tType,X: B,B4: multiset(B),X5: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),B4)),X5)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),X5),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))))
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))),X5) ) ) ).

% mset_le_subtract_add_mset_left
tff(fact_3743_mset__le__subtract__add__mset__right,axiom,
    ! [B: $tType,X: B,B4: multiset(B),X5: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),B4)),X5)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),X5),B4))
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),X5) ) ) ).

% mset_le_subtract_add_mset_right
tff(fact_3744_mset__unplusm__dist__cases2,axiom,
    ! [B: $tType,B4: multiset(B),C6: multiset(B),S2: B,A4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6) = 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),S2),zero_zero(multiset(B)))),A4) )
     => ( ( ( B4 = 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),S2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) )
         => ( A4 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))),C6) ) )
       => ~ ( ( C6 = 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),S2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) )
           => ( A4 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) ) ) ) ) ).

% mset_unplusm_dist_cases2
tff(fact_3745_mset__unplusm__dist__cases,axiom,
    ! [B: $tType,S2: B,A4: multiset(B),B4: multiset(B),C6: multiset(B)] :
      ( ( 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),S2),zero_zero(multiset(B)))),A4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6) )
     => ( ( ( B4 = 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),S2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) )
         => ( A4 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))),C6) ) )
       => ~ ( ( C6 = 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),S2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) )
           => ( A4 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) ) ) ) ) ).

% mset_unplusm_dist_cases
tff(fact_3746_mset__single__cases2_H,axiom,
    ! [B: $tType,S2: B,C2: multiset(B),R3: B,C3: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),C2) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),C3) )
     => ( ( ( S2 = R3 )
         => ( C2 != C3 ) )
       => ~ ! [Cc: multiset(B)] :
              ( ( C3 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Cc),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B)))) )
             => ( ( C2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Cc),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B)))) )
               => ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B)))) = Cc )
                 => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases2'
tff(fact_3747_mset__single__cases2,axiom,
    ! [B: $tType,S2: B,C2: multiset(B),R3: B,C3: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),C2) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),C3) )
     => ( ( ( S2 = R3 )
         => ( C2 != C3 ) )
       => ~ ( ( C3 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B)))) )
           => ( ( C2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B))))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B)))) )
             => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B)))) != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B)))) ) ) ) ) ) ).

% mset_single_cases2
tff(fact_3748_mset__single__cases_H,axiom,
    ! [B: $tType,S2: B,C2: multiset(B),R3: B,C3: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),C2) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),C3) )
     => ( ( ( S2 = R3 )
         => ( C2 != C3 ) )
       => ~ ! [Cc: multiset(B)] :
              ( ( C3 = 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),S2),zero_zero(multiset(B)))),Cc) )
             => ( ( C2 = 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),R3),zero_zero(multiset(B)))),Cc) )
               => ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B)))) = Cc )
                 => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases'
tff(fact_3749_mset__single__cases,axiom,
    ! [B: $tType,S2: B,C2: multiset(B),R3: B,C3: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),C2) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),C3) )
     => ( ( ( S2 = R3 )
         => ( C2 != C3 ) )
       => ~ ( ( C3 = 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),S2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B))))) )
           => ( ( C2 = 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),R3),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B))))) )
             => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C2),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),R3),zero_zero(multiset(B)))) != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),S2),zero_zero(multiset(B)))) ) ) ) ) ) ).

% mset_single_cases
tff(fact_3750_size__1__singleton__mset,axiom,
    ! [B: $tType,M: multiset(B)] :
      ( ( aa(multiset(B),nat,size_size(multiset(B)),M) = one_one(nat) )
     => ? [A3: B] : M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),zero_zero(multiset(B))) ) ).

% size_1_singleton_mset
tff(fact_3751_size__single,axiom,
    ! [B: $tType,B2: B] : aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B)))) = one_one(nat) ).

% size_single
tff(fact_3752_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_3753_size__mset__SucE,axiom,
    ! [B: $tType,A4: multiset(B),N: nat] :
      ( ( aa(multiset(B),nat,size_size(multiset(B)),A4) = aa(nat,nat,suc,N) )
     => ~ ! [A3: B,B10: multiset(B)] :
            ( ( A4 = 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)))),B10) )
           => ( aa(multiset(B),nat,size_size(multiset(B)),B10) != N ) ) ) ).

% size_mset_SucE
tff(fact_3754_size__Diff1__le,axiom,
    ! [B: $tType,M: multiset(B),X: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))))),aa(multiset(B),nat,size_size(multiset(B)),M)) ).

% size_Diff1_le
tff(fact_3755_sum_Ohead,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or3652927894154168847AtMost(nat,Ma,N))) ) ) ) ).

% sum.head
tff(fact_3756_prod_Ohead,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,Ma)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or3652927894154168847AtMost(nat,Ma,N))) ) ) ) ).

% prod.head
tff(fact_3757_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or3652927894154168847AtMost(B,A2,B2)),set_or1337092689740270186AtMost(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_3758_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or3652927894154168847AtMost(B,A2,B2)),set_or7035219750837199246ssThan(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_3759_subset__mset_Osum__nonneg,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,multiset(C))] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,X2)) )
     => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,A4)) ) ).

% subset_mset.sum_nonneg
tff(fact_3760_subset__mset_Osum__nonpos,axiom,
    ! [B: $tType,C: $tType,A4: set(B),F: fun(B,multiset(C))] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),aa(B,multiset(C),F,X2)),zero_zero(multiset(C))) )
     => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,A4)),zero_zero(multiset(C))) ) ).

% subset_mset.sum_nonpos
tff(fact_3761_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] : set_or3652927894154168847AtMost(B,A2,B2) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_3762_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [B: $tType] :
      ( dense_linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or5935395276787703475ssThan(B,A2,B2)),set_or3652927894154168847AtMost(B,C2,D2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_3763_subset__mset_Osum__mono,axiom,
    ! [C: $tType,B: $tType,K3: set(B),F: fun(B,multiset(C)),G: fun(B,multiset(C))] :
      ( ! [I2: B] :
          ( aa(set(B),$o,member(B,I2),K3)
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),aa(B,multiset(C),F,I2)),aa(B,multiset(C),G,I2)) )
     => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,K3)),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),G,K3)) ) ).

% subset_mset.sum_mono
tff(fact_3764_mset__size__le1__cases,axiom,
    ! [B: $tType,M: multiset(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(B),nat,size_size(multiset(B)),M)),aa(nat,nat,suc,zero_zero(nat)))
     => ( ( M != zero_zero(multiset(B)) )
       => ~ ! [M2: B] : M != aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),M2),zero_zero(multiset(B))) ) ) ).

% mset_size_le1_cases
tff(fact_3765_subset__mset_Osum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,C: $tType,A4: set(B),F: fun(B,multiset(C))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,X2)) )
       => ( ( groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,A4) = zero_zero(multiset(C)) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A4)
             => ( aa(B,multiset(C),F,X3) = zero_zero(multiset(C)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_eq_0_iff
tff(fact_3766_subset__mset_Osum__le__included,axiom,
    ! [B: $tType,D: $tType,C: $tType,S2: set(B),Ta: set(C),G: fun(C,multiset(D)),I: fun(C,B),F: fun(B,multiset(D))] :
      ( aa(set(B),$o,finite_finite2(B),S2)
     => ( aa(set(C),$o,finite_finite2(C),Ta)
       => ( ! [X2: C] :
              ( aa(set(C),$o,member(C,X2),Ta)
             => aa(multiset(D),$o,aa(multiset(D),fun(multiset(D),$o),subseteq_mset(D),zero_zero(multiset(D))),aa(C,multiset(D),G,X2)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),S2)
               => ? [Xa: C] :
                    ( aa(set(C),$o,member(C,Xa),Ta)
                    & ( aa(C,B,I,Xa) = X2 )
                    & aa(multiset(D),$o,aa(multiset(D),fun(multiset(D),$o),subseteq_mset(D),aa(B,multiset(D),F,X2)),aa(C,multiset(D),G,Xa)) ) )
           => aa(multiset(D),$o,aa(multiset(D),fun(multiset(D),$o),subseteq_mset(D),groups3894954378712506084id_sum(multiset(D),B,plus_plus(multiset(D)),zero_zero(multiset(D)),F,S2)),groups3894954378712506084id_sum(multiset(D),C,plus_plus(multiset(D)),zero_zero(multiset(D)),G,Ta)) ) ) ) ) ).

% subset_mset.sum_le_included
tff(fact_3767_subset__mset_Osum__nonneg__0,axiom,
    ! [B: $tType,C: $tType,S2: set(B),F: fun(B,multiset(C)),I: B] :
      ( aa(set(B),$o,finite_finite2(B),S2)
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),S2)
           => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,S2) = zero_zero(multiset(C)) )
         => ( aa(set(B),$o,member(B,I),S2)
           => ( aa(B,multiset(C),F,I) = zero_zero(multiset(C)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_0
tff(fact_3768_subset__mset_Osum__nonneg__leq__bound,axiom,
    ! [B: $tType,C: $tType,S2: set(B),F: fun(B,multiset(C)),B4: multiset(C),I: B] :
      ( aa(set(B),$o,finite_finite2(B),S2)
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),S2)
           => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,S2) = B4 )
         => ( aa(set(B),$o,member(B,I),S2)
           => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),aa(B,multiset(C),F,I)),B4) ) ) ) ) ).

% subset_mset.sum_nonneg_leq_bound
tff(fact_3769_and__minus__numerals_I7_J,axiom,
    ! [N: num,Ma: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),Ma)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_3770_and__minus__numerals_I3_J,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),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(Ma,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_3771_and__minus__numerals_I8_J,axiom,
    ! [N: num,Ma: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),Ma)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,aa(num,num,bit0,N))) ).

% and_minus_numerals(8)
tff(fact_3772_and__minus__numerals_I4_J,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),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(Ma,aa(num,num,bit0,N))) ).

% and_minus_numerals(4)
tff(fact_3773_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or3652927894154168847AtMost(int,L,U)) ).

% finite_greaterThanAtMost_int
tff(fact_3774_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_3775_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_greaterThanAtMost_int
tff(fact_3776_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_3777_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_3778_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_3779_int__numeral__not__and__num,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Ma))),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,Ma)) ).

% int_numeral_not_and_num
tff(fact_3780_int__numeral__and__not__num,axiom,
    ! [Ma: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Ma,N)) ).

% int_numeral_and_not_num
tff(fact_3781_sorted__list__of__multiset__insert,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,M: multiset(B)] : linord6283353356039996273ltiset(B,aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),M)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),linord6283353356039996273ltiset(B,M)) ) ).

% sorted_list_of_multiset_insert
tff(fact_3782_wcount__add__mset,axiom,
    ! [B: $tType,F: fun(B,nat),X: B,M: multiset(B),A2: B] :
      wcount(B,F,aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),M),A2) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          $ite(X = A2,aa(nat,nat,suc,aa(B,nat,F,A2)),zero_zero(nat))),
        wcount(B,F,M,A2)) ).

% wcount_add_mset
tff(fact_3783_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),X)
     => ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Y)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(code_integer,nat,code_nat_of_integer,X)),aa(code_integer,nat,code_nat_of_integer,Y))
        <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_3784_distinct__foldl__invar,axiom,
    ! [C: $tType,B: $tType,S: list(B),I5: fun(set(B),fun(C,$o)),Sigma_0: C,F: fun(C,fun(B,C))] :
      ( distinct(B,S)
     => ( aa(C,$o,aa(set(B),fun(C,$o),I5,aa(list(B),set(B),set2(B),S)),Sigma_0)
       => ( ! [X2: B,It: set(B),Sigma: C] :
              ( aa(set(B),$o,member(B,X2),It)
             => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),It),aa(list(B),set(B),set2(B),S))
               => ( aa(C,$o,aa(set(B),fun(C,$o),I5,It),Sigma)
                 => aa(C,$o,aa(set(B),fun(C,$o),I5,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),It),aa(set(B),set(B),insert(B,X2),bot_bot(set(B))))),aa(B,C,aa(C,fun(B,C),F,Sigma),X2)) ) ) )
         => aa(C,$o,aa(set(B),fun(C,$o),I5,bot_bot(set(B))),aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,F),Sigma_0),S)) ) ) ) ).

% distinct_foldl_invar
tff(fact_3785_nat__of__integer__of__nat,axiom,
    ! [N: nat] : aa(code_integer,nat,code_nat_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),N)) = N ).

% nat_of_integer_of_nat
tff(fact_3786_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa(nat,code_integer,semiring_1_of_nat(code_integer),aa(code_integer,nat,code_nat_of_integer,K)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).

% of_nat_of_integer
tff(fact_3787_foldl__length,axiom,
    ! [B: $tType,L: list(B)] : aa(list(B),nat,aa(nat,fun(list(B),nat),foldl(nat,B,aTP_Lamp_ht(nat,fun(B,nat))),zero_zero(nat)),L) = aa(list(B),nat,size_size(list(B)),L) ).

% foldl_length
tff(fact_3788_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer))
     => ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_3789_foldl__A1__eq,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),N: B,I: B,Ww: list(B)] :
      ( ! [E4: B] : aa(B,B,aa(B,fun(B,B),F,N),E4) = E4
     => ( ! [E4: B] : aa(B,B,aa(B,fun(B,B),F,E4),N) = E4
       => ( ! [A3: B,B3: B,C4: B] : aa(B,B,aa(B,fun(B,B),F,A3),aa(B,B,aa(B,fun(B,B),F,B3),C4)) = aa(B,B,aa(B,fun(B,B),F,aa(B,B,aa(B,fun(B,B),F,A3),B3)),C4)
         => ( aa(list(B),B,aa(B,fun(list(B),B),foldl(B,B,F),I),Ww) = aa(B,B,aa(B,fun(B,B),F,I),aa(list(B),B,aa(B,fun(list(B),B),foldl(B,B,F),N),Ww)) ) ) ) ) ).

% foldl_A1_eq
tff(fact_3790_foldl__absorb1,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [X: B,Zs: list(B)] : aa(B,B,aa(B,fun(B,B),times_times(B),X),aa(list(B),B,aa(B,fun(list(B),B),foldl(B,B,times_times(B)),one_one(B)),Zs)) = aa(list(B),B,aa(B,fun(list(B),B),foldl(B,B,times_times(B)),X),Zs) ) ).

% foldl_absorb1
tff(fact_3791_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,nat,code_nat_of_integer,aa(int,code_integer,code_integer_of_int,X)) = aa(int,nat,nat2,X) ).

% nat_of_integer.abs_eq
tff(fact_3792_foldl__length__aux,axiom,
    ! [B: $tType,A2: nat,L: list(B)] : aa(list(B),nat,aa(nat,fun(list(B),nat),foldl(nat,B,aTP_Lamp_ht(nat,fun(B,nat))),A2),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(B),nat,size_size(list(B)),L)) ).

% foldl_length_aux
tff(fact_3793_wcount__union,axiom,
    ! [B: $tType,F: fun(B,nat),M: multiset(B),N5: multiset(B),A2: B] : wcount(B,F,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5),A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),wcount(B,F,M,A2)),wcount(B,F,N5,A2)) ).

% wcount_union
tff(fact_3794_length__n__lists,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(list(B)),nat,size_size(list(list(B))),n_lists(B,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(B),nat,size_size(list(B)),Xs)),N) ).

% length_n_lists
tff(fact_3795_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_3796_prod_Oinsert_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(B),P2: fun(B,C),I: B] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fp(set(B),fun(fun(B,C),fun(B,$o)),I5),P2)))
         => ( groups1962203154675924110t_prod(B,C,P2,aa(set(B),set(B),insert(B,I),I5)) = $ite(aa(set(B),$o,member(B,I),I5),groups1962203154675924110t_prod(B,C,P2,I5),aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,P2,I)),groups1962203154675924110t_prod(B,C,P2,I5))) ) ) ) ).

% prod.insert'
tff(fact_3797_set__n__lists,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(list(B)),set(list(B)),set2(list(B)),n_lists(B,N,Xs)) = aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(list(B),fun(list(B),$o),aTP_Lamp_hu(nat,fun(list(B),fun(list(B),$o)),N),Xs)) ).

% set_n_lists
tff(fact_3798_empty__append__eq__id,axiom,
    ! [B: $tType,X4: list(B)] : aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),nil(B)),X4) = X4 ).

% empty_append_eq_id
tff(fact_3799_append__eq__append__conv,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Us: list(B),Vs: list(B)] :
      ( ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        | ( aa(list(B),nat,size_size(list(B)),Us) = aa(list(B),nat,size_size(list(B)),Vs) ) )
     => ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Us) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_3800_list__e__eq__lel_I2_J,axiom,
    ! [B: $tType,L1: list(B),E3: B,L22: list(B),E2: B] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E3),L22)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),nil(B)) )
    <=> ( ( L1 = nil(B) )
        & ( E3 = E2 )
        & ( L22 = nil(B) ) ) ) ).

% list_e_eq_lel(2)
tff(fact_3801_list__e__eq__lel_I1_J,axiom,
    ! [B: $tType,E2: B,L1: list(B),E3: B,L22: list(B)] :
      ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),nil(B)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E3),L22)) )
    <=> ( ( L1 = nil(B) )
        & ( E3 = E2 )
        & ( L22 = nil(B) ) ) ) ).

% list_e_eq_lel(1)
tff(fact_3802_list__se__match_I4_J,axiom,
    ! [B: $tType,L22: list(B),A2: B,L1: list(B)] :
      ( ( L22 != nil(B) )
     => ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),L22) )
      <=> ( ( L1 = nil(B) )
          & ( L22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) ) ) ) ) ).

% list_se_match(4)
tff(fact_3803_list__se__match_I3_J,axiom,
    ! [B: $tType,L1: list(B),A2: B,L22: list(B)] :
      ( ( L1 != nil(B) )
     => ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),L22) )
      <=> ( ( L1 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) )
          & ( L22 = nil(B) ) ) ) ) ).

% list_se_match(3)
tff(fact_3804_list__se__match_I2_J,axiom,
    ! [B: $tType,L22: list(B),L1: list(B),A2: B] :
      ( ( L22 != nil(B) )
     => ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),L22) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) )
      <=> ( ( L1 = nil(B) )
          & ( L22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) ) ) ) ) ).

% list_se_match(2)
tff(fact_3805_list__se__match_I1_J,axiom,
    ! [B: $tType,L1: list(B),L22: list(B),A2: B] :
      ( ( L1 != nil(B) )
     => ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),L22) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) )
      <=> ( ( L1 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B)) )
          & ( L22 = nil(B) ) ) ) ) ).

% list_se_match(1)
tff(fact_3806_list__ee__eq__leel_I2_J,axiom,
    ! [B: $tType,L1: list(B),E1: B,E22: B,L22: list(B),E12: B,E23: B] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E22),L22))) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E23),nil(B))) )
    <=> ( ( L1 = nil(B) )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil(B) ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_3807_list__ee__eq__leel_I1_J,axiom,
    ! [B: $tType,E12: B,E23: B,L1: list(B),E1: B,E22: B,L22: list(B)] :
      ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E23),nil(B))) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E22),L22))) )
    <=> ( ( L1 = nil(B) )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L22 = nil(B) ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_3808_length__append,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_append
tff(fact_3809_prod_Oempty_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [P2: fun(C,B)] : groups1962203154675924110t_prod(C,B,P2,bot_bot(set(C))) = one_one(B) ) ).

% prod.empty'
tff(fact_3810_n__lists__Nil,axiom,
    ! [B: $tType,N: nat] :
      n_lists(B,N,nil(B)) = $ite(N = zero_zero(nat),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),nil(list(B))),nil(list(B))) ).

% n_lists_Nil
tff(fact_3811_nth__append__length,axiom,
    ! [B: $tType,Xs: list(B),X: B,Ys: list(B)] : aa(nat,B,nth(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Ys))),aa(list(B),nat,size_size(list(B)),Xs)) = X ).

% nth_append_length
tff(fact_3812_nth__append__first,axiom,
    ! [B: $tType,I: nat,L: list(B),L3: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
     => ( aa(nat,B,nth(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L3)),I) = aa(nat,B,nth(B,L),I) ) ) ).

% nth_append_first
tff(fact_3813_nth__append__length__plus,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),N: nat] : aa(nat,B,nth(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs)),N)) = aa(nat,B,nth(B,Ys),N) ).

% nth_append_length_plus
tff(fact_3814_list__update__length,axiom,
    ! [B: $tType,Xs: list(B),X: B,Ys: list(B),Y: B] : list_update(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Ys)),aa(list(B),nat,size_size(list(B)),Xs),Y) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) ).

% list_update_length
tff(fact_3815_foldl__conc__empty__eq,axiom,
    ! [B: $tType,I: list(B),Ww: list(list(B))] : aa(list(list(B)),list(B),aa(list(B),fun(list(list(B)),list(B)),foldl(list(B),list(B),append(B)),I),Ww) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),I),aa(list(list(B)),list(B),aa(list(B),fun(list(list(B)),list(B)),foldl(list(B),list(B),append(B)),nil(B)),Ww)) ).

% foldl_conc_empty_eq
tff(fact_3816_list__match__lel__lel,axiom,
    ! [B: $tType,C1: list(B),Qs: B,C22: list(B),C12: list(B),Qs2: B,C23: list(B)] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),C1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Qs),C22)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),C12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Qs2),C23)) )
     => ( ! [C21: list(B)] :
            ( ( C1 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),C12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Qs2),C21)) )
           => ( C23 != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),C21),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list(B)] :
                ( ( C12 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),C1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Qs),C212)) )
               => ( C22 != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),C212),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_3817_prod_Onon__neutral_H,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,B),I5: set(C)] : groups1962203154675924110t_prod(C,B,G,aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aTP_Lamp_hv(fun(C,B),fun(set(C),fun(C,$o)),G),I5))) = groups1962203154675924110t_prod(C,B,G,I5) ) ).

% prod.non_neutral'
tff(fact_3818_neq__Nil__revE,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( L != nil(B) )
     => ~ ! [Ll2: list(B),E4: B] : L != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ll2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E4),nil(B))) ) ).

% neq_Nil_revE
tff(fact_3819_rev__induct2_H,axiom,
    ! [B: $tType,C: $tType,Pa: fun(list(B),fun(list(C),$o)),Xs: list(B),Ys: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,nil(B)),nil(C))
     => ( ! [X2: B,Xs2: list(B)] : aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))),nil(C))
       => ( ! [Y2: C,Ys3: list(C)] : aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,nil(B)),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys3),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),nil(C))))
         => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C)] :
                ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs2),Ys3)
               => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys3),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),nil(C)))) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_3820_neq__Nil__rev__conv,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( L != nil(B) )
    <=> ? [Xs3: list(B),X3: B] : L = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),nil(B))) ) ).

% neq_Nil_rev_conv
tff(fact_3821_rev__nonempty__induct2_H,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Pa: fun(list(B),fun(list(C),$o))] :
      ( ( Xs != nil(B) )
     => ( ( Ys != nil(C) )
       => ( ! [X2: B,Y2: C] : aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B))),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),nil(C)))
         => ( ! [X2: B,Xs2: list(B),Y2: C] :
                ( ( Xs2 != nil(B) )
               => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),nil(C))) )
           => ( ! [X2: B,Y2: C,Ys3: list(C)] :
                  ( ( Ys3 != nil(C) )
                 => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B))),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys3),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),nil(C)))) )
             => ( ! [X2: B,Xs2: list(B),Y2: C,Ys3: list(C)] :
                    ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs2),Ys3)
                   => ( ( Xs2 != nil(B) )
                     => ( ( Ys3 != nil(C) )
                       => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys3),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),nil(C)))) ) ) )
               => aa(list(C),$o,aa(list(B),fun(list(C),$o),Pa,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_3822_list__Cons__eq__append__cases,axiom,
    ! [B: $tType,X: B,Xs: list(B),Ys: list(B),Zs: list(B)] :
      ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs) )
     => ( ( ( Ys = nil(B) )
         => ( Zs != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) ) )
       => ~ ! [Ys5: list(B)] :
              ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Ys5) )
             => ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),Zs) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_3823_list__append__eq__Cons__cases,axiom,
    ! [B: $tType,Ys: list(B),Zs: list(B),X: B,Xs: list(B)] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) )
     => ( ( ( Ys = nil(B) )
         => ( Zs != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) ) )
       => ~ ! [Ys5: list(B)] :
              ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Ys5) )
             => ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),Zs) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_3824_in__set__list__format,axiom,
    ! [B: $tType,E2: B,L: list(B)] :
      ( aa(set(B),$o,member(B,E2),aa(list(B),set(B),set2(B),L))
     => ~ ! [L12: list(B),L23: list(B)] : L != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),L23)) ) ).

% in_set_list_format
tff(fact_3825_xy__in__set__cases,axiom,
    ! [B: $tType,X: B,L: list(B),Y: B] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),L))
     => ( aa(set(B),$o,member(B,Y),aa(list(B),set(B),set2(B),L))
       => ( ( ( X = Y )
           => ! [L12: list(B),L23: list(B)] : L != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),L23)) )
         => ( ( ( X != Y )
             => ! [L12: list(B),L23: list(B),L32: list(B)] : L != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L23),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),L32)))) )
           => ~ ( ( X != Y )
               => ! [L12: list(B),L23: list(B),L32: list(B)] : L != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L23),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L32)))) ) ) ) ) ) ).

% xy_in_set_cases
tff(fact_3826_list__rest__coinc,axiom,
    ! [B: $tType,S22: list(B),S1: list(B),R1: list(B),R22: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),S22)),aa(list(B),nat,size_size(list(B)),S1))
     => ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),S1),R1) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),S22),R22) )
       => ? [R1p: list(B)] : R22 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),R1p),R1) ) ) ).

% list_rest_coinc
tff(fact_3827_distinct__match,axiom,
    ! [B: $tType,Al: list(B),E2: B,Bl: list(B),Al2: list(B),Bl2: list(B)] :
      ( distinct(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Al),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),Bl)))
     => ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Al),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),Bl)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Al2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),Bl2)) )
      <=> ( ( Al = Al2 )
          & ( Bl = Bl2 ) ) ) ) ).

% distinct_match
tff(fact_3828_rotate__append,axiom,
    ! [B: $tType,L: list(B),Q2: list(B)] : aa(list(B),list(B),rotate(B,aa(list(B),nat,size_size(list(B)),L)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),Q2)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Q2),L) ).

% rotate_append
tff(fact_3829_length__n__lists__elem,axiom,
    ! [B: $tType,Ys: list(B),N: nat,Xs: list(B)] :
      ( aa(set(list(B)),$o,member(list(B),Ys),aa(list(list(B)),set(list(B)),set2(list(B)),n_lists(B,N,Xs)))
     => ( aa(list(B),nat,size_size(list(B)),Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_3830_same__length__different,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( Xs != Ys )
     => ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
       => ? [Pre: list(B),X2: B,Xs4: list(B),Y2: B,Ys5: list(B)] :
            ( ( X2 != Y2 )
            & ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Pre),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B))),Xs4)) )
            & ( Ys = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Pre),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_3831_prod_Odistrib__triv_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( groups1962203154675924110t_prod(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_hw(fun(B,C),fun(fun(B,C),fun(B,C)),G),H),I5) = aa(C,C,aa(C,fun(C,C),times_times(C),groups1962203154675924110t_prod(B,C,G,I5)),groups1962203154675924110t_prod(B,C,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_3832_sorted__append,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Ys: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys))
        <=> ( sorted_wrt(B,ord_less_eq(B),Xs)
            & sorted_wrt(B,ord_less_eq(B),Ys)
            & ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Xs))
               => ! [Xa4: B] :
                    ( aa(set(B),$o,member(B,Xa4),aa(list(B),set(B),set2(B),Ys))
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Xa4) ) ) ) ) ) ).

% sorted_append
tff(fact_3833_list__update__append1,axiom,
    ! [B: $tType,I: nat,Xs: list(B),Ys: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( list_update(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys),I,X) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),list_update(B,Xs,I,X)),Ys) ) ) ).

% list_update_append1
tff(fact_3834_foldl__rule,axiom,
    ! [B: $tType,C: $tType,I5: fun(B,fun(list(C),fun(list(C),$o))),Sigma_0: B,L0: list(C),F: fun(B,fun(C,B))] :
      ( aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,Sigma_0),nil(C)),L0)
     => ( ! [L12: list(C),L23: list(C),X2: C,Sigma: B] :
            ( ( L0 = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23)) )
           => ( aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,Sigma),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23))
             => aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,aa(C,B,aa(B,fun(C,B),F,Sigma),X2)),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),nil(C)))),L23) ) )
       => aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),Sigma_0),L0)),L0),nil(C)) ) ) ).

% foldl_rule
tff(fact_3835_foldl__rule__P,axiom,
    ! [B: $tType,C: $tType,I5: fun(B,fun(list(C),fun(list(C),$o))),Sigma_0: B,L0: list(C),F: fun(B,fun(C,B)),Pa: fun(B,$o)] :
      ( aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,Sigma_0),nil(C)),L0)
     => ( ! [L12: list(C),L23: list(C),X2: C,Sigma: B] :
            ( ( L0 = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23)) )
           => ( aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,Sigma),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23))
             => aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,aa(C,B,aa(B,fun(C,B),F,Sigma),X2)),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),nil(C)))),L23) ) )
       => ( ! [Sigma: B] :
              ( aa(list(C),$o,aa(list(C),fun(list(C),$o),aa(B,fun(list(C),fun(list(C),$o)),I5,Sigma),L0),nil(C))
             => aa(B,$o,Pa,Sigma) )
         => aa(B,$o,Pa,aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),Sigma_0),L0)) ) ) ) ).

% foldl_rule_P
tff(fact_3836_foldl__rule__aux,axiom,
    ! [B: $tType,C: $tType,I5: fun(B,fun(list(C),$o)),Sigma_0: B,L0: list(C),F: fun(B,fun(C,B))] :
      ( aa(list(C),$o,aa(B,fun(list(C),$o),I5,Sigma_0),L0)
     => ( ! [L12: list(C),L23: list(C),X2: C,Sigma: B] :
            ( ( L0 = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23)) )
           => ( aa(list(C),$o,aa(B,fun(list(C),$o),I5,Sigma),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23))
             => aa(list(C),$o,aa(B,fun(list(C),$o),I5,aa(C,B,aa(B,fun(C,B),F,Sigma),X2)),L23) ) )
       => aa(list(C),$o,aa(B,fun(list(C),$o),I5,aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),Sigma_0),L0)),nil(C)) ) ) ).

% foldl_rule_aux
tff(fact_3837_foldl__rule__aux__P,axiom,
    ! [B: $tType,C: $tType,I5: fun(B,fun(list(C),$o)),Sigma_0: B,L0: list(C),F: fun(B,fun(C,B)),Pa: fun(B,$o)] :
      ( aa(list(C),$o,aa(B,fun(list(C),$o),I5,Sigma_0),L0)
     => ( ! [L12: list(C),L23: list(C),X2: C,Sigma: B] :
            ( ( L0 = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L12),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23)) )
           => ( aa(list(C),$o,aa(B,fun(list(C),$o),I5,Sigma),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X2),L23))
             => aa(list(C),$o,aa(B,fun(list(C),$o),I5,aa(C,B,aa(B,fun(C,B),F,Sigma),X2)),L23) ) )
       => ( ! [Sigma: B] :
              ( aa(list(C),$o,aa(B,fun(list(C),$o),I5,Sigma),nil(C))
             => aa(B,$o,Pa,Sigma) )
         => aa(B,$o,Pa,aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),Sigma_0),L0)) ) ) ) ).

% foldl_rule_aux_P
tff(fact_3838_prod_Omono__neutral__cong__right_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),T4: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,G,X2) = one_one(C) ) )
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),S)
                 => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
             => ( groups1962203154675924110t_prod(B,C,G,T4) = groups1962203154675924110t_prod(B,C,H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3839_prod_Omono__neutral__cong__left_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),T4: set(B),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,H,I2) = one_one(C) ) )
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),S)
                 => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
             => ( groups1962203154675924110t_prod(B,C,G,S) = groups1962203154675924110t_prod(B,C,H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3840_prod_Omono__neutral__right_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),T4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,G,X2) = one_one(C) ) )
           => ( groups1962203154675924110t_prod(B,C,G,T4) = groups1962203154675924110t_prod(B,C,G,S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3841_prod_Omono__neutral__left_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),T4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),T4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
               => ( aa(B,C,G,X2) = one_one(C) ) )
           => ( groups1962203154675924110t_prod(B,C,G,S) = groups1962203154675924110t_prod(B,C,G,T4) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3842_length__Suc__conv__rev,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y5: B,Ys4: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B))) )
          & ( aa(list(B),nat,size_size(list(B)),Ys4) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3843_length__Suc__rev__conv,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Ys4: list(B),Y5: B] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B))) )
          & ( aa(list(B),nat,size_size(list(B)),Ys4) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_3844_length__append__singleton,axiom,
    ! [B: $tType,Xs: list(B),X: B] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)))) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_append_singleton
tff(fact_3845_length__compl__rev__induct,axiom,
    ! [B: $tType,Pa: fun(list(B),$o),L: list(B)] :
      ( aa(list(B),$o,Pa,nil(B))
     => ( ! [L2: list(B),E4: B] :
            ( ! [Ll: list(B)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Ll)),aa(list(B),nat,size_size(list(B)),L2))
               => aa(list(B),$o,Pa,Ll) )
           => aa(list(B),$o,Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E4),nil(B)))) )
       => aa(list(B),$o,Pa,L) ) ) ).

% length_compl_rev_induct
tff(fact_3846_not__distinct__split__distinct,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ~ distinct(B,Xs)
     => ~ ! [Y2: B,Ys3: list(B)] :
            ( distinct(B,Ys3)
           => ( aa(set(B),$o,member(B,Y2),aa(list(B),set(B),set2(B),Ys3))
             => ! [Zs2: list(B)] : Xs != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))),Zs2)) ) ) ) ).

% not_distinct_split_distinct
tff(fact_3847_nth__append,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),N: nat] :
      aa(nat,B,nth(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),N) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),aa(nat,B,nth(B,Xs),N),aa(nat,B,nth(B,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(B),nat,size_size(list(B)),Xs)))) ).

% nth_append
tff(fact_3848_list__update__append,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),N: nat,X: B] :
      list_update(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys),N,X) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),list_update(B,Xs,N,X)),Ys),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),list_update(B,Ys,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),X))) ).

% list_update_append
tff(fact_3849_n__lists_Osimps_I1_J,axiom,
    ! [B: $tType,Xs: list(B)] : n_lists(B,zero_zero(nat),Xs) = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),nil(list(B))) ).

% n_lists.simps(1)
tff(fact_3850_prod_Odistrib_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(B),G: fun(B,C),H: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fp(set(B),fun(fun(B,C),fun(B,$o)),I5),G)))
         => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_fp(set(B),fun(fun(B,C),fun(B,$o)),I5),H)))
           => ( groups1962203154675924110t_prod(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_hw(fun(B,C),fun(fun(B,C),fun(B,C)),G),H),I5) = aa(C,C,aa(C,fun(C,C),times_times(C),groups1962203154675924110t_prod(B,C,G,I5)),groups1962203154675924110t_prod(B,C,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_3851_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_3852_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_3853_prod_OG__def,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [P2: fun(C,B),I5: set(C)] :
          groups1962203154675924110t_prod(C,B,P2,I5) = $ite(aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aTP_Lamp_hv(fun(C,B),fun(set(C),fun(C,$o)),P2),I5))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),P2),aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aTP_Lamp_hv(fun(C,B),fun(set(C),fun(C,$o)),P2),I5))),one_one(B)) ) ).

% prod.G_def
tff(fact_3854_slice__prepend,axiom,
    ! [B: $tType,I: nat,K: nat,Xs: list(B),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(B),nat,size_size(list(B)),Xs))
       => $let(
            p2: nat,
            p2:= aa(list(B),nat,size_size(list(B)),Ys),
            slice(B,I,K,Xs) = slice(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),p2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Xs)) ) ) ) ).

% slice_prepend
tff(fact_3855_sorted__append__bigger,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Y: B] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y) )
           => sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),nil(B)))) ) ) ) ).

% sorted_append_bigger
tff(fact_3856_horner__sum__append,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_1(B)
     => ! [F: fun(C,B),A2: B,Xs: list(C),Ys: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Xs),Ys)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),Xs)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),aa(list(C),nat,size_size(list(C)),Xs))),aa(list(C),B,aa(B,fun(list(C),B),aa(fun(C,B),fun(B,fun(list(C),B)),groups4207007520872428315er_sum(C,B),F),A2),Ys))) ) ).

% horner_sum_append
tff(fact_3857_sorted__insort__is__snoc,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),A2: B] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),A2) )
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),A2),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_3858_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_3859_Succ__def,axiom,
    ! [B: $tType,Kl: set(list(B)),Kl2: list(B)] : bNF_Greatest_Succ(B,Kl,Kl2) = aa(fun(B,$o),set(B),collect(B),aa(list(B),fun(B,$o),aTP_Lamp_hx(set(list(B)),fun(list(B),fun(B,$o)),Kl),Kl2)) ).

% Succ_def
tff(fact_3860_SuccI,axiom,
    ! [B: $tType,Kl2: list(B),K: B,Kl: set(list(B))] :
      ( aa(set(list(B)),$o,member(list(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Kl2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),K),nil(B)))),Kl)
     => aa(set(B),$o,member(B,K),bNF_Greatest_Succ(B,Kl,Kl2)) ) ).

% SuccI
tff(fact_3861_SuccD,axiom,
    ! [B: $tType,K: B,Kl: set(list(B)),Kl2: list(B)] :
      ( aa(set(B),$o,member(B,K),bNF_Greatest_Succ(B,Kl,Kl2))
     => aa(set(list(B)),$o,member(list(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Kl2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),K),nil(B)))),Kl) ) ).

% SuccD
tff(fact_3862_Gcd__remove0__nat,axiom,
    ! [M: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M)
     => ( gcd_Gcd(nat,M) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_3863_Gcd__empty,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ( gcd_Gcd(B,bot_bot(set(B))) = zero_zero(B) ) ) ).

% Gcd_empty
tff(fact_3864_Gcd__0__iff,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A4: set(B)] :
          ( ( gcd_Gcd(B,A4) = zero_zero(B) )
        <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(B),set(B),insert(B,zero_zero(B)),bot_bot(set(B)))) ) ) ).

% Gcd_0_iff
tff(fact_3865_Gcd__nat__eq__one,axiom,
    ! [N5: set(nat)] :
      ( aa(set(nat),$o,member(nat,one_one(nat)),N5)
     => ( gcd_Gcd(nat,N5) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_3866_Gcd__dvd__nat,axiom,
    ! [A2: nat,A4: set(nat)] :
      ( aa(set(nat),$o,member(nat,A2),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),gcd_Gcd(nat,A4)),A2) ) ).

% Gcd_dvd_nat
tff(fact_3867_Gcd__greatest__nat,axiom,
    ! [A4: set(nat),A2: nat] :
      ( ! [B3: nat] :
          ( aa(set(nat),$o,member(nat,B3),A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B3) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),gcd_Gcd(nat,A4)) ) ).

% Gcd_greatest_nat
tff(fact_3868_Gcd__1,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,member(B,one_one(B)),A4)
         => ( gcd_Gcd(B,A4) = one_one(B) ) ) ) ).

% Gcd_1
tff(fact_3869_Gcd__dvd,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A2: B,A4: set(B)] :
          ( aa(set(B),$o,member(B,A2),A4)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_Gcd(B,A4)),A2) ) ) ).

% Gcd_dvd
tff(fact_3870_dvd__GcdD,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [X: B,A4: set(B),Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),gcd_Gcd(B,A4))
         => ( aa(set(B),$o,member(B,Y),A4)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),Y) ) ) ) ).

% dvd_GcdD
tff(fact_3871_dvd__Gcd__iff,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [X: B,A4: set(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),gcd_Gcd(B,A4))
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),X3) ) ) ) ).

% dvd_Gcd_iff
tff(fact_3872_Gcd__greatest,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A4: set(B),A2: B] :
          ( ! [B3: B] :
              ( aa(set(B),$o,member(B,B3),A4)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B3) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),gcd_Gcd(B,A4)) ) ) ).

% Gcd_greatest
tff(fact_3873_Gcd__eq__1__I,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A2: B,A4: set(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(set(B),$o,member(B,A2),A4)
           => ( gcd_Gcd(B,A4) = one_one(B) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_3874_semiring__char__def,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Uu: itself(B)] : semiri4206861660011772517g_char(B,Uu) = gcd_Gcd(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_hy(nat,$o))) ) ).

% semiring_char_def
tff(fact_3875_empty__Shift,axiom,
    ! [B: $tType,Kl: set(list(B)),K: B] :
      ( aa(set(list(B)),$o,member(list(B),nil(B)),Kl)
     => ( aa(set(B),$o,member(B,K),bNF_Greatest_Succ(B,Kl,nil(B)))
       => aa(set(list(B)),$o,member(list(B),nil(B)),bNF_Greatest_Shift(B,Kl,K)) ) ) ).

% empty_Shift
tff(fact_3876_Succ__Shift,axiom,
    ! [B: $tType,Kl: set(list(B)),K: B,Kl2: list(B)] : bNF_Greatest_Succ(B,bNF_Greatest_Shift(B,Kl,K),Kl2) = bNF_Greatest_Succ(B,Kl,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),K),Kl2)) ).

% Succ_Shift
tff(fact_3877_one__diff__power__eq_H,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,N: nat] : aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_hz(B,fun(nat,fun(nat,B)),X),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq'
tff(fact_3878_abs__Gcd__eq,axiom,
    ! [K3: set(int)] : aa(int,int,abs_abs(int),gcd_Gcd(int,K3)) = gcd_Gcd(int,K3) ).

% abs_Gcd_eq
tff(fact_3879_lessThan__eq__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,set(B),set_ord_lessThan(B),X) = aa(B,set(B),set_ord_lessThan(B),Y) )
        <=> ( X = Y ) ) ) ).

% lessThan_eq_iff
tff(fact_3880_lessThan__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,K: B] :
          ( aa(set(B),$o,member(B,I),aa(B,set(B),set_ord_lessThan(B),K))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),I),K) ) ) ).

% lessThan_iff
tff(fact_3881_finite__lessThan,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% finite_lessThan
tff(fact_3882_card__lessThan,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_lessThan(nat),U)) = U ).

% card_lessThan
tff(fact_3883_lessThan__subset__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_lessThan(B),X)),aa(B,set(B),set_ord_lessThan(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% lessThan_subset_iff
tff(fact_3884_lessThan__minus__lessThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [N: B,Ma: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(B,set(B),set_ord_lessThan(B),N)),aa(B,set(B),set_ord_lessThan(B),Ma)) = set_or7035219750837199246ssThan(B,Ma,N) ) ).

% lessThan_minus_lessThan
tff(fact_3885_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3886_single__Diff__lessThan,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [K: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),insert(B,K),bot_bot(set(B)))),aa(B,set(B),set_ord_lessThan(B),K)) = aa(set(B),set(B),insert(B,K),bot_bot(set(B))) ) ).

% single_Diff_lessThan
tff(fact_3887_sum_OlessThan__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,B,G,N)) ) ).

% sum.lessThan_Suc
tff(fact_3888_prod_OlessThan__Suc,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,B,G,N)) ) ).

% prod.lessThan_Suc
tff(fact_3889_infinite__Iio,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [A2: B] : ~ aa(set(B),$o,finite_finite2(B),aa(B,set(B),set_ord_lessThan(B),A2)) ) ).

% infinite_Iio
tff(fact_3890_lessThan__non__empty,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [X: B] : aa(B,set(B),set_ord_lessThan(B),X) != bot_bot(set(B)) ) ).

% lessThan_non_empty
tff(fact_3891_Gcd__dvd__int,axiom,
    ! [A2: int,A4: set(int)] :
      ( aa(set(int),$o,member(int,A2),A4)
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),gcd_Gcd(int,A4)),A2) ) ).

% Gcd_dvd_int
tff(fact_3892_Gcd__greatest__int,axiom,
    ! [A4: set(int),A2: int] :
      ( ! [B3: int] :
          ( aa(set(int),$o,member(int,B3),A4)
         => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),B3) )
     => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),gcd_Gcd(int,A4)) ) ).

% Gcd_greatest_int
tff(fact_3893_lessThan__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [U: B] : aa(B,set(B),set_ord_lessThan(B),U) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ia(B,fun(B,$o),U)) ) ).

% lessThan_def
tff(fact_3894_lessThan__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_3895_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_atMost(nat),K) ).

% lessThan_Suc_atMost
tff(fact_3896_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3897_lessThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_3898_Iio__eq__empty__iff,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & order_bot(B) )
     => ! [N: B] :
          ( ( aa(B,set(B),set_ord_lessThan(B),N) = bot_bot(set(B)) )
        <=> ( N = bot_bot(B) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_3899_lessThan__strict__subset__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ma: B,N: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(B,set(B),set_ord_lessThan(B),Ma)),aa(B,set(B),set_ord_lessThan(B),N))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),N) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3900_Gcd__int__greater__eq__0,axiom,
    ! [K3: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K3)) ).

% Gcd_int_greater_eq_0
tff(fact_3901_lessThan__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_3902_sum_Onat__diff__reindex,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ib(fun(nat,B),fun(nat,fun(nat,B)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nat_diff_reindex
tff(fact_3903_ShiftD,axiom,
    ! [B: $tType,Kl2: list(B),Kl: set(list(B)),K: B] :
      ( aa(set(list(B)),$o,member(list(B),Kl2),bNF_Greatest_Shift(B,Kl,K))
     => aa(set(list(B)),$o,member(list(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),K),Kl2)),Kl) ) ).

% ShiftD
tff(fact_3904_prod_Onat__diff__reindex,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ic(fun(nat,B),fun(nat,fun(nat,B)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nat_diff_reindex
tff(fact_3905_Iic__subset__Iio__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atMost(B),A2)),aa(B,set(B),set_ord_lessThan(B),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3906_sum__diff__distrib,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Q: fun(B,nat),Pa: fun(B,nat),N: B] :
          ( ! [X2: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,Q,X2)),aa(B,nat,Pa,X2))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),Pa),aa(B,set(B),set_ord_lessThan(B),N))),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),Q),aa(B,set(B),set_ord_lessThan(B),N))) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(fun(B,nat),fun(B,nat),aTP_Lamp_id(fun(B,nat),fun(fun(B,nat),fun(B,nat)),Q),Pa)),aa(B,set(B),set_ord_lessThan(B),N)) ) ) ) ).

% sum_diff_distrib
tff(fact_3907_sum_OlessThan__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3908_sum__lessThan__telescope,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(nat,B),Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cg(fun(nat,B),fun(nat,B),F)),aa(nat,set(nat),set_ord_lessThan(nat),Ma)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,Ma)),aa(nat,B,F,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3909_sum__lessThan__telescope_H,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(nat,B),Ma: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cf(fun(nat,B),fun(nat,B),F)),aa(nat,set(nat),set_ord_lessThan(nat),Ma)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,F,zero_zero(nat))),aa(nat,B,F,Ma)) ) ).

% sum_lessThan_telescope'
tff(fact_3910_prod_OlessThan__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3911_sum_OatLeast1__atMost__eq,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3912_prod_OatLeast1__atMost__eq,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3913_sum__bounds__lt__plus1,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [F: fun(nat,B),Mm: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),F)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3914_sum_Onat__group,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ie(fun(nat,B),fun(nat,fun(nat,B)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_3915_Shift__def,axiom,
    ! [B: $tType,Kl: set(list(B)),K: B] : bNF_Greatest_Shift(B,Kl,K) = aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(B,fun(list(B),$o),aTP_Lamp_if(set(list(B)),fun(B,fun(list(B),$o)),Kl),K)) ).

% Shift_def
tff(fact_3916_prod_Onat__group,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ig(fun(nat,B),fun(nat,fun(nat,B)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_3917_sum_Onested__swap_H,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: fun(nat,fun(nat,B)),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_ih(fun(nat,fun(nat,B)),fun(nat,B),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_el(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),A2),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nested_swap'
tff(fact_3918_prod_Onested__swap_H,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: fun(nat,fun(nat,B)),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_ii(fun(nat,fun(nat,B)),fun(nat,B),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_eb(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),A2),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nested_swap'
tff(fact_3919_one__diff__power__eq,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,N: nat] : aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq
tff(fact_3920_power__diff__1__eq,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,N: nat] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)),one_one(B)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),one_one(B))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_1_eq
tff(fact_3921_geometric__sum,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B,N: nat] :
          ( ( X != one_one(B) )
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)),one_one(B))),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),one_one(B))) ) ) ) ).

% geometric_sum
tff(fact_3922_sum_OatMost__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_cb(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.atMost_shift
tff(fact_3923_prod_OatMost__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_dq(fun(nat,B),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.atMost_shift
tff(fact_3924_power__diff__sumr2,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,N: nat,Y: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aa(nat,fun(B,fun(nat,B)),aTP_Lamp_ij(B,fun(nat,fun(B,fun(nat,B))),X),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_sumr2
tff(fact_3925_diff__power__eq__sum,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [X: B,N: nat,Y: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,suc,N))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Y),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Y)),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),aa(nat,fun(B,fun(nat,B)),aTP_Lamp_ik(B,fun(nat,fun(B,fun(nat,B))),X),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_3926_sum__gp__strict,axiom,
    ! [B: $tType] :
      ( ( division_ring(B)
        & comm_ring(B) )
     => ! [X: B,N: nat] :
          aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(B,fun(nat,B),power_power(B),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = $ite(X = one_one(B),aa(nat,B,semiring_1_of_nat(B),N),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N))),aa(B,B,aa(B,fun(B,B),minus_minus(B),one_one(B)),X))) ) ).

% sum_gp_strict
tff(fact_3927_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_lessThan(nat),N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3928_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),U)
     => ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image2(nat,code_integer,semiring_1_of_nat(code_integer)),aa(nat,set(nat),set_ord_lessThan(nat),aa(code_integer,nat,code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_3929_take__bit__num__simps_I7_J,axiom,
    ! [R: num,Ma: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit1,Ma)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R),Ma))) ).

% take_bit_num_simps(7)
tff(fact_3930_sorted__list__of__set__def,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( linord4507533701916653071of_set(B) = linord144544945434240204of_set(B,B,aTP_Lamp_hr(B,B)) ) ) ).

% sorted_list_of_set_def
tff(fact_3931_in__set__product__lists__length,axiom,
    ! [B: $tType,Xs: list(B),Xss: list(list(B))] :
      ( aa(set(list(B)),$o,member(list(B),Xs),aa(list(list(B)),set(list(B)),set2(list(B)),product_lists(B,Xss)))
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(list(B)),nat,size_size(list(list(B))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_3932_image__ident,axiom,
    ! [B: $tType,Y4: set(B)] : aa(set(B),set(B),image2(B,B,aTP_Lamp_il(B,B)),Y4) = Y4 ).

% image_ident
tff(fact_3933_image__add__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(B)] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),zero_zero(B))),S) = S ) ).

% image_add_0
tff(fact_3934_image__add__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: B,I: B,J: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),K)),set_or1337092689740270186AtMost(B,I,J)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_3935_image__diff__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [D2: B,A2: B,B2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),minus_minus(B),D2)),set_or1337092689740270186AtMost(B,A2,B2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),D2),B2),aa(B,B,aa(B,fun(B,B),minus_minus(B),D2),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_3936_image__add__atLeastLessThan,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: B,I: B,J: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),K)),set_or7035219750837199246ssThan(B,I,J)) = set_or7035219750837199246ssThan(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_3937_image__uminus__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B,Y: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),set_or1337092689740270186AtMost(B,X,Y)) = set_or1337092689740270186AtMost(B,aa(B,B,uminus_uminus(B),Y),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_atLeastAtMost
tff(fact_3938_image__add__atMost,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [C2: B,A2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),C2)),aa(B,set(B),set_ord_atMost(B),A2)) = aa(B,set(B),set_ord_atMost(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2)) ) ).

% image_add_atMost
tff(fact_3939_image__add__greaterThanAtMost,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [C2: B,A2: B,B2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),C2)),set_or3652927894154168847AtMost(B,A2,B2)) = set_or3652927894154168847AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_3940_image__uminus__greaterThanLessThan,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B,Y: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),set_or5935395276787703475ssThan(B,X,Y)) = set_or5935395276787703475ssThan(B,aa(B,B,uminus_uminus(B),Y),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_greaterThanLessThan
tff(fact_3941_image__add__atLeastAtMost_H,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: B,I: B,J: B] : aa(set(B),set(B),image2(B,B,aTP_Lamp_im(B,fun(B,B),K)),set_or1337092689740270186AtMost(B,I,J)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_3942_image__minus__const__atLeastAtMost_H,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [D2: B,A2: B,B2: B] : aa(set(B),set(B),image2(B,B,aTP_Lamp_in(B,fun(B,B),D2)),set_or1337092689740270186AtMost(B,A2,B2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),D2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_3943_image__add__atLeastLessThan_H,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: B,I: B,J: B] : aa(set(B),set(B),image2(B,B,aTP_Lamp_im(B,fun(B,B),K)),set_or7035219750837199246ssThan(B,I,J)) = set_or7035219750837199246ssThan(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),I),K),aa(B,B,aa(B,fun(B,B),plus_plus(B),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_3944_image__minus__const__greaterThanAtMost,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,A2: B,B2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),minus_minus(B),C2)),set_or3652927894154168847AtMost(B,A2,B2)) = set_or7035219750837199246ssThan(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_3945_image__diff__atLeastLessThan,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,A2: B,B2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),minus_minus(B),C2)),set_or7035219750837199246ssThan(B,A2,B2)) = set_or3652927894154168847AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_3946_image__uminus__atLeastLessThan,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B,Y: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),set_or7035219750837199246ssThan(B,X,Y)) = set_or3652927894154168847AtMost(B,aa(B,B,uminus_uminus(B),Y),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_atLeastLessThan
tff(fact_3947_image__uminus__greaterThanAtMost,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B,Y: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),set_or3652927894154168847AtMost(B,X,Y)) = set_or7035219750837199246ssThan(B,aa(B,B,uminus_uminus(B),Y),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_greaterThanAtMost
tff(fact_3948_image__mult__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [D2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),D2)
         => ( aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),times_times(B),D2)),set_or1337092689740270186AtMost(B,A2,B2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),times_times(B),D2),A2),aa(B,B,aa(B,fun(B,B),times_times(B),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_3949_image__divide__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [D2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),D2)
         => ( aa(set(B),set(B),image2(B,B,aTP_Lamp_io(B,fun(B,B),D2)),set_or1337092689740270186AtMost(B,A2,B2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),D2),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_3950_nth__image__indices,axiom,
    ! [B: $tType,L: list(B)] : aa(set(nat),set(B),image2(nat,B,nth(B,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),L))) = aa(list(B),set(B),set2(B),L) ).

% nth_image_indices
tff(fact_3951_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,Ma: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,Ma)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N,Ma))) ).

% take_bit_num_simps(4)
tff(fact_3952_imageE,axiom,
    ! [B: $tType,C: $tType,B2: B,F: fun(C,B),A4: set(C)] :
      ( aa(set(B),$o,member(B,B2),aa(set(C),set(B),image2(C,B,F),A4))
     => ~ ! [X2: C] :
            ( ( B2 = aa(C,B,F,X2) )
           => ~ aa(set(C),$o,member(C,X2),A4) ) ) ).

% imageE
tff(fact_3953_image__image,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,B),G: fun(D,C),A4: set(D)] : aa(set(C),set(B),image2(C,B,F),aa(set(D),set(C),image2(D,C,G),A4)) = aa(set(D),set(B),image2(D,B,aa(fun(D,C),fun(D,B),aTP_Lamp_ip(fun(C,B),fun(fun(D,C),fun(D,B)),F),G)),A4) ).

% image_image
tff(fact_3954_Compr__image__eq,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),A4: set(C),Pa: fun(B,$o)] : aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aa(set(C),fun(fun(B,$o),fun(B,$o)),aTP_Lamp_iq(fun(C,B),fun(set(C),fun(fun(B,$o),fun(B,$o))),F),A4),Pa)) = aa(set(C),set(B),image2(C,B,F),aa(fun(C,$o),set(C),collect(C),aa(fun(B,$o),fun(C,$o),aa(set(C),fun(fun(B,$o),fun(C,$o)),aTP_Lamp_ir(fun(C,B),fun(set(C),fun(fun(B,$o),fun(C,$o))),F),A4),Pa))) ).

% Compr_image_eq
tff(fact_3955_le__some__optE,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [Ma: B,X: option(B)] :
          ( aa(option(B),$o,aa(option(B),fun(option(B),$o),ord_less_eq(option(B)),aa(B,option(B),some(B),Ma)),X)
         => ~ ! [M9: B] :
                ( ( X = aa(B,option(B),some(B),M9) )
               => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),M9) ) ) ) ).

% le_some_optE
tff(fact_3956_pigeonhole__infinite,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C)] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,F),A4))
       => ? [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
            & ~ aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aa(fun(B,C),fun(B,fun(B,$o)),aTP_Lamp_is(set(B),fun(fun(B,C),fun(B,fun(B,$o))),A4),F),X2))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3957_image__constant__conv,axiom,
    ! [C: $tType,B: $tType,C2: B,A4: set(C)] :
      aa(set(C),set(B),image2(C,B,aa(B,fun(C,B),aTP_Lamp_it(B,fun(C,B)),C2)),A4) = $ite(A4 = bot_bot(set(C)),bot_bot(set(B)),aa(set(B),set(B),insert(B,C2),bot_bot(set(B)))) ).

% image_constant_conv
tff(fact_3958_image__constant,axiom,
    ! [B: $tType,C: $tType,X: B,A4: set(B),C2: C] :
      ( aa(set(B),$o,member(B,X),A4)
     => ( aa(set(B),set(C),image2(B,C,aTP_Lamp_iu(C,fun(B,C),C2)),A4) = aa(set(C),set(C),insert(C,C2),bot_bot(set(C))) ) ) ).

% image_constant
tff(fact_3959_finite__conv__nat__seg__image,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
    <=> ? [N4: nat,F5: fun(nat,B)] : A4 = aa(set(nat),set(B),image2(nat,B,F5),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N4))) ) ).

% finite_conv_nat_seg_image
tff(fact_3960_nat__seg__image__imp__finite,axiom,
    ! [B: $tType,A4: set(B),F: fun(nat,B),N: nat] :
      ( ( A4 = aa(set(nat),set(B),image2(nat,B,F),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N))) )
     => aa(set(B),$o,finite_finite2(B),A4) ) ).

% nat_seg_image_imp_finite
tff(fact_3961_sum_Oimage__gen,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(B),H: fun(B,C),G: fun(B,D)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),S) = aa(set(D),C,aa(fun(D,C),fun(set(D),C),groups7311177749621191930dd_sum(D,C),aa(fun(B,D),fun(D,C),aa(fun(B,C),fun(fun(B,D),fun(D,C)),aTP_Lamp_iw(set(B),fun(fun(B,C),fun(fun(B,D),fun(D,C))),S),H),G)),aa(set(B),set(D),image2(B,D,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_3962_prod_Oimage__gen,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(B),H: fun(B,C),G: fun(B,D)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),S) = aa(set(D),C,aa(fun(D,C),fun(set(D),C),groups7121269368397514597t_prod(D,C),aa(fun(B,D),fun(D,C),aa(fun(B,C),fun(fun(B,D),fun(D,C)),aTP_Lamp_ix(set(B),fun(fun(B,C),fun(fun(B,D),fun(D,C))),S),H),G)),aa(set(B),set(D),image2(B,D,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_3963_Gcd__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_Gcd(C)
     => ! [A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),dvd_dvd(C),aa(B,C,F,X2)),aa(B,C,G,X2)) )
         => aa(C,$o,aa(C,fun(C,$o),dvd_dvd(C),gcd_Gcd(C,aa(set(B),set(C),image2(B,C,F),A4))),gcd_Gcd(C,aa(set(B),set(C),image2(B,C,G),A4))) ) ) ).

% Gcd_mono
tff(fact_3964_card__image__le,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(C),nat,finite_card(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(B),nat,finite_card(B),A4)) ) ).

% card_image_le
tff(fact_3965_sum_Ogroup,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [S: set(B),T4: set(C),G: fun(B,C),H: fun(B,D)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(C),$o,finite_finite2(C),T4)
           => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G),S)),T4)
             => ( aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),aa(fun(B,D),fun(C,D),aa(fun(B,C),fun(fun(B,D),fun(C,D)),aTP_Lamp_iz(set(B),fun(fun(B,C),fun(fun(B,D),fun(C,D))),S),G),H)),T4) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),H),S) ) ) ) ) ) ).

% sum.group
tff(fact_3966_prod_Ogroup,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [S: set(B),T4: set(C),G: fun(B,C),H: fun(B,D)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(C),$o,finite_finite2(C),T4)
           => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G),S)),T4)
             => ( aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),aa(fun(B,D),fun(C,D),aa(fun(B,C),fun(fun(B,D),fun(C,D)),aTP_Lamp_ja(set(B),fun(fun(B,C),fun(fun(B,D),fun(C,D))),S),G),H)),T4) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),H),S) ) ) ) ) ) ).

% prod.group
tff(fact_3967_surj__card__le,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(C),F: fun(B,C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),B4),aa(set(B),set(C),image2(B,C,F),A4))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(C),nat,finite_card(C),B4)),aa(set(B),nat,finite_card(B),A4)) ) ) ).

% surj_card_le
tff(fact_3968_set__image__eq__pointwiseI,axiom,
    ! [C: $tType,B: $tType,L: list(B),L3: list(B),F: fun(B,C)] :
      ( ( aa(list(B),nat,size_size(list(B)),L) = aa(list(B),nat,size_size(list(B)),L3) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L))
           => ( aa(B,C,F,aa(nat,B,nth(B,L),I2)) = aa(B,C,F,aa(nat,B,nth(B,L3),I2)) ) )
       => ( aa(set(B),set(C),image2(B,C,F),aa(list(B),set(B),set2(B),L)) = aa(set(B),set(C),image2(B,C,F),aa(list(B),set(B),set2(B),L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_3969_in__set__image__conv__nth,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),X: C,L: list(C)] :
      ( aa(set(B),$o,member(B,aa(C,B,F,X)),aa(set(C),set(B),image2(C,B,F),aa(list(C),set(C),set2(C),L)))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(C),nat,size_size(list(C)),L))
          & ( aa(C,B,F,aa(nat,C,nth(C,L),I3)) = aa(C,B,F,X) ) ) ) ).

% in_set_image_conv_nth
tff(fact_3970_and__not__num_Osimps_I8_J,axiom,
    ! [Ma: num,N: num] : bit_and_not_num(aa(num,num,bit1,Ma),aa(num,num,bit0,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_jb(num,option(num)),bit_and_not_num(Ma,N)) ).

% and_not_num.simps(8)
tff(fact_3971_image__mult__atLeastAtMost__if,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,X: B,Y: B] :
          aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),times_times(B),C2)),set_or1337092689740270186AtMost(B,X,Y)) = $ite(
            aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),
            set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),X),aa(B,B,aa(B,fun(B,B),times_times(B),C2),Y)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),Y),aa(B,B,aa(B,fun(B,B),times_times(B),C2),X)),bot_bot(set(B))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_3972_image__mult__atLeastAtMost__if_H,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [C2: B,X: B,Y: B] :
          aa(set(B),set(B),image2(B,B,aTP_Lamp_jc(B,fun(B,B),C2)),set_or1337092689740270186AtMost(B,X,Y)) = $ite(
            aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),times_times(B),X),C2),aa(B,B,aa(B,fun(B,B),times_times(B),Y),C2)),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),times_times(B),Y),C2),aa(B,B,aa(B,fun(B,B),times_times(B),X),C2))),
            bot_bot(set(B)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_3973_image__affinity__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Ma: B,C2: B,A2: B,B2: B] :
          aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),aTP_Lamp_jd(B,fun(B,fun(B,B)),Ma),C2)),set_or1337092689740270186AtMost(B,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(B,A2,B2) = bot_bot(set(B)),
            bot_bot(set(B)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Ma),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),A2)),C2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),B2)),C2)),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),B2)),C2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_3974_image__affinity__atLeastAtMost__diff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Ma: B,C2: B,A2: B,B2: B] :
          aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),aTP_Lamp_je(B,fun(B,fun(B,B)),Ma),C2)),set_or1337092689740270186AtMost(B,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(B,A2,B2) = bot_bot(set(B)),
            bot_bot(set(B)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Ma),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),A2)),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),B2)),C2)),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),B2)),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Ma),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_3975_image__affinity__atLeastAtMost__div,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Ma: B,C2: B,A2: B,B2: B] :
          aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),aTP_Lamp_jf(B,fun(B,fun(B,B)),Ma),C2)),set_or1337092689740270186AtMost(B,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(B,A2,B2) = bot_bot(set(B)),
            bot_bot(set(B)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Ma),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Ma)),C2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),Ma)),C2)),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),Ma)),C2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Ma)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_3976_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Ma: B,C2: B,A2: B,B2: B] :
          aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),aTP_Lamp_jg(B,fun(B,fun(B,B)),Ma),C2)),set_or1337092689740270186AtMost(B,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(B,A2,B2) = bot_bot(set(B)),
            bot_bot(set(B)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Ma),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Ma)),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),Ma)),C2)),set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),Ma)),C2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),Ma)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_3977_sum__fun__comp,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( semiring_1(D)
     => ! [S: set(B),R2: set(C),G: fun(B,C),F: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(set(C),$o,finite_finite2(C),R2)
           => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G),S)),R2)
             => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_jh(fun(B,C),fun(fun(C,D),fun(B,D)),G),F)),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),aa(fun(C,D),fun(C,D),aa(fun(B,C),fun(fun(C,D),fun(C,D)),aTP_Lamp_ji(set(B),fun(fun(B,C),fun(fun(C,D),fun(C,D))),S),G),F)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_3978_less__eq__option__Some,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(option(B),$o,aa(option(B),fun(option(B),$o),ord_less_eq(option(B)),aa(B,option(B),some(B),X)),aa(B,option(B),some(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ).

% less_eq_option_Some
tff(fact_3979_translation__subtract__diff,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,S2: set(B),Ta: set(B)] : aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),Ta)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),S2)),aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),Ta)) ) ).

% translation_subtract_diff
tff(fact_3980_option_Osize_I4_J,axiom,
    ! [B: $tType,X22: B] : aa(option(B),nat,size_size(option(B)),aa(B,option(B),some(B),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_3981_translation__diff,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,S2: set(B),Ta: set(B)] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),Ta)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),S2)),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),Ta)) ) ).

% translation_diff
tff(fact_3982_op__conc__empty__img__id,axiom,
    ! [B: $tType,L4: set(list(B))] : aa(set(list(B)),set(list(B)),image2(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),nil(B))),L4) = L4 ).

% op_conc_empty_img_id
tff(fact_3983_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_3984_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_3985_Gcd__abs__eq,axiom,
    ! [K3: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K3)) = gcd_Gcd(int,K3) ).

% Gcd_abs_eq
tff(fact_3986_Gcd__int__eq,axiom,
    ! [N5: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N5)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N5)) ).

% Gcd_int_eq
tff(fact_3987_Gcd__nat__abs__eq,axiom,
    ! [K3: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_jk(int,nat)),K3)) = aa(int,nat,nat2,gcd_Gcd(int,K3)) ).

% Gcd_nat_abs_eq
tff(fact_3988_zero__notin__Suc__image,axiom,
    ! [A4: set(nat)] : ~ aa(set(nat),$o,member(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),A4)) ).

% zero_notin_Suc_image
tff(fact_3989_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_3990_option_Osize__neq,axiom,
    ! [B: $tType,X: option(B)] : aa(option(B),nat,size_size(option(B)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_3991_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_3992_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_3993_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_3994_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_3995_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_3996_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N))) ).

% atMost_Suc_eq_insert_0
tff(fact_3997_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_3998_finite__set__image,axiom,
    ! [B: $tType,A4: set(list(B))] :
      ( aa(set(set(B)),$o,finite_finite2(set(B)),aa(set(list(B)),set(set(B)),image2(list(B),set(B),set2(B)),A4))
     => ( ! [Xs2: list(B)] :
            ( aa(set(list(B)),$o,member(list(B),Xs2),A4)
           => distinct(B,Xs2) )
       => aa(set(list(B)),$o,finite_finite2(list(B)),A4) ) ) ).

% finite_set_image
tff(fact_3999_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_jl(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_4000_option_Ocase__distrib,axiom,
    ! [C: $tType,B: $tType,D: $tType,H: fun(C,B),F1: C,F22: fun(D,C),Option: option(D)] : aa(C,B,H,case_option(C,D,F1,F22,Option)) = case_option(B,D,aa(C,B,H,F1),aa(fun(D,C),fun(D,B),aTP_Lamp_ip(fun(C,B),fun(fun(D,C),fun(D,B)),H),F22),Option) ).

% option.case_distrib
tff(fact_4001_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_jm(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_4002_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),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_4003_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,X: nat,Y: nat] :
      aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_jn(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4004_translation__Compl,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,Ta: set(B)] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),aa(set(B),set(B),uminus_uminus(set(B)),Ta)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),Ta)) ) ).

% translation_Compl
tff(fact_4005_translation__subtract__Compl,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,Ta: set(B)] : aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),aa(set(B),set(B),uminus_uminus(set(B)),Ta)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),Ta)) ) ).

% translation_subtract_Compl
tff(fact_4006_option_Osize__gen_I2_J,axiom,
    ! [B: $tType,X: fun(B,nat),X22: B] : size_option(B,X,aa(B,option(B),some(B),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_4007_take__bit__num__simps_I6_J,axiom,
    ! [R: num,Ma: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit0,Ma)) = case_option(option(num),num,none(num),aTP_Lamp_jo(num,option(num)),bit_take_bit_num(pred_numeral(R),Ma)) ).

% take_bit_num_simps(6)
tff(fact_4008_set__Cons__sing__Nil,axiom,
    ! [B: $tType,A4: set(B)] : set_Cons(B,A4,aa(set(list(B)),set(list(B)),insert(list(B),nil(B)),bot_bot(set(list(B))))) = aa(set(B),set(list(B)),image2(B,list(B),aTP_Lamp_jp(B,list(B))),A4) ).

% set_Cons_sing_Nil
tff(fact_4009_image__Collect__subsetI,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),F: fun(B,C),B4: set(C)] :
      ( ! [X2: B] :
          ( aa(B,$o,Pa,X2)
         => aa(set(C),$o,member(C,aa(B,C,F,X2)),B4) )
     => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F),aa(fun(B,$o),set(B),collect(B),Pa))),B4) ) ).

% image_Collect_subsetI
tff(fact_4010_take__bit__num__simps_I1_J,axiom,
    ! [Ma: num] : bit_take_bit_num(zero_zero(nat),Ma) = none(num) ).

% take_bit_num_simps(1)
tff(fact_4011_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,Ma: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,Ma)) = case_option(option(num),num,none(num),aTP_Lamp_jo(num,option(num)),bit_take_bit_num(N,Ma)) ).

% take_bit_num_simps(3)
tff(fact_4012_case__optionE,axiom,
    ! [B: $tType,Pa: $o,Q: fun(B,$o),X: option(B)] :
      ( case_option($o,B,(Pa),Q,X)
     => ( ( ( X = none(B) )
         => ~ (Pa) )
       => ~ ! [Y2: B] :
              ( ( X = aa(B,option(B),some(B),Y2) )
             => ~ aa(B,$o,Q,Y2) ) ) ) ).

% case_optionE
tff(fact_4013_option_Odisc__eq__case_I1_J,axiom,
    ! [B: $tType,Option: option(B)] :
      ( ( Option = none(B) )
    <=> case_option($o,B,$true,aTP_Lamp_hh(B,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_4014_option_Odisc__eq__case_I2_J,axiom,
    ! [B: $tType,Option: option(B)] :
      ( ( Option != none(B) )
    <=> case_option($o,B,$false,aTP_Lamp_jq(B,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_4015_uminus__set__def,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(B),set(B),uminus_uminus(set(B)),A4) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),uminus_uminus(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4))) ).

% uminus_set_def
tff(fact_4016_Collect__neg__eq,axiom,
    ! [B: $tType,Pa: fun(B,$o)] : aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,$o),set(B),collect(B),Pa)) ).

% Collect_neg_eq
tff(fact_4017_Compl__eq,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(B),set(B),uminus_uminus(set(B)),A4) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_jr(set(B),fun(B,$o),A4)) ).

% Compl_eq
tff(fact_4018_option_Osize__gen_I1_J,axiom,
    ! [B: $tType,X: fun(B,nat)] : size_option(B,X,none(B)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_4019_less__option__def,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: option(B),Y: option(B)] :
          ( aa(option(B),$o,aa(option(B),fun(option(B),$o),ord_less(option(B)),X),Y)
        <=> case_option($o,B,$false,aTP_Lamp_jt(option(B),fun(B,$o),X),Y) ) ) ).

% less_option_def
tff(fact_4020_refines__option,axiom,
    ! [C: $tType,B: $tType,A2: option(B),A5: option(B),M1: heap_Time_Heap(C),M12: heap_Time_Heap(C),M22: fun(B,heap_Time_Heap(C)),M23: fun(B,heap_Time_Heap(C))] :
      ( ( A2 = A5 )
     => ( refine_Imp_refines(C,M1,M12)
       => ( ! [X2: B] : refine_Imp_refines(C,aa(B,heap_Time_Heap(C),M22,X2),aa(B,heap_Time_Heap(C),M23,X2))
         => refine_Imp_refines(C,case_option(heap_Time_Heap(C),B,M1,M22,A2),case_option(heap_Time_Heap(C),B,M12,M23,A5)) ) ) ) ).

% refines_option
tff(fact_4021_less__eq__option__def,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: option(B),Y: option(B)] :
          ( aa(option(B),$o,aa(option(B),fun(option(B),$o),ord_less_eq(option(B)),X),Y)
        <=> case_option($o,B,$true,aTP_Lamp_ju(option(B),fun(B,$o),Y),X) ) ) ).

% less_eq_option_def
tff(fact_4022_option_Osize_I3_J,axiom,
    ! [B: $tType] : aa(option(B),nat,size_size(option(B)),none(B)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(3)
tff(fact_4023_take__bit__num__eq__None__imp,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: num] :
          ( ( bit_take_bit_num(Ma,N) = none(num) )
         => ( aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),Ma),aa(num,B,numeral_numeral(B),N)) = zero_zero(B) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_4024_and__not__num__eq__None__iff,axiom,
    ! [Ma: num,N: num] :
      ( ( bit_and_not_num(Ma,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Ma)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_4025_take__bit__num__def,axiom,
    ! [N: nat,Ma: num] :
      bit_take_bit_num(N,Ma) = $ite(aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(num,nat,numeral_numeral(nat),Ma)) = zero_zero(nat),none(num),aa(num,option(num),some(num),aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(num,nat,numeral_numeral(nat),Ma))))) ).

% take_bit_num_def
tff(fact_4026_rel__of__empty,axiom,
    ! [C: $tType,B: $tType,Pa: fun(product_prod(B,C),$o)] : rel_of(B,C,aTP_Lamp_jv(B,option(C)),Pa) = bot_bot(set(product_prod(B,C))) ).

% rel_of_empty
tff(fact_4027_ran__nth__set__encoding__conv,axiom,
    ! [B: $tType,L: list(B)] : ran(nat,B,aTP_Lamp_jw(list(B),fun(nat,option(B)),L)) = aa(list(B),set(B),set2(B),L) ).

% ran_nth_set_encoding_conv
tff(fact_4028_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa2 = aa(num,num,bit0,N2)
             => ( Y != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa2 = aa(num,num,bit1,N2)
               => ( Y != none(num) ) ) )
           => ( ! [M2: num] :
                  ( ( X = aa(num,num,bit0,M2) )
                 => ( ( Xa2 = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( X = aa(num,num,bit0,M2) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M2,N2)) ) ) )
               => ( ! [M2: num] :
                      ( ( X = aa(num,num,bit0,M2) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit1,N2) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M2,N2)) ) ) )
                 => ( ! [M2: num] :
                        ( ( X = aa(num,num,bit1,M2) )
                       => ( ( Xa2 = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( X = aa(num,num,bit1,M2) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa(num,num,bit0,N2) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_jb(num,option(num)),bit_and_not_num(M2,N2)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( X = aa(num,num,bit1,M2) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit1,N2) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M2,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_4029_Gcd__int__def,axiom,
    ! [K3: set(int)] : gcd_Gcd(int,K3) = 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),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))),K3))) ).

% Gcd_int_def
tff(fact_4030_map__update__eta__repair_I2_J,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B)),K: C,V2: B] :
      ( ( aa(C,option(B),Ma,K) = none(B) )
     => ( ran(C,B,aa(B,fun(C,option(B)),aa(C,fun(B,fun(C,option(B))),aTP_Lamp_jx(fun(C,option(B)),fun(C,fun(B,fun(C,option(B)))),Ma),K),V2)) = aa(set(B),set(B),insert(B,V2),ran(C,B,Ma)) ) ) ).

% map_update_eta_repair(2)
tff(fact_4031_fun__comp__eq__conv,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(D,C),G: fun(B,D),Fg: fun(B,C)] :
      ( ( aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),F),G) = Fg )
    <=> ! [X3: B] : aa(D,C,F,aa(B,D,G,X3)) = aa(B,C,Fg,X3) ) ).

% fun_comp_eq_conv
tff(fact_4032_comp__cong__right,axiom,
    ! [D: $tType,C: $tType,B: $tType,X: fun(B,C),Y: fun(B,C),F: fun(C,D)] :
      ( ( X = Y )
     => ( aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),F),X) = aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),F),Y) ) ) ).

% comp_cong_right
tff(fact_4033_comp__cong__left,axiom,
    ! [C: $tType,B: $tType,D: $tType,X: fun(B,C),Y: fun(B,C),F: fun(D,B)] :
      ( ( X = Y )
     => ( aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),X),F) = aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),Y),F) ) ) ).

% comp_cong_left
tff(fact_4034_rewriteR__comp__comp2,axiom,
    ! [D: $tType,C: $tType,F6: $tType,E: $tType,B: $tType,G: fun(D,C),H: fun(B,D),R1: fun(E,C),R22: fun(B,E),F: fun(C,F6),L: fun(E,F6)] :
      ( ( aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),G),H) = aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),R1),R22) )
     => ( ( aa(fun(E,C),fun(E,F6),aa(fun(C,F6),fun(fun(E,C),fun(E,F6)),comp(C,F6,E),F),R1) = L )
       => ( aa(fun(B,D),fun(B,F6),aa(fun(D,F6),fun(fun(B,D),fun(B,F6)),comp(D,F6,B),aa(fun(D,C),fun(D,F6),aa(fun(C,F6),fun(fun(D,C),fun(D,F6)),comp(C,F6,D),F),G)),H) = aa(fun(B,E),fun(B,F6),aa(fun(E,F6),fun(fun(B,E),fun(B,F6)),comp(E,F6,B),L),R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_4035_rewriteL__comp__comp2,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType,F6: $tType,F: fun(D,C),G: fun(B,D),L1: fun(E,C),L22: fun(B,E),H: fun(F6,B),R: fun(F6,E)] :
      ( ( aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),F),G) = aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),L1),L22) )
     => ( ( aa(fun(F6,B),fun(F6,E),aa(fun(B,E),fun(fun(F6,B),fun(F6,E)),comp(B,E,F6),L22),H) = R )
       => ( aa(fun(F6,D),fun(F6,C),aa(fun(D,C),fun(fun(F6,D),fun(F6,C)),comp(D,C,F6),F),aa(fun(F6,B),fun(F6,D),aa(fun(B,D),fun(fun(F6,B),fun(F6,D)),comp(B,D,F6),G),H)) = aa(fun(F6,E),fun(F6,C),aa(fun(E,C),fun(fun(F6,E),fun(F6,C)),comp(E,C,F6),L1),R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_4036_rewriteR__comp__comp,axiom,
    ! [D: $tType,E: $tType,C: $tType,B: $tType,G: fun(D,C),H: fun(B,D),R: fun(B,C),F: fun(C,E)] :
      ( ( aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),G),H) = R )
     => ( aa(fun(B,D),fun(B,E),aa(fun(D,E),fun(fun(B,D),fun(B,E)),comp(D,E,B),aa(fun(D,C),fun(D,E),aa(fun(C,E),fun(fun(D,C),fun(D,E)),comp(C,E,D),F),G)),H) = aa(fun(B,C),fun(B,E),aa(fun(C,E),fun(fun(B,C),fun(B,E)),comp(C,E,B),F),R) ) ) ).

% rewriteR_comp_comp
tff(fact_4037_rewriteL__comp__comp,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,F: fun(D,C),G: fun(B,D),L: fun(B,C),H: fun(E,B)] :
      ( ( aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),F),G) = L )
     => ( aa(fun(E,D),fun(E,C),aa(fun(D,C),fun(fun(E,D),fun(E,C)),comp(D,C,E),F),aa(fun(E,B),fun(E,D),aa(fun(B,D),fun(fun(E,B),fun(E,D)),comp(B,D,E),G),H)) = aa(fun(E,B),fun(E,C),aa(fun(B,C),fun(fun(E,B),fun(E,C)),comp(B,C,E),L),H) ) ) ).

% rewriteL_comp_comp
tff(fact_4038_comp__funpow,axiom,
    ! [B: $tType,C: $tType,N: nat,F: fun(C,C)] : aa(fun(fun(B,C),fun(B,C)),fun(fun(B,C),fun(B,C)),aa(nat,fun(fun(fun(B,C),fun(B,C)),fun(fun(B,C),fun(B,C))),compow(fun(fun(B,C),fun(B,C))),N),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),F)) = aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(fun(C,C),fun(C,C),aa(nat,fun(fun(C,C),fun(C,C)),compow(fun(C,C)),N),F)) ).

% comp_funpow
tff(fact_4039_option_Omap__ident,axiom,
    ! [B: $tType,Ta: option(B)] : aa(option(B),option(B),map_option(B,B,aTP_Lamp_il(B,B)),Ta) = Ta ).

% option.map_ident
tff(fact_4040_timeFrame_Osimps_I2_J,axiom,
    ! [B: $tType,N: nat] : heap_Time_timeFrame(B,N,none(product_prod(B,product_prod(heap_ext(product_unit),nat)))) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ).

% timeFrame.simps(2)
tff(fact_4041_funpow_Osimps_I2_J,axiom,
    ! [B: $tType,N: nat,F: fun(B,B)] : aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,suc,N)),F) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),F),aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F)) ).

% funpow.simps(2)
tff(fact_4042_funpow__Suc__right,axiom,
    ! [B: $tType,N: nat,F: fun(B,B)] : aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,suc,N)),F) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F)),F) ).

% funpow_Suc_right
tff(fact_4043_funpow__add,axiom,
    ! [B: $tType,Ma: nat,N: nat,F: fun(B,B)] : aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),F) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Ma),F)),aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F)) ).

% funpow_add
tff(fact_4044_sum__comp__morphism,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( ( comm_monoid_add(C)
        & comm_monoid_add(B) )
     => ! [H: fun(C,B),G: fun(D,C),A4: set(D)] :
          ( ( aa(C,B,H,zero_zero(C)) = zero_zero(B) )
         => ( ! [X2: C,Y2: C] : aa(C,B,H,aa(C,C,aa(C,fun(C,C),plus_plus(C),X2),Y2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(C,B,H,X2)),aa(C,B,H,Y2))
           => ( aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7311177749621191930dd_sum(D,B),aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),H),G)),A4) = aa(C,B,H,aa(set(D),C,aa(fun(D,C),fun(set(D),C),groups7311177749621191930dd_sum(D,C),G),A4)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_4045_map__option__case,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Y: option(C)] : aa(option(C),option(B),map_option(C,B,F),Y) = case_option(option(B),C,none(B),aTP_Lamp_jy(fun(C,B),fun(C,option(B)),F),Y) ).

% map_option_case
tff(fact_4046_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [A4: set(B),H: fun(B,C),G: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [X2: B,Y2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => ( aa(set(B),$o,member(B,Y2),A4)
                 => ( ( X2 != Y2 )
                   => ( ( aa(B,C,H,X2) = aa(B,C,H,Y2) )
                     => ( aa(C,D,G,aa(B,C,H,X2)) = zero_zero(D) ) ) ) ) )
           => ( aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),G),H)),A4) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_4047_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [A4: set(B),H: fun(B,C),G: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [X2: B,Y2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => ( aa(set(B),$o,member(B,Y2),A4)
                 => ( ( X2 != Y2 )
                   => ( ( aa(B,C,H,X2) = aa(B,C,H,Y2) )
                     => ( aa(C,D,G,aa(B,C,H,X2)) = one_one(D) ) ) ) ) )
           => ( aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),G),H)),A4) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_4048_sum__image__le,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [I5: set(B),G: fun(D,C),F: fun(B,D)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),I5)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(D,C,G,aa(B,D,F,I2))) )
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(D),C,aa(fun(D,C),fun(set(D),C),groups7311177749621191930dd_sum(D,C),G),aa(set(B),set(D),image2(B,D,F),I5))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),G),F)),I5)) ) ) ) ).

% sum_image_le
tff(fact_4049_ran__empty,axiom,
    ! [C: $tType,B: $tType] : ran(C,B,aTP_Lamp_jz(C,option(B))) = bot_bot(set(B)) ).

% ran_empty
tff(fact_4050_ran__map__option,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(D,B),Ma: fun(C,option(D))] : ran(C,B,aa(fun(C,option(D)),fun(C,option(B)),aTP_Lamp_ka(fun(D,B),fun(fun(C,option(D)),fun(C,option(B))),F),Ma)) = aa(set(D),set(B),image2(D,B,F),ran(C,D,Ma)) ).

% ran_map_option
tff(fact_4051_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa2 = aa(num,num,bit0,N2)
             => ( Y != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa2 = aa(num,num,bit1,N2)
               => ( Y != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M2: num] : X = aa(num,num,bit0,M2)
               => ( ( Xa2 = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M2: num] :
                    ( ( X = aa(num,num,bit0,M2) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M2,N2)) ) ) )
               => ( ! [M2: num] :
                      ( ( X = aa(num,num,bit0,M2) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit1,N2) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M2,N2)) ) ) )
                 => ( ( ? [M2: num] : X = aa(num,num,bit1,M2)
                     => ( ( Xa2 = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M2: num] :
                          ( ( X = aa(num,num,bit1,M2) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa(num,num,bit0,N2) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M2,N2)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( X = aa(num,num,bit1,M2) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit1,N2) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_jb(num,option(num)),bit_un7362597486090784418nd_num(M2,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_4052_disjE__realizer2,axiom,
    ! [C: $tType,B: $tType,Pa: $o,Q: fun(B,$o),X: option(B),R2: fun(C,$o),F: C,G: fun(B,C)] :
      ( case_option($o,B,(Pa),Q,X)
     => ( ( (Pa)
         => aa(C,$o,R2,F) )
       => ( ! [Q4: B] :
              ( aa(B,$o,Q,Q4)
             => aa(C,$o,R2,aa(B,C,G,Q4)) )
         => aa(C,$o,R2,case_option(C,B,F,G,X)) ) ) ) ).

% disjE_realizer2
tff(fact_4053_map__option__o__empty,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(D,C),X4: B] : aa(B,option(C),aa(fun(B,option(D)),fun(B,option(C)),aa(fun(option(D),option(C)),fun(fun(B,option(D)),fun(B,option(C))),comp(option(D),option(C),B),map_option(D,C,F)),aTP_Lamp_kb(B,option(D))),X4) = none(C) ).

% map_option_o_empty
tff(fact_4054_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Y: B,X: B] : aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),Y)),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X)),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4055_rotate__add,axiom,
    ! [B: $tType,Ma: nat,N: nat] : rotate(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),rotate(B,Ma)),rotate(B,N)) ).

% rotate_add
tff(fact_4056_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_4057_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_4058_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_4059_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_4060_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_4061_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_4062_lift__def,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C)] : heap_Time_lift(B,C,F) = aa(fun(B,C),fun(B,heap_Time_Heap(C)),aa(fun(C,heap_Time_Heap(C)),fun(fun(B,C),fun(B,heap_Time_Heap(C))),comp(C,heap_Time_Heap(C),B),heap_Time_return(C)),F) ).

% lift_def
tff(fact_4063_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_4064_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,K: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_4065_bit__drop__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [N: nat,A2: B] : aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),N),A2)) = aa(fun(nat,nat),fun(nat,$o),aa(fun(nat,$o),fun(fun(nat,nat),fun(nat,$o)),comp(nat,$o,nat),aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2)),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_4066_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_4067_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_4068_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_4069_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,G,zero_zero(nat))),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_4070_sum_OatLeastLessThan__shift__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_4071_prod_OatLeastLessThan__shift__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_4072_sum_OatLeast__atMost__pred__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aTP_Lamp_kc(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_4073_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aTP_Lamp_kc(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_4074_prod_OatLeast__atMost__pred__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aTP_Lamp_kc(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_4075_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,B),Ma: nat,N: nat] : aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aTP_Lamp_kc(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_4076_and__num__eq__None__iff,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Ma: num,N: num] :
          ( ( bit_un7362597486090784418nd_num(Ma,N) = none(num) )
        <=> ( aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = zero_zero(B) ) ) ) ).

% and_num_eq_None_iff
tff(fact_4077_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(int,B),Ma: nat,N: nat] : aa(set(int),B,aa(fun(int,B),fun(set(int),B),groups7311177749621191930dd_sum(int,B),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,int),fun(nat,B),aa(fun(int,B),fun(fun(nat,int),fun(nat,B)),comp(int,B,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_4078_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(int,B),Ma: nat,N: nat] : aa(set(int),B,aa(fun(int,B),fun(set(int),B),groups7121269368397514597t_prod(int,B),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,int),fun(nat,B),aa(fun(int,B),fun(fun(nat,int),fun(nat,B)),comp(int,B,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Ma,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_4079_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(int,B),Ma: nat,N: nat] : aa(set(int),B,aa(fun(int,B),fun(set(int),B),groups7311177749621191930dd_sum(int,B),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,int),fun(nat,B),aa(fun(int,B),fun(fun(nat,int),fun(nat,B)),comp(int,B,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_4080_sum_OatLeastAtMost__shift__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_4081_prod_OatLeastAtMost__shift__0,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Ma: nat,N: nat,G: fun(nat,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),G),set_or1337092689740270186AtMost(nat,Ma,N)) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,nat),fun(nat,B),aa(fun(nat,B),fun(fun(nat,nat),fun(nat,B)),comp(nat,B,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),Ma))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_4082_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(int,B),Ma: nat,N: nat] : aa(set(int),B,aa(fun(int,B),fun(set(int),B),groups7121269368397514597t_prod(int,B),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,int),fun(nat,B),aa(fun(int,B),fun(fun(nat,int),fun(nat,B)),comp(int,B,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Ma,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_4083_numeral__and__num,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344872417868541ns_and(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = case_option(B,num,zero_zero(B),numeral_numeral(B),bit_un7362597486090784418nd_num(Ma,N)) ) ).

% numeral_and_num
tff(fact_4084_and__num_Osimps_I9_J,axiom,
    ! [Ma: num,N: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,Ma),aa(num,num,bit1,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_jb(num,option(num)),bit_un7362597486090784418nd_num(Ma,N)) ).

% and_num.simps(9)
tff(fact_4085_make_H__def,axiom,
    ! [B: $tType,C: $tType] :
      ( ( semiring_1(C)
        & heap(B) )
     => ! [I: code_integer,F: fun(C,B)] : array_make2(C,B,I,F) = array_make(B,aa(code_integer,nat,code_nat_of_integer,I),aa(fun(nat,C),fun(nat,B),aa(fun(C,B),fun(fun(nat,C),fun(nat,B)),comp(C,B,nat),F),semiring_1_of_nat(C))) ) ).

% make'_def
tff(fact_4086_Code__Numeral_Onegative__def,axiom,
    code_negative = aa(fun(num,code_integer),fun(num,code_integer),aa(fun(code_integer,code_integer),fun(fun(num,code_integer),fun(num,code_integer)),comp(code_integer,code_integer,num),uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_4087_empty__natural,axiom,
    ! [D: $tType,C: $tType,E: $tType,B: $tType,F: fun(B,D),G: fun(E,C)] : aa(fun(B,D),fun(B,set(C)),aa(fun(D,set(C)),fun(fun(B,D),fun(B,set(C))),comp(D,set(C),B),aTP_Lamp_kd(D,set(C))),F) = aa(fun(B,set(E)),fun(B,set(C)),aa(fun(set(E),set(C)),fun(fun(B,set(E)),fun(B,set(C))),comp(set(E),set(C),B),image2(E,C,G)),aTP_Lamp_ke(B,set(E))) ).

% empty_natural
tff(fact_4088_measure__function__int,axiom,
    fun_is_measure(int,aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))) ).

% measure_function_int
tff(fact_4089_conj__comp__iff,axiom,
    ! [C: $tType,B: $tType,Pa: fun(C,$o),Q: fun(C,$o),G: fun(B,C),X4: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(C,$o),fun(fun(B,C),fun(B,$o)),comp(C,$o,B),aa(fun(C,$o),fun(C,$o),aTP_Lamp_kf(fun(C,$o),fun(fun(C,$o),fun(C,$o)),Pa),Q)),G),X4)
    <=> ( aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(C,$o),fun(fun(B,C),fun(B,$o)),comp(C,$o,B),Pa),G),X4)
        & aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(C,$o),fun(fun(B,C),fun(B,$o)),comp(C,$o,B),Q),G),X4) ) ) ).

% conj_comp_iff
tff(fact_4090_measure__size,axiom,
    ! [B: $tType] :
      ( size(B)
     => fun_is_measure(B,size_size(B)) ) ).

% measure_size
tff(fact_4091_sum__of__bool__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [A4: set(B),Pa: fun(B,$o)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),A4)
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aTP_Lamp_kg(fun(B,$o),fun(B,C),Pa)),A4) = aa(nat,C,semiring_1_of_nat(C),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_4092_nth__image,axiom,
    ! [B: $tType,L: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(set(nat),set(B),image2(nat,B,nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(B),set(B),set2(B),take(B,L,Xs)) ) ) ).

% nth_image
tff(fact_4093_foldr__max__sorted,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Y: B] :
          ( sorted_wrt(B,ord_less_eq(B),rev(B,Xs))
         => ( aa(B,B,foldr(B,B,ord_max(B),Xs),Y) = $ite(Xs = nil(B),Y,aa(B,B,aa(B,fun(B,B),ord_max(B),aa(nat,B,nth(B,Xs),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_4094_mset__update,axiom,
    ! [B: $tType,I: nat,Ls: list(B),V2: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ls))
     => ( mset(B,list_update(B,Ls,I,V2)) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),V2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),mset(B,Ls)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),aa(nat,B,nth(B,Ls),I)),zero_zero(multiset(B))))) ) ) ).

% mset_update
tff(fact_4095_inf_Obounded__iff,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),inf_inf(B),B2),C2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_4096_le__inf__iff,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),inf_inf(B),Y),Z))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z) ) ) ) ).

% le_inf_iff
tff(fact_4097_length__rev,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),rev(B,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_rev
tff(fact_4098_take__update,axiom,
    ! [B: $tType,N: nat,L: list(B),I: nat,X: B] : take(B,N,list_update(B,L,I,X)) = list_update(B,take(B,N,L),I,X) ).

% take_update
tff(fact_4099_take0,axiom,
    ! [B: $tType,X4: list(B)] : take(B,zero_zero(nat),X4) = nil(B) ).

% take0
tff(fact_4100_take__eq__Nil,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( take(B,N,Xs) = nil(B) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(B) ) ) ) ).

% take_eq_Nil
tff(fact_4101_take__eq__Nil2,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( nil(B) = take(B,N,Xs) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(B) ) ) ) ).

% take_eq_Nil2
tff(fact_4102_take__all,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),N)
     => ( take(B,N,Xs) = Xs ) ) ).

% take_all
tff(fact_4103_take__all__iff,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( take(B,N,Xs) = Xs )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),N) ) ).

% take_all_iff
tff(fact_4104_mset__append,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] : mset(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),mset(B,Xs)),mset(B,Ys)) ).

% mset_append
tff(fact_4105_inter__compl__diff__conv,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),B4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4) ).

% inter_compl_diff_conv
tff(fact_4106_take__update__cancel,axiom,
    ! [B: $tType,N: nat,Ma: nat,Xs: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( take(B,N,list_update(B,Xs,Ma,Y)) = take(B,N,Xs) ) ) ).

% take_update_cancel
tff(fact_4107_size__mset,axiom,
    ! [B: $tType,Xs: list(B)] : aa(multiset(B),nat,size_size(multiset(B)),mset(B,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% size_mset
tff(fact_4108_sorted__wrt__rev__linord,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B)] :
          ( sorted_wrt(B,aTP_Lamp_kh(B,fun(B,$o)),L)
        <=> sorted_wrt(B,ord_less_eq(B),rev(B,L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_4109_Multiset_Omset__insort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Xs: list(B)] : mset(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),Xs)) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),mset(B,Xs)) ) ).

% Multiset.mset_insort
tff(fact_4110_take__append,axiom,
    ! [B: $tType,N: nat,Xs: list(B),Ys: list(B)] : take(B,N,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,N,Xs)),take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),Ys)) ).

% take_append
tff(fact_4111_take__Cons__numeral,axiom,
    ! [B: $tType,V2: num,X: B,Xs: list(B)] : take(B,aa(num,nat,numeral_numeral(nat),V2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_4112_sum__mult__of__bool__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [A4: set(B),F: fun(B,C),Pa: fun(B,$o)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,$o),fun(B,C),aTP_Lamp_ki(fun(B,C),fun(fun(B,$o),fun(B,C)),F),Pa)),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_4113_sum__of__bool__mult__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [A4: set(B),Pa: fun(B,$o),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aTP_Lamp_kj(fun(B,$o),fun(fun(B,C),fun(B,C)),Pa),F)),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_4114_disjointI,axiom,
    ! [B: $tType,A2: set(B),B2: set(B)] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A2)
         => ~ aa(set(B),$o,member(B,X2),B2) )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A2),B2) = bot_bot(set(B)) ) ) ).

% disjointI
tff(fact_4115_mset__eq__length,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( mset(B,Xs) = mset(B,Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% mset_eq_length
tff(fact_4116_inf_OcoboundedI2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_4117_inf_OcoboundedI1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_4118_inf_Oabsorb__iff2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_4119_inf_Oabsorb__iff1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_4120_inf_Ocobounded2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),B2) ) ).

% inf.cobounded2
tff(fact_4121_inf_Ocobounded1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),A2) ) ).

% inf.cobounded1
tff(fact_4122_inf_Oorder__iff,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( A2 = aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_4123_inf__greatest,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),inf_inf(B),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_4124_inf_OboundedI,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),inf_inf(B),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_4125_inf_OboundedE,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),inf_inf(B),B2),C2))
         => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% inf.boundedE
tff(fact_4126_inf__absorb2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_4127_inf__absorb1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_4128_inf_Oabsorb2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_4129_inf_Oabsorb1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_4130_le__iff__inf,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
        <=> ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_4131_inf__unique,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [F: fun(B,fun(B,B)),X: B,Y: B] :
          ( ! [X2: B,Y2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),F,X2),Y2)),X2)
         => ( ! [X2: B,Y2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),F,X2),Y2)),Y2)
           => ( ! [X2: B,Y2: B,Z2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Z2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),aa(B,B,aa(B,fun(B,B),F,Y2),Z2)) ) )
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y) = aa(B,B,aa(B,fun(B,B),F,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_4132_inf_OorderI,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% inf.orderI
tff(fact_4133_inf_OorderE,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( A2 = aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_4134_le__infI2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [B2: B,X: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),X) ) ) ).

% le_infI2
tff(fact_4135_le__infI1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,X: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),X) ) ) ).

% le_infI1
tff(fact_4136_inf__mono,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)),aa(B,B,aa(B,fun(B,B),inf_inf(B),C2),D2)) ) ) ) ).

% inf_mono
tff(fact_4137_le__infI,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2)) ) ) ) ).

% le_infI
tff(fact_4138_le__infE,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),B2))
         => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A2)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),B2) ) ) ) ).

% le_infE
tff(fact_4139_inf__le2,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y)),Y) ) ).

% inf_le2
tff(fact_4140_inf__le1,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y)),X) ) ).

% inf_le1
tff(fact_4141_inf__sup__ord_I1_J,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y)),X) ) ).

% inf_sup_ord(1)
tff(fact_4142_inf__sup__ord_I2_J,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_4143_inter__eq__subsetI,axiom,
    ! [B: $tType,S: set(B),S5: set(B),A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),S5)
     => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),S5) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),S5) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),S) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_4144_Int__def,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_kk(set(B),fun(set(B),fun(B,$o)),A4),B4)) ).

% Int_def
tff(fact_4145_Int__Collect,axiom,
    ! [B: $tType,X: B,A4: set(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,member(B,X),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa)))
    <=> ( aa(set(B),$o,member(B,X),A4)
        & aa(B,$o,Pa,X) ) ) ).

% Int_Collect
tff(fact_4146_Collect__conj__eq,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] : aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_gd(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(fun(B,$o),set(B),collect(B),Pa)),aa(fun(B,$o),set(B),collect(B),Q)) ).

% Collect_conj_eq
tff(fact_4147_sorted__wrt__rev,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),Xs: list(B)] :
      ( sorted_wrt(B,Pa,rev(B,Xs))
    <=> sorted_wrt(B,aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),Pa),Xs) ) ).

% sorted_wrt_rev
tff(fact_4148_mset__eq__finite,axiom,
    ! [B: $tType,Xs: list(B)] : aa(set(list(B)),$o,finite_finite2(list(B)),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aTP_Lamp_km(list(B),fun(list(B),$o),Xs))) ).

% mset_eq_finite
tff(fact_4149_inf__option__def,axiom,
    ! [B: $tType] :
      ( inf(B)
     => ! [X: option(B),Y: option(B)] : aa(option(B),option(B),aa(option(B),fun(option(B),option(B)),inf_inf(option(B)),X),Y) = case_option(option(B),B,none(B),aTP_Lamp_ko(option(B),fun(B,option(B)),Y),X) ) ).

% inf_option_def
tff(fact_4150_take__0,axiom,
    ! [B: $tType,Xs: list(B)] : take(B,zero_zero(nat),Xs) = nil(B) ).

% take_0
tff(fact_4151_sorted__take,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),N: nat] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),take(B,N,Xs)) ) ) ).

% sorted_take
tff(fact_4152_translation__Int,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,S2: set(B),Ta: set(B)] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),Ta)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),S2)),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),Ta)) ) ).

% translation_Int
tff(fact_4153_disjoint__mono,axiom,
    ! [B: $tType,A2: set(B),A5: set(B),B2: set(B),B5: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),B5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5) = bot_bot(set(B)) )
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A2),B2) = bot_bot(set(B)) ) ) ) ) ).

% disjoint_mono
tff(fact_4154_disjoint__alt__simp1,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4) = A4 )
    <=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = bot_bot(set(B)) ) ) ).

% disjoint_alt_simp1
tff(fact_4155_disjoint__alt__simp2,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4) != A4 )
    <=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) != bot_bot(set(B)) ) ) ).

% disjoint_alt_simp2
tff(fact_4156_ivl__disj__int__two_I3_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or7035219750837199246ssThan(B,L,Ma)),set_or7035219750837199246ssThan(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(3)
tff(fact_4157_ivl__disj__int__two_I6_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or3652927894154168847AtMost(B,L,Ma)),set_or3652927894154168847AtMost(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(6)
tff(fact_4158_translation__subtract__Int,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B,S2: set(B),Ta: set(B)] : aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),Ta)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),S2)),aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),Ta)) ) ).

% translation_subtract_Int
tff(fact_4159_foldr__conv__foldl,axiom,
    ! [B: $tType,C: $tType,F: fun(C,fun(B,B)),Xs: list(C),A2: B] : aa(B,B,foldr(C,B,F,Xs),A2) = aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,aTP_Lamp_kp(fun(C,fun(B,B)),fun(B,fun(C,B)),F)),A2),rev(C,Xs)) ).

% foldr_conv_foldl
tff(fact_4160_foldl__conv__foldr,axiom,
    ! [C: $tType,B: $tType,F: fun(B,fun(C,B)),A2: B,Xs: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),A2),Xs) = aa(B,B,foldr(C,B,aTP_Lamp_kq(fun(B,fun(C,B)),fun(C,fun(B,B)),F),rev(C,Xs)),A2) ).

% foldl_conv_foldr
tff(fact_4161_set__take__subset__set__take,axiom,
    ! [B: $tType,Ma: nat,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),take(B,Ma,Xs))),aa(list(B),set(B),set2(B),take(B,N,Xs))) ) ).

% set_take_subset_set_take
tff(fact_4162_inf__shunt,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y) = bot_bot(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,uminus_uminus(B),Y)) ) ) ).

% inf_shunt
tff(fact_4163_ivl__disj__int__two_I7_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or7035219750837199246ssThan(B,L,Ma)),set_or1337092689740270186AtMost(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(7)
tff(fact_4164_Ioc__disjoint,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or3652927894154168847AtMost(B,A2,B2)),set_or3652927894154168847AtMost(B,C2,D2)) = bot_bot(set(B)) )
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D2),C2)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D2),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_4165_disjoint__alt__simp3,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4)),A4)
    <=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) != bot_bot(set(B)) ) ) ).

% disjoint_alt_simp3
tff(fact_4166_ivl__disj__int__one_I4_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_lessThan(B),L)),set_or1337092689740270186AtMost(B,L,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(4)
tff(fact_4167_ivl__disj__int__one_I2_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_lessThan(B),L)),set_or7035219750837199246ssThan(B,L,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(2)
tff(fact_4168_ivl__disj__int__two_I8_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or1337092689740270186AtMost(B,L,Ma)),set_or3652927894154168847AtMost(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(8)
tff(fact_4169_ivl__disj__int__two_I4_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or1337092689740270186AtMost(B,L,Ma)),set_or5935395276787703475ssThan(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(4)
tff(fact_4170_ivl__disj__int__two_I5_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or5935395276787703475ssThan(B,L,Ma)),set_or1337092689740270186AtMost(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(5)
tff(fact_4171_ivl__disj__int__two_I1_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or5935395276787703475ssThan(B,L,Ma)),set_or7035219750837199246ssThan(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(1)
tff(fact_4172_ivl__disj__int__one_I3_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atMost(B),L)),set_or3652927894154168847AtMost(B,L,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(3)
tff(fact_4173_ivl__disj__int__one_I1_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atMost(B),L)),set_or5935395276787703475ssThan(B,L,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(1)
tff(fact_4174_ivl__disj__int__two_I2_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,Ma: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or3652927894154168847AtMost(B,L,Ma)),set_or5935395276787703475ssThan(B,Ma,U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_two(2)
tff(fact_4175_sum_Ointer__restrict,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(set(B),fun(B,C),aTP_Lamp_kr(fun(B,C),fun(set(B),fun(B,C)),G),B4)),A4) ) ) ) ).

% sum.inter_restrict
tff(fact_4176_prod_Ointer__restrict,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),G: fun(B,C),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(set(B),fun(B,C),aTP_Lamp_ks(fun(B,C),fun(set(B),fun(B,C)),G),B4)),A4) ) ) ) ).

% prod.inter_restrict
tff(fact_4177_nth__take__lemma,axiom,
    ! [B: $tType,K: nat,Xs: list(B),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(B),nat,size_size(list(B)),Ys))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,B,nth(B,Xs),I2) = aa(nat,B,nth(B,Ys),I2) ) )
         => ( take(B,K,Xs) = take(B,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_4178_mset__swap,axiom,
    ! [B: $tType,I: nat,Ls: list(B),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ls))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Ls))
       => ( mset(B,list_update(B,list_update(B,Ls,J,aa(nat,B,nth(B,Ls),I)),I,aa(nat,B,nth(B,Ls),J))) = mset(B,Ls) ) ) ) ).

% mset_swap
tff(fact_4179_take__Cons,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : take(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_nat(list(B),nil(B),aa(list(B),fun(nat,list(B)),aTP_Lamp_kt(B,fun(list(B),fun(nat,list(B))),X),Xs),N) ).

% take_Cons
tff(fact_4180_rotate__rev,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),list(B),rotate(B,N),rev(B,Xs)) = rev(B,aa(list(B),list(B),rotate(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_4181_sum_Omono__neutral__cong,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [T4: set(B),S: set(B),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,finite_finite2(B),S)
           => ( ! [I2: B] :
                  ( aa(set(B),$o,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,H,I2) = zero_zero(C) ) )
             => ( ! [I2: B] :
                    ( aa(set(B),$o,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),T4))
                   => ( aa(B,C,G,I2) = zero_zero(C) ) )
               => ( ! [X2: B] :
                      ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T4))
                     => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
                 => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T4) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_4182_sum_OInt__Diff,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),G: fun(B,C),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))) ) ) ) ).

% sum.Int_Diff
tff(fact_4183_prod_Omono__neutral__cong,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [T4: set(B),S: set(B),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),T4)
         => ( aa(set(B),$o,finite_finite2(B),S)
           => ( ! [I2: B] :
                  ( aa(set(B),$o,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S))
                 => ( aa(B,C,H,I2) = one_one(C) ) )
             => ( ! [I2: B] :
                    ( aa(set(B),$o,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),T4))
                   => ( aa(B,C,G,I2) = one_one(C) ) )
               => ( ! [X2: B] :
                      ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T4))
                     => ( aa(B,C,G,X2) = aa(B,C,H,X2) ) )
                 => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T4) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_4184_Iio__Int__singleton,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [K: B,X: B] :
          aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_lessThan(B),K)),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),K),aa(set(B),set(B),insert(B,X),bot_bot(set(B))),bot_bot(set(B))) ) ).

% Iio_Int_singleton
tff(fact_4185_sum_OIf__cases,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),Pa: fun(B,$o),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_ku(fun(B,$o),fun(fun(B,C),fun(fun(B,C),fun(B,C))),Pa),H),G)),A4) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa)))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,$o),set(B),collect(B),Pa))))) ) ) ) ).

% sum.If_cases
tff(fact_4186_prod_OIf__cases,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),Pa: fun(B,$o),H: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_kv(fun(B,$o),fun(fun(B,C),fun(fun(B,C),fun(B,C))),Pa),H),G)),A4) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa)))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,$o),set(B),collect(B),Pa))))) ) ) ) ).

% prod.If_cases
tff(fact_4187_take__Cons_H,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] :
      take(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = $ite(N = zero_zero(nat),nil(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs))) ).

% take_Cons'
tff(fact_4188_rev__nth,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,B,nth(B,rev(B,Xs)),N) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(nat,nat,suc,N))) ) ) ).

% rev_nth
tff(fact_4189_rev__update,axiom,
    ! [B: $tType,K: nat,Xs: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(B),nat,size_size(list(B)),Xs))
     => ( rev(B,list_update(B,Xs,K,Y)) = list_update(B,rev(B,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(B),nat,size_size(list(B)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_4190_sum__div__partition,axiom,
    ! [B: $tType,C: $tType] :
      ( euclid4440199948858584721cancel(C)
     => ! [A4: set(B),F: fun(B,C),B2: C] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),B2) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(C,fun(B,C),aTP_Lamp_kw(fun(B,C),fun(C,fun(B,C)),F),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aTP_Lamp_kx(fun(B,C),fun(C,fun(B,$o)),F),B2))))),aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aTP_Lamp_ky(fun(B,C),fun(C,fun(B,$o)),F),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_4191_sorted__rev__iff__nth__Suc,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),rev(B,Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(B),nat,size_size(list(B)),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),aa(nat,nat,suc,I3))),aa(nat,B,nth(B,Xs),I3)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_4192_sorted__rev__iff__nth__mono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),rev(B,Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(B),nat,size_size(list(B)),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),J3)),aa(nat,B,nth(B,Xs),I3)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_4193_sorted__rev__nth__mono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),I: nat,J: nat] :
          ( sorted_wrt(B,ord_less_eq(B),rev(B,Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Xs),J)),aa(nat,B,nth(B,Xs),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_4194_take__Suc__conv__app__nth,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( take(B,aa(nat,nat,suc,I),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,I,Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(nat,B,nth(B,Xs),I)),nil(B))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_4195_take__update__last,axiom,
    ! [B: $tType,N: nat,List: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),List))
     => ( list_update(B,take(B,aa(nat,nat,suc,N),List),N,X) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,N,List)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ) ) ).

% take_update_last
tff(fact_4196_butlast__upd__last__eq,axiom,
    ! [B: $tType,L: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(list(B),nat,size_size(list(B)),L))
     => ( list_update(B,aa(list(B),list(B),butlast(B),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),L)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ) ) ).

% butlast_upd_last_eq
tff(fact_4197_card__disjoint__shuffles,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)) = bot_bot(set(B)) )
     => ( aa(set(list(B)),nat,finite_card(list(B)),shuffles(B,Xs,Ys)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))),aa(list(B),nat,size_size(list(B)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_4198_last__take__nth__conv,axiom,
    ! [B: $tType,N: nat,L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(B),nat,size_size(list(B)),L))
     => ( ( N != zero_zero(nat) )
       => ( last(B,take(B,N,L)) = aa(nat,B,nth(B,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_4199_take__butlast__conv,axiom,
    ! [B: $tType,L: list(B)] : take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(B),list(B),butlast(B),L) ).

% take_butlast_conv
tff(fact_4200_Misc_Olast__in__set,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( L != nil(B) )
     => aa(set(B),$o,member(B,last(B,L)),aa(list(B),set(B),set2(B),L)) ) ).

% Misc.last_in_set
tff(fact_4201_length__butlast,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),butlast(B),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)) ).

% length_butlast
tff(fact_4202_inf__set__def,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),inf_inf(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),B4))) ).

% inf_set_def
tff(fact_4203_butlast__update_H,axiom,
    ! [B: $tType,L: list(B),I: nat,X: B] : list_update(B,aa(list(B),list(B),butlast(B),L),I,X) = aa(list(B),list(B),butlast(B),list_update(B,L,I,X)) ).

% butlast_update'
tff(fact_4204_distinct__butlast__swap,axiom,
    ! [B: $tType,Pq: list(B),I: nat] :
      ( distinct(B,Pq)
     => distinct(B,aa(list(B),list(B),butlast(B),list_update(B,Pq,I,last(B,Pq)))) ) ).

% distinct_butlast_swap
tff(fact_4205_snoc__eq__iff__butlast_H,axiom,
    ! [B: $tType,Ys: list(B),Xs: list(B),X: B] :
      ( ( Ys = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) )
    <=> ( ( Ys != nil(B) )
        & ( aa(list(B),list(B),butlast(B),Ys) = Xs )
        & ( last(B,Ys) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_4206_length__shuffles,axiom,
    ! [B: $tType,Zs: list(B),Xs: list(B),Ys: list(B)] :
      ( aa(set(list(B)),$o,member(list(B),Zs),shuffles(B,Xs,Ys))
     => ( aa(list(B),nat,size_size(list(B)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ) ) ).

% length_shuffles
tff(fact_4207_mset__shuffles,axiom,
    ! [B: $tType,Zs: list(B),Xs: list(B),Ys: list(B)] :
      ( aa(set(list(B)),$o,member(list(B),Zs),shuffles(B,Xs,Ys))
     => ( mset(B,Zs) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),mset(B,Xs)),mset(B,Ys)) ) ) ).

% mset_shuffles
tff(fact_4208_butlast__subset,axiom,
    ! [B: $tType,Xs: list(B),A4: set(B)] :
      ( ( Xs != nil(B) )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),A4)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),aa(list(B),list(B),butlast(B),Xs))),A4) ) ) ).

% butlast_subset
tff(fact_4209_butlast__eq__consE,axiom,
    ! [B: $tType,L: list(B),X: B,Xs: list(B)] :
      ( ( aa(list(B),list(B),butlast(B),L) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) )
     => ~ ! [Xl: B] : L != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xl),nil(B)))) ) ).

% butlast_eq_consE
tff(fact_4210_butlast__eq__cons__conv,axiom,
    ! [B: $tType,L: list(B),X: B,Xs: list(B)] :
      ( ( aa(list(B),list(B),butlast(B),L) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) )
    <=> ? [Xl2: B] : L = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xl2),nil(B)))) ) ).

% butlast_eq_cons_conv
tff(fact_4211_sorted__butlast,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( ( Xs != nil(B) )
         => ( sorted_wrt(B,ord_less_eq(B),Xs)
           => sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),butlast(B),Xs)) ) ) ) ).

% sorted_butlast
tff(fact_4212_nth__butlast,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),butlast(B),Xs)))
     => ( aa(nat,B,nth(B,aa(list(B),list(B),butlast(B),Xs)),N) = aa(nat,B,nth(B,Xs),N) ) ) ).

% nth_butlast
tff(fact_4213_take__butlast,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( take(B,N,aa(list(B),list(B),butlast(B),Xs)) = take(B,N,Xs) ) ) ).

% take_butlast
tff(fact_4214_butlast__power,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),list(B),aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(nat,fun(fun(list(B),list(B)),fun(list(B),list(B))),compow(fun(list(B),list(B))),N),butlast(B)),Xs) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),N),Xs) ).

% butlast_power
tff(fact_4215_butlast__conv__take,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),list(B),butlast(B),Xs) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_4216_butlast__list__update,axiom,
    ! [B: $tType,Xs: list(B),K: nat,X: B] :
      aa(list(B),list(B),butlast(B),list_update(B,Xs,K,X)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)),aa(list(B),list(B),butlast(B),Xs),list_update(B,aa(list(B),list(B),butlast(B),Xs),K,X)) ).

% butlast_list_update
tff(fact_4217_last__conv__nth,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( last(B,Xs) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_4218_last__list__update,axiom,
    ! [B: $tType,Xs: list(B),K: nat,X: B] :
      ( ( Xs != nil(B) )
     => ( last(B,list_update(B,Xs,K,X)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)),X,last(B,Xs)) ) ) ).

% last_list_update
tff(fact_4219_butlast__take,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(B),list(B),butlast(B),take(B,N,Xs)) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_4220_take__minus__one__conv__butlast,axiom,
    ! [B: $tType,N: nat,L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(B),nat,size_size(list(B)),L))
     => ( take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(B),list(B),butlast(B),take(B,N,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_4221_drop__last__conv,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( L != nil(B) )
     => ( drop(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),last(B,L)),nil(B)) ) ) ).

% drop_last_conv
tff(fact_4222_upd__conv__take__nth__drop,axiom,
    ! [B: $tType,I: nat,Xs: list(B),A2: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( list_update(B,Xs,I,A2) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,I,Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),drop(B,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_4223_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,nat,code_nat_of_integer,K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,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_kz(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% nat_of_integer_code
tff(fact_4224_revg__fun,axiom,
    ! [B: $tType,A2: list(B),B2: list(B)] : revg(B,A2,B2) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),rev(B,A2)),B2) ).

% revg_fun
tff(fact_4225_drop0,axiom,
    ! [B: $tType,X4: list(B)] : drop(B,zero_zero(nat),X4) = X4 ).

% drop0
tff(fact_4226_drop__drop,axiom,
    ! [B: $tType,N: nat,Ma: nat,Xs: list(B)] : drop(B,N,drop(B,Ma,Xs)) = drop(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma),Xs) ).

% drop_drop
tff(fact_4227_length__drop,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),drop(B,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),N) ).

% length_drop
tff(fact_4228_drop__upd__irrelevant,axiom,
    ! [B: $tType,Ma: nat,N: nat,L: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
     => ( drop(B,N,list_update(B,L,Ma,X)) = drop(B,N,L) ) ) ).

% drop_upd_irrelevant
tff(fact_4229_drop__all,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),N)
     => ( drop(B,N,Xs) = nil(B) ) ) ).

% drop_all
tff(fact_4230_drop__eq__Nil,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( drop(B,N,Xs) = nil(B) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),N) ) ).

% drop_eq_Nil
tff(fact_4231_drop__eq__Nil2,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( ( nil(B) = drop(B,N,Xs) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),N) ) ).

% drop_eq_Nil2
tff(fact_4232_drop__append,axiom,
    ! [B: $tType,N: nat,Xs: list(B),Ys: list(B)] : drop(B,N,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),drop(B,N,Xs)),drop(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),Ys)) ).

% drop_append
tff(fact_4233_last__drop,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( last(B,drop(B,N,Xs)) = last(B,Xs) ) ) ).

% last_drop
tff(fact_4234_drop__Cons__numeral,axiom,
    ! [B: $tType,V2: num,X: B,Xs: list(B)] : drop(B,aa(num,nat,numeral_numeral(nat),V2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = drop(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_4235_nth__drop,axiom,
    ! [B: $tType,N: nat,Xs: list(B),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,B,nth(B,drop(B,N,Xs)),I) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ) ) ).

% nth_drop
tff(fact_4236_drop__0,axiom,
    ! [B: $tType,Xs: list(B)] : drop(B,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_4237_case__prod__app,axiom,
    ! [E: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,fun(D,fun(E,B))),X: product_prod(C,D),Y: E] : aa(E,B,aa(product_prod(C,D),fun(E,B),aa(fun(C,fun(D,fun(E,B))),fun(product_prod(C,D),fun(E,B)),product_case_prod(C,D,fun(E,B)),F),X),Y) = aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),aa(E,fun(C,fun(D,B)),aTP_Lamp_la(fun(C,fun(D,fun(E,B))),fun(E,fun(C,fun(D,B))),F),Y)),X) ).

% case_prod_app
tff(fact_4238_nested__case__prod__simp,axiom,
    ! [E: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,fun(D,fun(E,B))),X: product_prod(C,D),Y: E] : aa(E,B,aa(product_prod(C,D),fun(E,B),aa(fun(C,fun(D,fun(E,B))),fun(product_prod(C,D),fun(E,B)),product_case_prod(C,D,fun(E,B)),F),X),Y) = aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),aa(E,fun(C,fun(D,B)),aTP_Lamp_la(fun(C,fun(D,fun(E,B))),fun(E,fun(C,fun(D,B))),F),Y)),X) ).

% nested_case_prod_simp
tff(fact_4239_drop__eq__ConsD,axiom,
    ! [B: $tType,N: nat,Xs: list(B),X: B,Xs5: list(B)] :
      ( ( drop(B,N,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs5) )
     => ( drop(B,aa(nat,nat,suc,N),Xs) = Xs5 ) ) ).

% drop_eq_ConsD
tff(fact_4240_sorted__drop,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),N: nat] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),drop(B,N,Xs)) ) ) ).

% sorted_drop
tff(fact_4241_take__drop,axiom,
    ! [B: $tType,N: nat,Ma: nat,Xs: list(B)] : take(B,N,drop(B,Ma,Xs)) = drop(B,Ma,take(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma),Xs)) ).

% take_drop
tff(fact_4242_revg_Osimps_I2_J,axiom,
    ! [B: $tType,A2: B,As: list(B),B2: list(B)] : revg(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),As),B2) = revg(B,As,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),B2)) ).

% revg.simps(2)
tff(fact_4243_revg_Osimps_I1_J,axiom,
    ! [B: $tType,B2: list(B)] : revg(B,nil(B),B2) = B2 ).

% revg.simps(1)
tff(fact_4244_set__drop__subset__set__drop,axiom,
    ! [B: $tType,N: nat,Ma: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),drop(B,Ma,Xs))),aa(list(B),set(B),set2(B),drop(B,N,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_4245_append__eq__conv__conj,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Zs: list(B)] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys) = Zs )
    <=> ( ( Xs = take(B,aa(list(B),nat,size_size(list(B)),Xs),Zs) )
        & ( Ys = drop(B,aa(list(B),nat,size_size(list(B)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_4246_drop__take__drop__unsplit,axiom,
    ! [B: $tType,I: nat,J: nat,L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),drop(B,I,take(B,J,L))),drop(B,J,L)) = drop(B,I,L) ) ) ).

% drop_take_drop_unsplit
tff(fact_4247_take__add,axiom,
    ! [B: $tType,I: nat,J: nat,Xs: list(B)] : take(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,I,Xs)),take(B,J,drop(B,I,Xs))) ).

% take_add
tff(fact_4248_drop__update__swap,axiom,
    ! [B: $tType,Ma: nat,N: nat,Xs: list(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( drop(B,Ma,list_update(B,Xs,N,X)) = list_update(B,drop(B,Ma,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma),X) ) ) ).

% drop_update_swap
tff(fact_4249_drop__Cons,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : drop(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_nat(list(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),aTP_Lamp_lb(list(B),fun(nat,list(B)),Xs),N) ).

% drop_Cons
tff(fact_4250_slice__def,axiom,
    ! [B: $tType,From: nat,To: nat,List: list(B)] : slice(B,From,To,List) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From),drop(B,From,List)) ).

% slice_def
tff(fact_4251_drop__Cons_H,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] :
      drop(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = $ite(N = zero_zero(nat),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),drop(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs)) ).

% drop_Cons'
tff(fact_4252_append__eq__append__conv__if,axiom,
    ! [B: $tType,Xs_1: list(B),Xs_2: list(B),Ys_1: list(B),Ys_2: list(B)] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs_1),Xs_2) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys_1),Ys_2) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs_1)),aa(list(B),nat,size_size(list(B)),Ys_1)),
          ( ( Xs_1 = take(B,aa(list(B),nat,size_size(list(B)),Xs_1),Ys_1) )
          & ( Xs_2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),drop(B,aa(list(B),nat,size_size(list(B)),Xs_1),Ys_1)),Ys_2) ) ),
          ( ( take(B,aa(list(B),nat,size_size(list(B)),Ys_1),Xs_1) = Ys_1 )
          & ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),drop(B,aa(list(B),nat,size_size(list(B)),Ys_1),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_4253_take__rev,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : take(B,N,rev(B,Xs)) = rev(B,drop(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),N),Xs)) ).

% take_rev
tff(fact_4254_rev__take,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] : rev(B,take(B,I,Xs)) = drop(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),I),rev(B,Xs)) ).

% rev_take
tff(fact_4255_rev__drop,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] : rev(B,drop(B,I,Xs)) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),I),rev(B,Xs)) ).

% rev_drop
tff(fact_4256_drop__rev,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : drop(B,N,rev(B,Xs)) = rev(B,take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),N),Xs)) ).

% drop_rev
tff(fact_4257_revg_Oelims,axiom,
    ! [B: $tType,X: list(B),Xa2: list(B),Y: list(B)] :
      ( ( revg(B,X,Xa2) = Y )
     => ( ( ( X = nil(B) )
         => ( Y != Xa2 ) )
       => ~ ! [A3: B,As2: list(B)] :
              ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),As2) )
             => ( Y != revg(B,As2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xa2)) ) ) ) ) ).

% revg.elims
tff(fact_4258_Cons__nth__drop__Suc,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(nat,B,nth(B,Xs),I)),drop(B,aa(nat,nat,suc,I),Xs)) = drop(B,I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_4259_in__set__drop__conv__nth,axiom,
    ! [B: $tType,X: B,N: nat,L: list(B)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),drop(B,N,L)))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),I3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),L))
          & ( X = aa(nat,B,nth(B,L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_4260_rotate__drop__take,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),list(B),rotate(B,N),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),drop(B,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),Xs)),take(B,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),Xs)) ).

% rotate_drop_take
tff(fact_4261_set__take__disj__set__drop__if__distinct,axiom,
    ! [B: $tType,Vs: list(B),I: nat,J: nat] :
      ( distinct(B,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),take(B,I,Vs))),aa(list(B),set(B),set2(B),drop(B,J,Vs))) = bot_bot(set(B)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_4262_foldl__list__update,axiom,
    ! [C: $tType,B: $tType,N: nat,Xs: list(B),F: fun(C,fun(B,C)),A2: C,X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,F),A2),list_update(B,Xs,N,X)) = aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,F),aa(B,C,aa(C,fun(B,C),F,aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,F),A2),take(B,N,Xs))),X)),drop(B,aa(nat,nat,suc,N),Xs)) ) ) ).

% foldl_list_update
tff(fact_4263_id__take__nth__drop,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,I,Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(nat,B,nth(B,Xs),I)),drop(B,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_4264_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,num,code_num_of_integer,K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,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_lc(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% num_of_integer_code
tff(fact_4265_inf__Int__eq,axiom,
    ! [B: $tType,R2: set(B),S: set(B),X4: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),inf_inf(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),R2)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S)),X4)
    <=> aa(set(B),$o,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),R2),S)) ) ).

% inf_Int_eq
tff(fact_4266_take__hd__drop,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,N,Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),drop(B,N,Xs))),nil(B))) = take(B,aa(nat,nat,suc,N),Xs) ) ) ).

% take_hd_drop
tff(fact_4267_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,int,code_int_of_integer,K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,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_ld(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ) ).

% int_of_integer_code
tff(fact_4268_int__of__integer__inverse,axiom,
    ! [X: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,X)) = X ).

% int_of_integer_inverse
tff(fact_4269_int__of__integer__integer__of__int,axiom,
    ! [K: int] : aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,K)) = K ).

% int_of_integer_integer_of_int
tff(fact_4270_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,K)) = K ).

% integer_of_int_int_of_integer
tff(fact_4271_int__of__integer__sub,axiom,
    ! [K: num,L: num] : aa(code_integer,int,code_int_of_integer,neg_numeral_sub(code_integer,K,L)) = neg_numeral_sub(int,K,L) ).

% int_of_integer_sub
tff(fact_4272_of__int__integer__of,axiom,
    ! [K: code_integer] : aa(int,code_integer,ring_1_of_int(code_integer),aa(code_integer,int,code_int_of_integer,K)) = K ).

% of_int_integer_of
tff(fact_4273_int__of__integer__of__int,axiom,
    ! [K: int] : aa(code_integer,int,code_int_of_integer,aa(int,code_integer,ring_1_of_int(code_integer),K)) = K ).

% int_of_integer_of_int
tff(fact_4274_zero__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_4275_int__of__integer__numeral,axiom,
    ! [K: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_4276_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_4277_one__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_4278_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),X)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,X)) ).

% uminus_integer.rep_eq
tff(fact_4279_times__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),times_times(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% times_integer.rep_eq
tff(fact_4280_minus__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),minus_minus(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% minus_integer.rep_eq
tff(fact_4281_abs__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,abs_abs(code_integer),X)) = aa(int,int,abs_abs(int),aa(code_integer,int,code_int_of_integer,X)) ).

% abs_integer.rep_eq
tff(fact_4282_divide__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),divide_divide(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% divide_integer.rep_eq
tff(fact_4283_modulo__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),modulo_modulo(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% modulo_integer.rep_eq
tff(fact_4284_int__of__integer__of__nat,axiom,
    ! [N: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),N)) = aa(nat,int,semiring_1_of_nat(int),N) ).

% int_of_integer_of_nat
tff(fact_4285_sgn__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,sgn_sgn(code_integer),X)) = aa(int,int,sgn_sgn(int),aa(code_integer,int,code_int_of_integer,X)) ).

% sgn_integer.rep_eq
tff(fact_4286_hd__take,axiom,
    ! [B: $tType,J: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( aa(list(B),B,hd(B),take(B,J,Xs)) = aa(list(B),B,hd(B),Xs) ) ) ).

% hd_take
tff(fact_4287_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),K),L)) = aa(int,int,aa(int,fun(int,int),ord_max(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ).

% int_of_integer_max
tff(fact_4288_integer__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( K = L )
    <=> ( aa(code_integer,int,code_int_of_integer,K) = aa(code_integer,int,code_int_of_integer,L) ) ) ).

% integer_eq_iff
tff(fact_4289_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( aa(code_integer,int,code_int_of_integer,K) = aa(code_integer,int,code_int_of_integer,L) )
     => ( K = L ) ) ).

% integer_eqI
tff(fact_4290_int__of__integer__inject,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa(code_integer,int,code_int_of_integer,X) = aa(code_integer,int,code_int_of_integer,Y) )
    <=> ( X = Y ) ) ).

% int_of_integer_inject
tff(fact_4291_prod__case__refines,axiom,
    ! [D: $tType,C: $tType,B: $tType,P2: product_prod(B,C),P4: product_prod(B,C),F: fun(B,fun(C,heap_Time_Heap(D))),F2: fun(B,fun(C,heap_Time_Heap(D)))] :
      ( ( P2 = P4 )
     => ( ! [A3: B,B3: C] : refine_Imp_refines(D,aa(C,heap_Time_Heap(D),aa(B,fun(C,heap_Time_Heap(D)),F,A3),B3),aa(C,heap_Time_Heap(D),aa(B,fun(C,heap_Time_Heap(D)),F2,A3),B3))
       => refine_Imp_refines(D,aa(product_prod(B,C),heap_Time_Heap(D),aa(fun(B,fun(C,heap_Time_Heap(D))),fun(product_prod(B,C),heap_Time_Heap(D)),product_case_prod(B,C,heap_Time_Heap(D)),F),P2),aa(product_prod(B,C),heap_Time_Heap(D),aa(fun(B,fun(C,heap_Time_Heap(D))),fun(product_prod(B,C),heap_Time_Heap(D)),product_case_prod(B,C,heap_Time_Heap(D)),F2),P4)) ) ) ).

% prod_case_refines
tff(fact_4292_refines__case__prod__right,axiom,
    ! [D: $tType,C: $tType,B: $tType,Ma: heap_Time_Heap(D),M3: fun(B,fun(C,heap_Time_Heap(D))),Ta: product_prod(B,C)] :
      ( ! [A3: B,B3: C] : refine_Imp_refines(D,Ma,aa(C,heap_Time_Heap(D),aa(B,fun(C,heap_Time_Heap(D)),M3,A3),B3))
     => refine_Imp_refines(D,Ma,aa(product_prod(B,C),heap_Time_Heap(D),aa(fun(B,fun(C,heap_Time_Heap(D))),fun(product_prod(B,C),heap_Time_Heap(D)),product_case_prod(B,C,heap_Time_Heap(D)),M3),Ta)) ) ).

% refines_case_prod_right
tff(fact_4293_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),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_4294_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_4295_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Y))
    <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Y) ) ).

% int_of_integer_less_iff
tff(fact_4296_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),X),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),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_4297_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_4298_nat__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,nat,code_nat_of_integer,X) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).

% nat_of_integer.rep_eq
tff(fact_4299_sum_Otriangle__reindex__eq,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,fun(nat,B)),N: nat] : aa(set(product_prod(nat,nat)),B,aa(fun(product_prod(nat,nat),B),fun(set(product_prod(nat,nat)),B),groups7311177749621191930dd_sum(product_prod(nat,nat),B),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_le(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_lg(fun(nat,fun(nat,B)),fun(nat,B),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.triangle_reindex_eq
tff(fact_4300_prod_Otriangle__reindex__eq,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,fun(nat,B)),N: nat] : aa(set(product_prod(nat,nat)),B,aa(fun(product_prod(nat,nat),B),fun(set(product_prod(nat,nat)),B),groups7121269368397514597t_prod(product_prod(nat,nat),B),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_le(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_li(fun(nat,fun(nat,B)),fun(nat,B),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.triangle_reindex_eq
tff(fact_4301_take__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),X),Xa2)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% take_bit_integer.rep_eq
tff(fact_4302_not__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),X)) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(code_integer,int,code_int_of_integer,X)) ).

% not_integer.rep_eq
tff(fact_4303_and__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),bit_se5824344872417868541ns_and(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% and_integer.rep_eq
tff(fact_4304_or__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),bit_se1065995026697491101ons_or(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% or_integer.rep_eq
tff(fact_4305_bit__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),X) = aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(code_integer,int,code_int_of_integer,X)) ).

% bit_integer.rep_eq
tff(fact_4306_xor__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),bit_se5824344971392196577ns_xor(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% xor_integer.rep_eq
tff(fact_4307_push__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),X),Xa2)) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% push_bit_integer.rep_eq
tff(fact_4308_drop__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),X),Xa2)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% drop_bit_integer.rep_eq
tff(fact_4309_mask__integer_Orep__eq,axiom,
    ! [X: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),X)) = aa(nat,int,bit_se2239418461657761734s_mask(int),X) ).

% mask_integer.rep_eq
tff(fact_4310_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),X),Xa2)) = aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% unset_bit_integer.rep_eq
tff(fact_4311_set__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),X),Xa2)) = aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% set_bit_integer.rep_eq
tff(fact_4312_sum_Otriangle__reindex,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(nat,fun(nat,B)),N: nat] : aa(set(product_prod(nat,nat)),B,aa(fun(product_prod(nat,nat),B),fun(set(product_prod(nat,nat)),B),groups7311177749621191930dd_sum(product_prod(nat,nat),B),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_lj(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aTP_Lamp_lg(fun(nat,fun(nat,B)),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.triangle_reindex
tff(fact_4313_flip__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),X),Xa2)) = aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% flip_bit_integer.rep_eq
tff(fact_4314_prod_Otriangle__reindex,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(nat,fun(nat,B)),N: nat] : aa(set(product_prod(nat,nat)),B,aa(fun(product_prod(nat,nat),B),fun(set(product_prod(nat,nat)),B),groups7121269368397514597t_prod(product_prod(nat,nat),B),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_lj(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aTP_Lamp_li(fun(nat,fun(nat,B)),fun(nat,B),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.triangle_reindex
tff(fact_4315_hd__conv__nth,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( aa(list(B),B,hd(B),Xs) = aa(nat,B,nth(B,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_4316_num__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,num,code_num_of_integer,X) = aa(nat,num,num_of_nat,aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X))) ).

% num_of_integer.rep_eq
tff(fact_4317_sorted__hd__min,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( ( Xs != nil(B) )
         => ( sorted_wrt(B,ord_less_eq(B),Xs)
           => ! [X4: B] :
                ( aa(set(B),$o,member(B,X4),aa(list(B),set(B),set2(B),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,hd(B),Xs)),X4) ) ) ) ) ).

% sorted_hd_min
tff(fact_4318_hd__drop__conv__nth,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(B),B,hd(B),drop(B,N,Xs)) = aa(nat,B,nth(B,Xs),N) ) ) ).

% hd_drop_conv_nth
tff(fact_4319_sorted__hd__last,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),L)
         => ( ( L != nil(B) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,hd(B),L)),last(B,L)) ) ) ) ).

% sorted_hd_last
tff(fact_4320_hd__last__singletonI,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( ( aa(list(B),B,hd(B),Xs) = last(B,Xs) )
       => ( distinct(B,Xs)
         => ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),Xs)),nil(B)) ) ) ) ) ).

% hd_last_singletonI
tff(fact_4321_hd__butlast,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(B),B,hd(B),aa(list(B),list(B),butlast(B),Xs)) = aa(list(B),B,hd(B),Xs) ) ) ).

% hd_butlast
tff(fact_4322_hd__rotate__conv__nth,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( ( Xs != nil(B) )
     => ( aa(list(B),B,hd(B),aa(list(B),list(B),rotate(B,N),Xs)) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_4323_slice__head,axiom,
    ! [B: $tType,From: nat,To: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(list(B),B,hd(B),slice(B,From,To,Xs)) = aa(nat,B,nth(B,Xs),From) ) ) ) ).

% slice_head
tff(fact_4324_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_lk(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_4325_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_ll(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_4326_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_lm(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_4327_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_ln(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4328_not__Some__eq2,axiom,
    ! [C: $tType,B: $tType,V2: option(product_prod(B,C))] :
      ( ! [X3: B,Y5: C] : V2 != aa(product_prod(B,C),option(product_prod(B,C)),some(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y5))
    <=> ( V2 = none(product_prod(B,C)) ) ) ).

% not_Some_eq2
tff(fact_4329_Eps__case__prod__eq,axiom,
    ! [B: $tType,C: $tType,X: B,Y: C] : fChoice(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(C,fun(B,fun(C,$o)),aTP_Lamp_lo(B,fun(C,fun(B,fun(C,$o))),X),Y))) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) ).

% Eps_case_prod_eq
tff(fact_4330_divmod__step__eq,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [L: num,Q2: B,R: B] :
          unique1321980374590559556d_step(B,L,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),R)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),L)),R),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Q2)),one_one(B))),aa(B,B,aa(B,fun(B,B),minus_minus(B),R),aa(num,B,numeral_numeral(B),L))),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Q2)),R)) ) ).

% divmod_step_eq
tff(fact_4331_inf__Int__eq2,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(B,C)),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(C,$o)),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),inf_inf(fun(B,fun(C,$o))),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),R2)),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),S)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),R2),S)) ) ).

% inf_Int_eq2
tff(fact_4332_split__paired__Eps,axiom,
    ! [C: $tType,B: $tType,Pa: fun(product_prod(B,C),$o)] : fChoice(product_prod(B,C),Pa) = fChoice(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_lq(fun(product_prod(B,C),$o),fun(B,fun(C,$o)),Pa))) ).

% split_paired_Eps
tff(fact_4333_case__prod__Pair__iden,axiom,
    ! [C: $tType,B: $tType,P2: product_prod(B,C)] : aa(product_prod(B,C),product_prod(B,C),aa(fun(B,fun(C,product_prod(B,C))),fun(product_prod(B,C),product_prod(B,C)),product_case_prod(B,C,product_prod(B,C)),product_Pair(B,C)),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_4334_pred__equals__eq2,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(B,C))] :
      ( ! [X3: B,Xa4: C] :
          ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Xa4)),R2)
        <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Xa4)),S) )
    <=> ( R2 = S ) ) ).

% pred_equals_eq2
tff(fact_4335_pred__subset__eq2,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(B,C))] :
      ( aa(fun(B,fun(C,$o)),$o,aa(fun(B,fun(C,$o)),fun(fun(B,fun(C,$o)),$o),ord_less_eq(fun(B,fun(C,$o))),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),R2)),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),S))
    <=> aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),R2),S) ) ).

% pred_subset_eq2
tff(fact_4336_pairself_Ocases,axiom,
    ! [C: $tType,B: $tType,X: product_prod(fun(B,C),product_prod(B,B))] :
      ~ ! [F4: fun(B,C),A3: B,B3: B] : X != aa(product_prod(B,B),product_prod(fun(B,C),product_prod(B,B)),aa(fun(B,C),fun(product_prod(B,B),product_prod(fun(B,C),product_prod(B,B))),product_Pair(fun(B,C),product_prod(B,B)),F4),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A3),B3)) ).

% pairself.cases
tff(fact_4337_bex2I,axiom,
    ! [B: $tType,C: $tType,A2: B,B2: C,S: set(product_prod(B,C)),Pa: fun(B,fun(C,$o))] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),S)
     => ( ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),S)
         => aa(C,$o,aa(B,fun(C,$o),Pa,A2),B2) )
       => ? [A3: B,B3: C] :
            ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)),S)
            & aa(C,$o,aa(B,fun(C,$o),Pa,A3),B3) ) ) ) ).

% bex2I
tff(fact_4338_bot__empty__eq2,axiom,
    ! [C: $tType,B: $tType,X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),bot_bot(fun(B,fun(C,$o))),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),bot_bot(set(product_prod(B,C)))) ) ).

% bot_empty_eq2
tff(fact_4339_rel__of__def,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),Pa: fun(product_prod(B,C),$o)] : rel_of(B,C,Ma,Pa) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(fun(product_prod(B,C),$o),fun(B,fun(C,$o)),aTP_Lamp_lr(fun(B,option(C)),fun(fun(product_prod(B,C),$o),fun(B,fun(C,$o))),Ma),Pa))) ).

% rel_of_def
tff(fact_4340_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L)) ).

% divmod_integer_def
tff(fact_4341_divmod__step__def,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [L: num,Qr: product_prod(B,B)] : unique1321980374590559556d_step(B,L,Qr) = aa(product_prod(B,B),product_prod(B,B),aa(fun(B,fun(B,product_prod(B,B))),fun(product_prod(B,B),product_prod(B,B)),product_case_prod(B,B,product_prod(B,B)),aTP_Lamp_ls(num,fun(B,fun(B,product_prod(B,B))),L)),Qr) ) ).

% divmod_step_def
tff(fact_4342_case__prodI2,axiom,
    ! [C: $tType,B: $tType,P2: product_prod(B,C),C2: fun(B,fun(C,$o))] :
      ( ! [A3: B,B3: C] :
          ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3) )
         => aa(C,$o,aa(B,fun(C,$o),C2,A3),B3) )
     => aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),C2),P2) ) ).

% case_prodI2
tff(fact_4343_case__prodI,axiom,
    ! [B: $tType,C: $tType,F: fun(B,fun(C,$o)),A2: B,B2: C] :
      ( aa(C,$o,aa(B,fun(C,$o),F,A2),B2)
     => aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) ) ).

% case_prodI
tff(fact_4344_case__prod__conv,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,fun(D,B)),A2: C,B2: D] : aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),F),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A2),B2)) = aa(D,B,aa(C,fun(D,B),F,A2),B2) ).

% case_prod_conv
tff(fact_4345_pair__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A2: B,B2: C,A4: set(product_prod(B,C)),F: fun(B,fun(C,D))] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),A4)
     => aa(set(D),$o,member(D,aa(C,D,aa(B,fun(C,D),F,A2),B2)),aa(set(product_prod(B,C)),set(D),image2(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),F)),A4)) ) ).

% pair_imageI
tff(fact_4346_case__prodI2_H,axiom,
    ! [B: $tType,C: $tType,D: $tType,P2: product_prod(B,C),C2: fun(B,fun(C,fun(D,$o))),X: D] :
      ( ! [A3: B,B3: C] :
          ( ( aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3) = P2 )
         => aa(D,$o,aa(C,fun(D,$o),aa(B,fun(C,fun(D,$o)),C2,A3),B3),X) )
     => aa(D,$o,aa(product_prod(B,C),fun(D,$o),aa(fun(B,fun(C,fun(D,$o))),fun(product_prod(B,C),fun(D,$o)),product_case_prod(B,C,fun(D,$o)),C2),P2),X) ) ).

% case_prodI2'
tff(fact_4347_mem__case__prodI,axiom,
    ! [B: $tType,C: $tType,D: $tType,Z: B,C2: fun(C,fun(D,set(B))),A2: C,B2: D] :
      ( aa(set(B),$o,member(B,Z),aa(D,set(B),aa(C,fun(D,set(B)),C2,A2),B2))
     => aa(set(B),$o,member(B,Z),aa(product_prod(C,D),set(B),aa(fun(C,fun(D,set(B))),fun(product_prod(C,D),set(B)),product_case_prod(C,D,set(B)),C2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A2),B2))) ) ).

% mem_case_prodI
tff(fact_4348_mem__case__prodI2,axiom,
    ! [D: $tType,C: $tType,B: $tType,P2: product_prod(B,C),Z: D,C2: fun(B,fun(C,set(D)))] :
      ( ! [A3: B,B3: C] :
          ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3) )
         => aa(set(D),$o,member(D,Z),aa(C,set(D),aa(B,fun(C,set(D)),C2,A3),B3)) )
     => aa(set(D),$o,member(D,Z),aa(product_prod(B,C),set(D),aa(fun(B,fun(C,set(D))),fun(product_prod(B,C),set(D)),product_case_prod(B,C,set(D)),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_4349_split__part,axiom,
    ! [C: $tType,B: $tType,Pa: $o,Q: fun(B,fun(C,$o)),X4: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_lt($o,fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),(Pa)),Q)),X4)
    <=> ( (Pa)
        & aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Q),X4) ) ) ).

% split_part
tff(fact_4350_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_4351_timeFrame_Osimps_I1_J,axiom,
    ! [B: $tType,N: nat,R: B,H: heap_ext(product_unit),N3: nat] : heap_Time_timeFrame(B,N,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)),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),H),N3)))) = 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)),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),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N3)))) ).

% timeFrame.simps(1)
tff(fact_4352_mergesort__by__rel__split_Ocases,axiom,
    ! [B: $tType,X: product_prod(product_prod(list(B),list(B)),list(B))] :
      ( ! [Xs1: list(B),Xs22: list(B)] : X != aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),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)),Xs1),Xs22)),nil(B))
     => ( ! [Xs1: list(B),Xs22: list(B),X2: B] : X != aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),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)),Xs1),Xs22)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))
       => ~ ! [Xs1: list(B),Xs22: list(B),X1: B,X23: B,Xs2: list(B)] : X != aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),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)),Xs1),Xs22)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_4353_partition__rev_Ocases,axiom,
    ! [B: $tType,X: product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B)))] :
      ( ! [P5: fun(B,$o),Yes: list(B),No: list(B)] : X != aa(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),aa(fun(B,$o),fun(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B)))),product_Pair(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),P5),aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Yes),No)),nil(B)))
     => ~ ! [P5: fun(B,$o),Yes: list(B),No: list(B),X2: B,Xs2: list(B)] : X != aa(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),aa(fun(B,$o),fun(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B)))),product_Pair(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),P5),aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),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)),Yes),No)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2))) ) ).

% partition_rev.cases
tff(fact_4354_list__all__zip_Ocases,axiom,
    ! [B: $tType,C: $tType,X: product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C)))] :
      ( ! [P5: fun(B,fun(C,$o))] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C))),aa(fun(B,fun(C,$o)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,$o)),product_prod(list(B),list(C))),P5),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),nil(C)))
     => ( ! [P5: fun(B,fun(C,$o)),A3: B,As2: list(B),B3: C,Bs2: list(C)] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C))),aa(fun(B,fun(C,$o)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,$o)),product_prod(list(B),list(C))),P5),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),As2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B3),Bs2)))
       => ( ! [P5: fun(B,fun(C,$o)),V4: B,Va: list(B)] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C))),aa(fun(B,fun(C,$o)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,$o)),product_prod(list(B),list(C))),P5),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va)),nil(C)))
         => ~ ! [P5: fun(B,fun(C,$o)),V4: C,Va: list(C)] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C))),aa(fun(B,fun(C,$o)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,$o)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,$o)),product_prod(list(B),list(C))),P5),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V4),Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_4355_quicksort__by__rel_Ocases,axiom,
    ! [B: $tType,X: product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))] :
      ( ! [R4: fun(B,fun(B,$o)),Sl: list(B)] : X != aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Sl),nil(B)))
     => ~ ! [R4: fun(B,fun(B,$o)),Sl: list(B),X2: B,Xs2: list(B)] : X != aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),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)),Sl),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_4356_mergesort__by__rel__merge_Ocases,axiom,
    ! [B: $tType,X: product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))] :
      ( ! [R4: fun(B,fun(B,$o)),X2: B,Xs2: list(B),Y2: B,Ys3: list(B)] : X != aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3)))
     => ( ! [R4: fun(B,fun(B,$o)),Xs2: list(B)] : X != aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),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)),Xs2),nil(B)))
       => ~ ! [R4: fun(B,fun(B,$o)),V4: B,Va: list(B)] : X != aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),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)),nil(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_4357_zipf_Ocases,axiom,
    ! [D: $tType,B: $tType,C: $tType,X: product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C)))] :
      ( ! [F4: fun(B,fun(C,D))] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C))),aa(fun(B,fun(C,D)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,D)),product_prod(list(B),list(C))),F4),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),nil(C)))
     => ( ! [F4: fun(B,fun(C,D)),A3: B,As2: list(B),B3: C,Bs2: list(C)] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C))),aa(fun(B,fun(C,D)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,D)),product_prod(list(B),list(C))),F4),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),As2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B3),Bs2)))
       => ( ! [A3: fun(B,fun(C,D)),V4: B,Va: list(B)] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C))),aa(fun(B,fun(C,D)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,D)),product_prod(list(B),list(C))),A3),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va)),nil(C)))
         => ~ ! [A3: fun(B,fun(C,D)),V4: C,Va: list(C)] : X != aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C))),aa(fun(B,fun(C,D)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,D)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,D)),product_prod(list(B),list(C))),A3),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V4),Va))) ) ) ) ).

% zipf.cases
tff(fact_4358_timeFrame_Ocases,axiom,
    ! [B: $tType,X: product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
      ( ! [N2: nat,R5: B,H2: heap_ext(product_unit),N7: nat] : X != aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),N2),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)),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),H2),N7))))
     => ~ ! [N2: nat] : X != aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),N2),none(product_prod(B,product_prod(heap_ext(product_unit),nat)))) ) ).

% timeFrame.cases
tff(fact_4359_timeFrame_Oelims,axiom,
    ! [B: $tType,X: nat,Xa2: option(product_prod(B,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(B,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(B,X,Xa2) = Y )
     => ( ! [R5: B,H2: heap_ext(product_unit),N7: nat] :
            ( ( Xa2 = 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)),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),H2),N7))) )
           => ( Y != 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)),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),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N7)))) ) )
       => ~ ( ( Xa2 = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) )
           => ( Y != none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).

% timeFrame.elims
tff(fact_4360_merge_Ocases,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: product_prod(list(B),list(B))] :
          ( ! [L23: list(B)] : X != 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)),nil(B)),L23)
         => ( ! [V4: B,Va: list(B)] : X != 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),V4),Va)),nil(B))
           => ~ ! [X1: B,L12: list(B),X23: B,L23: list(B)] : X != 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),X1),L12)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),L23)) ) ) ) ).

% merge.cases
tff(fact_4361_merge__list_Ocases,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: product_prod(list(list(B)),list(list(B)))] :
          ( ( X != aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),nil(list(B))) )
         => ( ! [L2: list(B)] : X != aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))))
           => ( ! [La: list(B),Acc2: list(list(B))] : X != aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),nil(list(B)))
             => ( ! [La: list(B),Acc2: list(list(B)),L2: list(B)] : X != aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))))
               => ~ ! [Acc2: list(list(B)),L12: list(B),L23: list(B),Ls2: list(list(B))] : X != aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),Acc2),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L12),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L23),Ls2))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_4362_case__prodD_H,axiom,
    ! [C: $tType,B: $tType,D: $tType,R2: fun(B,fun(C,fun(D,$o))),A2: B,B2: C,C2: D] :
      ( aa(D,$o,aa(product_prod(B,C),fun(D,$o),aa(fun(B,fun(C,fun(D,$o))),fun(product_prod(B,C),fun(D,$o)),product_case_prod(B,C,fun(D,$o)),R2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),C2)
     => aa(D,$o,aa(C,fun(D,$o),aa(B,fun(C,fun(D,$o)),R2,A2),B2),C2) ) ).

% case_prodD'
tff(fact_4363_case__prodE_H,axiom,
    ! [C: $tType,B: $tType,D: $tType,C2: fun(B,fun(C,fun(D,$o))),P2: product_prod(B,C),Z: D] :
      ( aa(D,$o,aa(product_prod(B,C),fun(D,$o),aa(fun(B,fun(C,fun(D,$o))),fun(product_prod(B,C),fun(D,$o)),product_case_prod(B,C,fun(D,$o)),C2),P2),Z)
     => ~ ! [X2: B,Y2: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ aa(D,$o,aa(C,fun(D,$o),aa(B,fun(C,fun(D,$o)),C2,X2),Y2),Z) ) ) ).

% case_prodE'
tff(fact_4364_prod_Ocase__distrib,axiom,
    ! [C: $tType,B: $tType,E: $tType,D: $tType,H: fun(C,B),F: fun(D,fun(E,C)),Prod: product_prod(D,E)] : aa(C,B,H,aa(product_prod(D,E),C,aa(fun(D,fun(E,C)),fun(product_prod(D,E),C),product_case_prod(D,E,C),F),Prod)) = aa(product_prod(D,E),B,aa(fun(D,fun(E,B)),fun(product_prod(D,E),B),product_case_prod(D,E,B),aa(fun(D,fun(E,C)),fun(D,fun(E,B)),aTP_Lamp_lu(fun(C,B),fun(fun(D,fun(E,C)),fun(D,fun(E,B))),H),F)),Prod) ).

% prod.case_distrib
tff(fact_4365_prod_Odisc__eq__case,axiom,
    ! [C: $tType,B: $tType,Prod: product_prod(B,C)] : aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_lv(B,fun(C,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_4366_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_lw(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_4367_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_lx(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_4368_case__prodE2,axiom,
    ! [C: $tType,B: $tType,D: $tType,Q: fun(B,$o),Pa: fun(C,fun(D,B)),Z: product_prod(C,D)] :
      ( aa(B,$o,Q,aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),Pa),Z))
     => ~ ! [X2: C,Y2: D] :
            ( ( Z = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X2),Y2) )
           => ~ aa(B,$o,Q,aa(D,B,aa(C,fun(D,B),Pa,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_4369_case__prod__eta,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(product_prod(B,C),D)] : aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),aTP_Lamp_ly(fun(product_prod(B,C),D),fun(B,fun(C,D)),F)) = F ).

% case_prod_eta
tff(fact_4370_cond__case__prod__eta,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,fun(C,D)),G: fun(product_prod(B,C),D)] :
      ( ! [X2: B,Y2: C] : aa(C,D,aa(B,fun(C,D),F,X2),Y2) = aa(product_prod(B,C),D,G,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2))
     => ( aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),F) = G ) ) ).

% cond_case_prod_eta
tff(fact_4371_case__prodD,axiom,
    ! [B: $tType,C: $tType,F: fun(B,fun(C,$o)),A2: B,B2: C] :
      ( aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2))
     => aa(C,$o,aa(B,fun(C,$o),F,A2),B2) ) ).

% case_prodD
tff(fact_4372_case__prodE,axiom,
    ! [B: $tType,C: $tType,C2: fun(B,fun(C,$o)),P2: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),C2),P2)
     => ~ ! [X2: B,Y2: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ aa(C,$o,aa(B,fun(C,$o),C2,X2),Y2) ) ) ).

% case_prodE
tff(fact_4373_take__bit__num__code,axiom,
    ! [N: nat,Ma: num] : bit_take_bit_num(N,Ma) = 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_mc(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),Ma)) ).

% take_bit_num_code
tff(fact_4374_divmod__algorithm__code_I6_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num,N: num] : unique8689654367752047608divmod(B,aa(num,num,bit1,Ma),aa(num,num,bit0,N)) = aa(product_prod(B,B),product_prod(B,B),aa(fun(B,fun(B,product_prod(B,B))),fun(product_prod(B,B),product_prod(B,B)),product_case_prod(B,B,product_prod(B,B)),aTP_Lamp_md(B,fun(B,product_prod(B,B)))),unique8689654367752047608divmod(B,Ma,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4375_divmod__algorithm__code_I7_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num,N: num] :
          unique8689654367752047608divmod(B,aa(num,num,bit0,Ma),aa(num,num,bit1,N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Ma),N),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),zero_zero(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,Ma))),unique1321980374590559556d_step(B,aa(num,num,bit1,N),unique8689654367752047608divmod(B,aa(num,num,bit0,Ma),aa(num,num,bit0,aa(num,num,bit1,N))))) ) ).

% divmod_algorithm_code(7)
tff(fact_4376_divmod__algorithm__code_I8_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num,N: num] :
          unique8689654367752047608divmod(B,aa(num,num,bit1,Ma),aa(num,num,bit1,N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),N),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),zero_zero(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit1,Ma))),unique1321980374590559556d_step(B,aa(num,num,bit1,N),unique8689654367752047608divmod(B,aa(num,num,bit1,Ma),aa(num,num,bit0,aa(num,num,bit1,N))))) ) ).

% divmod_algorithm_code(8)
tff(fact_4377_divmod__algorithm__code_I2_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num] : unique8689654367752047608divmod(B,Ma,one2) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(num,B,numeral_numeral(B),Ma)),zero_zero(B)) ) ).

% divmod_algorithm_code(2)
tff(fact_4378_divmod__algorithm__code_I3_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num] : unique8689654367752047608divmod(B,one2,aa(num,num,bit0,N)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),zero_zero(B)),aa(num,B,numeral_numeral(B),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_4379_divmod__algorithm__code_I4_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num] : unique8689654367752047608divmod(B,one2,aa(num,num,bit1,N)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),zero_zero(B)),aa(num,B,numeral_numeral(B),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_4380_minus__one__div__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_4381_one__div__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_4382_divmod__BitM__2__eq,axiom,
    ! [Ma: num] : unique8689654367752047608divmod(int,bitM(Ma),aa(num,num,bit0,one2)) = 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(num,int,numeral_numeral(int),Ma)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_4383_divmod__algorithm__code_I5_J,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num,N: num] : unique8689654367752047608divmod(B,aa(num,num,bit0,Ma),aa(num,num,bit0,N)) = aa(product_prod(B,B),product_prod(B,B),aa(fun(B,fun(B,product_prod(B,B))),fun(product_prod(B,B),product_prod(B,B)),product_case_prod(B,B,product_prod(B,B)),aTP_Lamp_me(B,fun(B,product_prod(B,B)))),unique8689654367752047608divmod(B,Ma,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_4384_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [B: $tType,X: product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))] :
      ~ ! [F4: fun(nat,fun(B,B)),A3: nat,B3: nat,Acc3: B] : X != aa(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),aa(fun(nat,fun(B,B)),fun(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))),product_Pair(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),F4),aa(product_prod(nat,B),product_prod(nat,product_prod(nat,B)),aa(nat,fun(product_prod(nat,B),product_prod(nat,product_prod(nat,B))),product_Pair(nat,product_prod(nat,B)),A3),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),B3),Acc3))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4385_num_Ocase__distrib,axiom,
    ! [C: $tType,B: $tType,H: fun(C,B),F1: C,F22: fun(num,C),F32: fun(num,C),Num: num] : aa(C,B,H,aa(num,C,aa(fun(num,C),fun(num,C),aa(fun(num,C),fun(fun(num,C),fun(num,C)),aa(C,fun(fun(num,C),fun(fun(num,C),fun(num,C))),case_num(C),F1),F22),F32),Num)) = aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),aa(C,B,H,F1)),aa(fun(num,C),fun(num,B),aTP_Lamp_mf(fun(C,B),fun(fun(num,C),fun(num,B)),H),F22)),aa(fun(num,C),fun(num,B),aTP_Lamp_mf(fun(C,B),fun(fun(num,C),fun(num,B)),H),F32)),Num) ).

% num.case_distrib
tff(fact_4386_verit__eq__simplify_I17_J,axiom,
    ! [B: $tType,F1: B,F22: fun(num,B),F32: fun(num,B),X22: num] : aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),F1),F22),F32),aa(num,num,bit0,X22)) = aa(num,B,F22,X22) ).

% verit_eq_simplify(17)
tff(fact_4387_verit__eq__simplify_I16_J,axiom,
    ! [B: $tType,F1: B,F22: fun(num,B),F32: fun(num,B)] : aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),F1),F22),F32),one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_4388_verit__eq__simplify_I18_J,axiom,
    ! [B: $tType,F1: B,F22: fun(num,B),F32: fun(num,B),X32: num] : aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),F1),F22),F32),aa(num,num,bit1,X32)) = aa(num,B,F32,X32) ).

% verit_eq_simplify(18)
tff(fact_4389_divmod__integer_H__def,axiom,
    ! [Ma: num,N: num] : unique8689654367752047608divmod(code_integer,Ma,N) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Ma)),aa(num,code_integer,numeral_numeral(code_integer),N))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Ma)),aa(num,code_integer,numeral_numeral(code_integer),N))) ).

% divmod_integer'_def
tff(fact_4390_divmod__divmod__step,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Ma: num,N: num] :
          unique8689654367752047608divmod(B,Ma,N) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Ma),N),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),zero_zero(B)),aa(num,B,numeral_numeral(B),Ma)),unique1321980374590559556d_step(B,N,unique8689654367752047608divmod(B,Ma,aa(num,num,bit0,N)))) ) ).

% divmod_divmod_step
tff(fact_4391_divmod__nat__if,axiom,
    ! [Ma: nat,N: nat] :
      divmod_nat(Ma,N) = $ite(
        ( ( N = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),Ma),
        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_mg(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N),N)) ) ).

% divmod_nat_if
tff(fact_4392_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => 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,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4393_divides__aux__eq,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Q2: B,R: B] :
          ( unique5940410009612947441es_aux(B,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),R))
        <=> ( R = zero_zero(B) ) ) ) ).

% divides_aux_eq
tff(fact_4394_product__nth,axiom,
    ! [B: $tType,C: $tType,N: nat,Xs: list(B),Ys: list(C)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(C),nat,size_size(list(C)),Ys)))
     => ( aa(nat,product_prod(B,C),nth(product_prod(B,C),product(B,C,Xs,Ys)),N) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(C),nat,size_size(list(C)),Ys)))),aa(nat,C,nth(C,Ys),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(C),nat,size_size(list(C)),Ys)))) ) ) ).

% product_nth
tff(fact_4395_length__product,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] : aa(list(product_prod(B,C)),nat,size_size(list(product_prod(B,C))),product(B,C,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(C),nat,size_size(list(C)),Ys)) ).

% length_product
tff(fact_4396_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_4397_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4398_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R = zero_zero(int),aa(int,int,uminus_uminus(int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),
              $ite(R = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_4399_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4400_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4401_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
    <=> ( ? [K2: int] :
            ( ( A12 = K2 )
            & ( A23 = 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)),K2) ) )
        | ? [L5: int,K2: int,Q6: int] :
            ( ( A12 = K2 )
            & ( A23 = L5 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
            & ( L5 != zero_zero(int) )
            & ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L5) ) )
        | ? [R6: int,L5: int,K2: int,Q6: int] :
            ( ( A12 = K2 )
            & ( A23 = L5 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R6) )
            & ( aa(int,int,sgn_sgn(int),R6) = aa(int,int,sgn_sgn(int),L5) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R6)),aa(int,int,abs_abs(int),L5))
            & ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L5)),R6) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_4402_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
     => ( ( ( A23 = 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)),A12) ) )
       => ( ! [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)) )
             => ( ( A23 != zero_zero(int) )
               => ( A12 != aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A23) ) ) )
         => ~ ! [R5: int,Q4: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R5) )
               => ( ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),A23) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),A23))
                   => ( A12 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A23)),R5) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_4403_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => 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,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4404_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( aa(set(int),$o,member(int,K),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_4405_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( Y = $ite(
                  ( aa(set(int),$o,member(int,X),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & aa(set(int),$o,member(int,Xa2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                      & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)
                        & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_4406_fun__of__rel__def,axiom,
    ! [B: $tType,C: $tType,R2: set(product_prod(C,B)),X: C] : fun_of_rel(C,B,R2,X) = fChoice(B,aa(C,fun(B,$o),aTP_Lamp_mh(set(product_prod(C,B)),fun(C,fun(B,$o)),R2),X)) ).

% fun_of_rel_def
tff(fact_4407_ureturn__def,axiom,
    ! [B: $tType,X: B] : aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X) = heap_Time_heap(B,aTP_Lamp_mi(B,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),X)) ).

% ureturn_def
tff(fact_4408_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,Pa: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A12))
     => ( ! [K4: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K4),L2))
           => ( ( ~ ( aa(set(int),$o,member(int,K4),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & aa(set(int),$o,member(int,L2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),Pa,aa(int,int,aa(int,fun(int,int),divide_divide(int),K4),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),Pa,K4),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),Pa,A0),A12) ) ) ).

% and_int.pinduct
tff(fact_4409_return__def,axiom,
    ! [B: $tType,X: B] : aa(B,heap_Time_Heap(B),heap_Time_return(B),X) = heap_Time_heap(B,aTP_Lamp_mj(B,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),X)) ).

% return_def
tff(fact_4410_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa2),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)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_4411_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_4412_upto_Opinduct,axiom,
    ! [A0: int,A12: int,Pa: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A12))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),Pa,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),Pa,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),Pa,A0),A12) ) ) ).

% upto.pinduct
tff(fact_4413_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          L = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
          aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),aa(fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(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))),aa(fun(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(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),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),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_mk(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_4414_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% divmod_abs_code(6)
tff(fact_4415_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).

% divmod_abs_code(5)
tff(fact_4416_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),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_ml(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),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_mm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_4417_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_4418_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_mn(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% bit_cut_integer_code
tff(fact_4419_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_4420_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),K)) ).

% bit_cut_integer_def
tff(fact_4421_lex__take__index,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),lex(B,R))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys))
             => ( ( take(B,I2,Xs) = take(B,I2,Ys) )
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,nth(B,Xs),I2)),aa(nat,B,nth(B,Ys),I2))),R) ) ) ) ) ).

% lex_take_index
tff(fact_4422_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_4423_Cons__in__lex,axiom,
    ! [B: $tType,X: B,Xs: list(B),Y: B,Ys: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),lex(B,R))
    <=> ( ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),R)
          & ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) )
        | ( ( X = Y )
          & aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),lex(B,R)) ) ) ) ).

% Cons_in_lex
tff(fact_4424_lex__append__rightI,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),R: set(product_prod(B,B)),Vs: list(B),Us: list(B)] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),lex(B,R))
     => ( ( aa(list(B),nat,size_size(list(B)),Vs) = aa(list(B),nat,size_size(list(B)),Us) )
       => aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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(list(B),fun(list(B),list(B)),append(B),Xs),Us)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Vs))),lex(B,R)) ) ) ).

% lex_append_rightI
tff(fact_4425_normalize__denom__pos,axiom,
    ! [R: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_4426_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P2: int,R: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4427_part__code_I2_J,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Pivot: C,X: B,Xs: list(B)] : linorder_part(B,C,F,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(C,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_mp(fun(B,C),fun(C,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),F),Pivot),X)),linorder_part(B,C,F,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_4428_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa2) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa2),X),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)),aa(nat,product_prod(nat,nat),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_4429_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Ma: nat] :
      aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),Ma) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Ma)),aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_4430_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_4431_extract__Cons__code,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,Xs: list(B)] :
      extract(B,Pa,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = $ite(aa(B,$o,Pa,X),aa(product_prod(list(B),product_prod(B,list(B))),option(product_prod(list(B),product_prod(B,list(B)))),some(product_prod(list(B),product_prod(B,list(B)))),aa(product_prod(B,list(B)),product_prod(list(B),product_prod(B,list(B))),aa(list(B),fun(product_prod(B,list(B)),product_prod(list(B),product_prod(B,list(B)))),product_Pair(list(B),product_prod(B,list(B))),nil(B)),aa(list(B),product_prod(B,list(B)),aa(B,fun(list(B),product_prod(B,list(B))),product_Pair(B,list(B)),X),Xs))),case_option(option(product_prod(list(B),product_prod(B,list(B)))),product_prod(list(B),product_prod(B,list(B))),none(product_prod(list(B),product_prod(B,list(B)))),aa(fun(list(B),fun(product_prod(B,list(B)),option(product_prod(list(B),product_prod(B,list(B)))))),fun(product_prod(list(B),product_prod(B,list(B))),option(product_prod(list(B),product_prod(B,list(B))))),product_case_prod(list(B),product_prod(B,list(B)),option(product_prod(list(B),product_prod(B,list(B))))),aTP_Lamp_mr(B,fun(list(B),fun(product_prod(B,list(B)),option(product_prod(list(B),product_prod(B,list(B)))))),X)),extract(B,Pa,Xs))) ).

% extract_Cons_code
tff(fact_4432_of__list__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B)] : array_of_list(B,Xs) = heap_Time_heap(array(B),aTP_Lamp_mt(list(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).

% of_list_def
tff(fact_4433_nth__enumerate__eq,axiom,
    ! [B: $tType,Ma: nat,Xs: list(B),N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,product_prod(nat,B),nth(product_prod(nat,B),enumerate(B,N,Xs)),Ma) = aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma)),aa(nat,B,nth(B,Xs),Ma)) ) ) ).

% nth_enumerate_eq
tff(fact_4434_timeFrame_Opelims,axiom,
    ! [B: $tType,X: nat,Xa2: option(product_prod(B,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(B,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(B,X,Xa2) = Y )
     => ( accp(product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(B),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),X),Xa2))
       => ( ! [R5: B,H2: heap_ext(product_unit),N7: nat] :
              ( ( Xa2 = 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)),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),H2),N7))) )
             => ( ( Y = 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)),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),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N7)))) )
               => ~ accp(product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(B),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),X),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)),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),H2),N7))))) ) )
         => ~ ( ( Xa2 = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) )
             => ( ( Y = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) )
               => ~ accp(product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(B),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),X),none(product_prod(B,product_prod(heap_ext(product_unit),nat))))) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_4435_length__enumerate,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(product_prod(nat,B)),nat,size_size(list(product_prod(nat,B))),enumerate(B,N,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_enumerate
tff(fact_4436_enumerate__append__eq,axiom,
    ! [B: $tType,N: nat,Xs: list(B),Ys: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(list(product_prod(nat,B)),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),append(product_prod(nat,B)),enumerate(B,N,Xs)),enumerate(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(B),nat,size_size(list(B)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_4437_new__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,X: B] : array_new(B,N,X) = heap_Time_heap(array(B),aa(B,fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mv(nat,fun(B,fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),N),X)) ) ).

% new_def
tff(fact_4438_execute__of__list,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(B),array_of_list(B,Xs)),H) = aa(product_prod(array(B),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(B),heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ms(list(B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Xs)),array_alloc(B,Xs,H))) ) ).

% execute_of_list
tff(fact_4439_subset__eq__mset__impl_Oelims,axiom,
    ! [B: $tType,X: list(B),Xa2: list(B),Y: option($o)] :
      ( ( subset_eq_mset_impl(B,X,Xa2) = Y )
     => ( ( ( X = nil(B) )
         => ( Y != aa($o,option($o),some($o),Xa2 != nil(B)) ) )
       => ~ ! [X2: B,Xs2: list(B)] :
              ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
             => ( Y != case_option(option($o),product_prod(list(B),product_prod(B,list(B))),none($o),aa(fun(list(B),fun(product_prod(B,list(B)),option($o))),fun(product_prod(list(B),product_prod(B,list(B))),option($o)),product_case_prod(list(B),product_prod(B,list(B)),option($o)),aTP_Lamp_mx(list(B),fun(list(B),fun(product_prod(B,list(B)),option($o))),Xs2)),extract(B,aa(B,fun(B,$o),fequal(B),X2),Xa2)) ) ) ) ) ).

% subset_eq_mset_impl.elims
tff(fact_4440_Cons__lenlex__iff,axiom,
    ! [B: $tType,Ma: B,Ms: list(B),N: B,Ns: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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),Ma),Ms)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),N),Ns))),lenlex(B,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Ms)),aa(list(B),nat,size_size(list(B)),Ns))
        | ( ( aa(list(B),nat,size_size(list(B)),Ms) = aa(list(B),nat,size_size(list(B)),Ns) )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Ma),N)),R) )
        | ( ( Ma = N )
          & aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Ms),Ns)),lenlex(B,R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4441_replicate__eq__replicate,axiom,
    ! [B: $tType,Ma: nat,X: B,N: nat,Y: B] :
      ( ( replicate(B,Ma,X) = replicate(B,N,Y) )
    <=> ( ( Ma = N )
        & ( ( Ma != zero_zero(nat) )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_4442_length__replicate,axiom,
    ! [B: $tType,N: nat,X: B] : aa(list(B),nat,size_size(list(B)),replicate(B,N,X)) = N ).

% length_replicate
tff(fact_4443_replicate__empty,axiom,
    ! [B: $tType,N: nat,X: B] :
      ( ( replicate(B,N,X) = nil(B) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4444_empty__replicate,axiom,
    ! [B: $tType,N: nat,X: B] :
      ( ( nil(B) = replicate(B,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4445_in__set__replicate,axiom,
    ! [B: $tType,X: B,N: nat,Y: B] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),replicate(B,N,Y)))
    <=> ( ( X = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_4446_Bex__set__replicate,axiom,
    ! [B: $tType,N: nat,A2: B,Pa: fun(B,$o)] :
      ( ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),replicate(B,N,A2)))
          & aa(B,$o,Pa,X3) )
    <=> ( aa(B,$o,Pa,A2)
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_4447_Ball__set__replicate,axiom,
    ! [B: $tType,N: nat,A2: B,Pa: fun(B,$o)] :
      ( ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),replicate(B,N,A2)))
         => aa(B,$o,Pa,X3) )
    <=> ( aa(B,$o,Pa,A2)
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_4448_hd__replicate,axiom,
    ! [B: $tType,N: nat,X: B] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(B),B,hd(B),replicate(B,N,X)) = X ) ) ).

% hd_replicate
tff(fact_4449_last__replicate,axiom,
    ! [B: $tType,N: nat,X: B] :
      ( ( N != zero_zero(nat) )
     => ( last(B,replicate(B,N,X)) = X ) ) ).

% last_replicate
tff(fact_4450_set__replicate,axiom,
    ! [B: $tType,N: nat,X: B] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(B),set(B),set2(B),replicate(B,N,X)) = aa(set(B),set(B),insert(B,X),bot_bot(set(B))) ) ) ).

% set_replicate
tff(fact_4451_Heap__eqI,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),G: heap_Time_Heap(B)] :
      ( ! [H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),H2) = aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,G),H2)
     => ( F = G ) ) ).

% Heap_eqI
tff(fact_4452_replicate__0,axiom,
    ! [B: $tType,X: B] : replicate(B,zero_zero(nat),X) = nil(B) ).

% replicate_0
tff(fact_4453_replicate__length__same,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
         => ( X2 = X ) )
     => ( replicate(B,aa(list(B),nat,size_size(list(B)),Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4454_replicate__eqI,axiom,
    ! [B: $tType,Xs: list(B),N: nat,X: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = N )
     => ( ! [Y2: B] :
            ( aa(set(B),$o,member(B,Y2),aa(list(B),set(B),set2(B),Xs))
           => ( Y2 = X ) )
       => ( Xs = replicate(B,N,X) ) ) ) ).

% replicate_eqI
tff(fact_4455_sorted__replicate,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [N: nat,X: B] : sorted_wrt(B,ord_less_eq(B),replicate(B,N,X)) ) ).

% sorted_replicate
tff(fact_4456_replicate__add,axiom,
    ! [B: $tType,N: nat,Ma: nat,X: B] : replicate(B,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma),X) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),replicate(B,N,X)),replicate(B,Ma,X)) ).

% replicate_add
tff(fact_4457_execute__bind_I2_J,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),G: fun(B,heap_Time_Heap(C))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) )
     => ( aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,heap_Time_bind(B,C,F,G)),H) = none(product_prod(C,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_bind(2)
tff(fact_4458_fails__def,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( time_fails(B,Ma,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Ma),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% fails_def
tff(fact_4459_execute__heap,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))] : heap_Time_execute(B,heap_Time_heap(B,F)) = aa(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(B,product_prod(heap_ext(product_unit),nat)))),F) ).

% execute_heap
tff(fact_4460_time__def,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),H: heap_ext(product_unit)] : time_time(B,C2,H) = case_option(nat,product_prod(B,product_prod(heap_ext(product_unit),nat)),zero_zero(nat),aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),nat),product_case_prod(B,product_prod(heap_ext(product_unit),nat),nat),aTP_Lamp_mz(B,fun(product_prod(heap_ext(product_unit),nat),nat))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,C2),H)) ).

% time_def
tff(fact_4461_execute__raise,axiom,
    ! [B: $tType,S2: list(char),X4: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_raise(B,S2)),X4) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ).

% execute_raise
tff(fact_4462_Heap__cases,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( ! [X2: B,H5: 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,F),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)),X2),H5))
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% Heap_cases
tff(fact_4463_set__replicate__conv__if,axiom,
    ! [B: $tType,N: nat,X: B] :
      aa(list(B),set(B),set2(B),replicate(B,N,X)) = $ite(N = zero_zero(nat),bot_bot(set(B)),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) ).

% set_replicate_conv_if
tff(fact_4464_replicate__Suc__conv__snoc,axiom,
    ! [B: $tType,N: nat,X: B] : replicate(B,aa(nat,nat,suc,N),X) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),replicate(B,N,X)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ).

% replicate_Suc_conv_snoc
tff(fact_4465_execute__bind__case,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(C),G: fun(C,heap_Time_Heap(B)),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(C,B,F,G)),H) = case_option(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(C,product_prod(heap_ext(product_unit),nat)),none(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(C,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(C,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(C,product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_nb(fun(C,heap_Time_Heap(B)),fun(C,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),G)),aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,F),H)) ).

% execute_bind_case
tff(fact_4466_execute__new,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,X: B,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(B),array_new(B,N,X)),H) = aa(product_prod(array(B),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(B),heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mu(nat,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(B,replicate(B,N,X),H))) ) ).

% execute_new
tff(fact_4467_lenlex__length,axiom,
    ! [B: $tType,Ms: list(B),Ns: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Ms),Ns)),lenlex(B,R))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Ms)),aa(list(B),nat,size_size(list(B)),Ns)) ) ).

% lenlex_length
tff(fact_4468_lenlex__append1,axiom,
    ! [B: $tType,Us: list(B),Xs: list(B),R2: set(product_prod(B,B)),Vs: list(B),Ys: list(B)] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Us),Xs)),lenlex(B,R2))
     => ( ( aa(list(B),nat,size_size(list(B)),Vs) = aa(list(B),nat,size_size(list(B)),Ys) )
       => aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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(list(B),fun(list(B),list(B)),append(B),Us),Vs)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys))),lenlex(B,R2)) ) ) ).

% lenlex_append1
tff(fact_4469_Cons__replicate__eq,axiom,
    ! [B: $tType,X: B,Xs: list(B),N: nat,Y: B] :
      ( ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs) = replicate(B,N,Y) )
    <=> ( ( X = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
        & ( Xs = replicate(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_4470_refines__def,axiom,
    ! [B: $tType,P2: heap_Time_Heap(B),Q2: heap_Time_Heap(B)] :
      ( refine_Imp_refines(B,P2,Q2)
    <=> ! [H3: heap_ext(product_unit)] : case_option($o,product_prod(B,product_prod(heap_ext(product_unit),nat)),$true,aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),$o)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),$o),product_case_prod(B,product_prod(heap_ext(product_unit),nat),$o),aa(heap_ext(product_unit),fun(B,fun(product_prod(heap_ext(product_unit),nat),$o)),aTP_Lamp_nd(heap_Time_Heap(B),fun(heap_ext(product_unit),fun(B,fun(product_prod(heap_ext(product_unit),nat),$o))),P2),H3)),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Q2),H3)) ) ).

% refines_def
tff(fact_4471_execute__bind__eq__SomeI,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),X: B,H4: heap_ext(product_unit),N: nat,G: fun(B,heap_Time_Heap(C)),Y: C,H6: heap_ext(product_unit),N3: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),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)),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),H4),N))) )
     => ( ( aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,aa(B,heap_Time_Heap(C),G,X)),H4) = aa(product_prod(C,product_prod(heap_ext(product_unit),nat)),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),some(product_prod(C,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(C,product_prod(heap_ext(product_unit),nat)),aa(C,fun(product_prod(heap_ext(product_unit),nat),product_prod(C,product_prod(heap_ext(product_unit),nat))),product_Pair(C,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),N3))) )
       => ( aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,heap_Time_bind(B,C,F,G)),H) = aa(product_prod(C,product_prod(heap_ext(product_unit),nat)),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),some(product_prod(C,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(C,product_prod(heap_ext(product_unit),nat)),aa(C,fun(product_prod(heap_ext(product_unit),nat),product_prod(C,product_prod(heap_ext(product_unit),nat))),product_Pair(C,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),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_4472_execute__bind_I1_J,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),X: B,H4: heap_ext(product_unit),N: nat,G: fun(B,heap_Time_Heap(C))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),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)),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),H4),N))) )
     => ( aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,heap_Time_bind(B,C,F,G)),H) = heap_Time_timeFrame(C,N,aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,aa(B,heap_Time_Heap(C),G,X)),H4)) ) ) ).

% execute_bind(1)
tff(fact_4473_execute__return,axiom,
    ! [B: $tType,X: B] : heap_Time_execute(B,aa(B,heap_Time_Heap(B),heap_Time_return(B),X)) = aa(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_mj(B,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_return
tff(fact_4474_execute__ureturn,axiom,
    ! [B: $tType,X: B] : heap_Time_execute(B,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X)) = aa(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_mi(B,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_ureturn
tff(fact_4475_lenlex__conv,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : lenlex(B,R) = aa(fun(product_prod(list(B),list(B)),$o),set(product_prod(list(B),list(B))),collect(product_prod(list(B),list(B))),aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_ne(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),R))) ).

% lenlex_conv
tff(fact_4476_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),Ys: list(B)] : subset_eq_mset_impl(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),Ys) = case_option(option($o),product_prod(list(B),product_prod(B,list(B))),none($o),aa(fun(list(B),fun(product_prod(B,list(B)),option($o))),fun(product_prod(list(B),product_prod(B,list(B))),option($o)),product_case_prod(list(B),product_prod(B,list(B)),option($o)),aTP_Lamp_mx(list(B),fun(list(B),fun(product_prod(B,list(B)),option($o))),Xs)),extract(B,aa(B,fun(B,$o),fequal(B),X),Ys)) ).

% subset_eq_mset_impl.simps(2)
tff(fact_4477_execute__tap,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),B),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_tap(B,F)),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)),aa(heap_ext(product_unit),B,F,H)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ).

% execute_tap
tff(fact_4478_subset__eq__mset__impl_Opelims,axiom,
    ! [B: $tType,X: list(B),Xa2: list(B),Y: option($o)] :
      ( ( subset_eq_mset_impl(B,X,Xa2) = Y )
     => ( accp(product_prod(list(B),list(B)),subset751672762298770561pl_rel(B),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)),X),Xa2))
       => ( ( ( X = nil(B) )
           => ( ( Y = aa($o,option($o),some($o),Xa2 != nil(B)) )
             => ~ accp(product_prod(list(B),list(B)),subset751672762298770561pl_rel(B),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),Xa2)) ) )
         => ~ ! [X2: B,Xs2: list(B)] :
                ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
               => ( ( Y = case_option(option($o),product_prod(list(B),product_prod(B,list(B))),none($o),aa(fun(list(B),fun(product_prod(B,list(B)),option($o))),fun(product_prod(list(B),product_prod(B,list(B))),option($o)),product_case_prod(list(B),product_prod(B,list(B)),option($o)),aTP_Lamp_mx(list(B),fun(list(B),fun(product_prod(B,list(B)),option($o))),Xs2)),extract(B,aa(B,fun(B,$o),fequal(B),X2),Xa2)) )
                 => ~ accp(product_prod(list(B),list(B)),subset751672762298770561pl_rel(B),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),X2),Xs2)),Xa2)) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_4479_execute__assert_I1_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,H: heap_ext(product_unit)] :
      ( aa(B,$o,Pa,X)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_assert(B,Pa,X)),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)),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_4480_execute__lookup,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [R: ref(B),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,ref_lookup(B,R)),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)),ref_get(B,H,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),H),one_one(nat)))) ) ).

% execute_lookup
tff(fact_4481_lookup__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [R: ref(B)] : ref_lookup(B,R) = heap_Time_tap(B,aTP_Lamp_nf(ref(B),fun(heap_ext(product_unit),B),R)) ) ).

% lookup_def
tff(fact_4482_assert__cong,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),P: fun(B,$o),F: fun(B,heap_Time_Heap(C)),F2: fun(B,heap_Time_Heap(C)),X: B] :
      ( ( Pa = P )
     => ( ! [X2: B] :
            ( aa(B,$o,P,X2)
           => ( aa(B,heap_Time_Heap(C),F,X2) = aa(B,heap_Time_Heap(C),F2,X2) ) )
       => ( heap_Time_bind(B,C,heap_Time_assert(B,Pa,X),F) = heap_Time_bind(B,C,heap_Time_assert(B,P,X),F2) ) ) ) ).

% assert_cong
tff(fact_4483_execute__assert_I2_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,H: heap_ext(product_unit)] :
      ( ~ aa(B,$o,Pa,X)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_assert(B,Pa,X)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_assert(2)
tff(fact_4484_revg_Opelims,axiom,
    ! [B: $tType,X: list(B),Xa2: list(B),Y: list(B)] :
      ( ( revg(B,X,Xa2) = Y )
     => ( accp(product_prod(list(B),list(B)),revg_rel(B),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)),X),Xa2))
       => ( ( ( X = nil(B) )
           => ( ( Y = Xa2 )
             => ~ accp(product_prod(list(B),list(B)),revg_rel(B),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)),nil(B)),Xa2)) ) )
         => ~ ! [A3: B,As2: list(B)] :
                ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),As2) )
               => ( ( Y = revg(B,As2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xa2)) )
                 => ~ accp(product_prod(list(B),list(B)),revg_rel(B),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),A3),As2)),Xa2)) ) ) ) ) ) ).

% revg.pelims
tff(fact_4485_Heap__Time__Monad_Obind__def,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(C),G: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,F,G) = heap_Time_Heap2(B,aa(fun(C,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ng(heap_Time_Heap(C),fun(fun(C,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_4486_the__res__def,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),H: heap_ext(product_unit)] : time_the_res(B,Ma,H) = case_option(B,product_prod(B,product_prod(heap_ext(product_unit),nat)),undefined(B),aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),B)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),B),product_case_prod(B,product_prod(heap_ext(product_unit),nat),B),aTP_Lamp_ni(B,fun(product_prod(heap_ext(product_unit),nat),B))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Ma),H)) ).

% the_res_def
tff(fact_4487_success__bind__executeI,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),X: B,H4: heap_ext(product_unit),N: nat,G: fun(B,heap_Time_Heap(C))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),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)),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),H4),N))) )
     => ( heap_Time_success(C,aa(B,heap_Time_Heap(C),G,X),H4)
       => heap_Time_success(C,heap_Time_bind(B,C,F,G),H) ) ) ).

% success_bind_executeI
tff(fact_4488_Heap_Oinject,axiom,
    ! [B: $tType,X: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),Ya: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
      ( ( heap_Time_Heap2(B,X) = heap_Time_Heap2(B,Ya) )
    <=> ( X = Ya ) ) ).

% Heap.inject
tff(fact_4489_Heap__execute,axiom,
    ! [B: $tType,F: heap_Time_Heap(B)] : heap_Time_Heap2(B,heap_Time_execute(B,F)) = F ).

% Heap_execute
tff(fact_4490_execute_Osimps,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] : heap_Time_execute(B,heap_Time_Heap2(B,F)) = F ).

% execute.simps
tff(fact_4491_success__ifI,axiom,
    ! [B: $tType,C2: $o,Ta: heap_Time_Heap(B),H: heap_ext(product_unit),E2: heap_Time_Heap(B)] :
      ( ( (C2)
       => heap_Time_success(B,Ta,H) )
     => ( ( ~ (C2)
         => heap_Time_success(B,E2,H) )
       => heap_Time_success(B,
            $ite((C2),Ta,E2),
            H) ) ) ).

% success_ifI
tff(fact_4492_Heap_Oexhaust,axiom,
    ! [B: $tType,Y: heap_Time_Heap(B)] :
      ~ ! [X2: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] : Y != heap_Time_Heap2(B,X2) ).

% Heap.exhaust
tff(fact_4493_success__LetI,axiom,
    ! [B: $tType,C: $tType,X: B,Ta: B,F: fun(B,heap_Time_Heap(C)),H: heap_ext(product_unit)] :
      ( ( X = Ta )
     => ( heap_Time_success(C,aa(B,heap_Time_Heap(C),F,X),H)
       => heap_Time_success(C,aa(B,heap_Time_Heap(C),F,Ta),H) ) ) ).

% success_LetI
tff(fact_4494_success__returnI,axiom,
    ! [B: $tType,X: B,H: heap_ext(product_unit)] : heap_Time_success(B,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),H) ).

% success_returnI
tff(fact_4495_success__heapI,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit)] : heap_Time_success(B,heap_Time_heap(B,F),H) ).

% success_heapI
tff(fact_4496_success__ureturnI,axiom,
    ! [B: $tType,X: B,H: heap_ext(product_unit)] : heap_Time_success(B,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X),H) ).

% success_ureturnI
tff(fact_4497_success__assertI,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,H: heap_ext(product_unit)] :
      ( aa(B,$o,Pa,X)
     => heap_Time_success(B,heap_Time_assert(B,Pa,X),H) ) ).

% success_assertI
tff(fact_4498_success__tapI,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),B),H: heap_ext(product_unit)] : heap_Time_success(B,heap_Time_tap(B,F),H) ).

% success_tapI
tff(fact_4499_successI,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),H) != none(product_prod(B,product_prod(heap_ext(product_unit),nat))) )
     => heap_Time_success(B,F,H) ) ).

% successI
tff(fact_4500_success__def,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( heap_Time_success(B,F,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),H) != none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% success_def
tff(fact_4501_refines__empty,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B)] : refine_Imp_refines(B,Ma,heap_Time_Heap2(B,aTP_Lamp_nj(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))) ).

% refines_empty
tff(fact_4502_TBOUND__empty,axiom,
    ! [B: $tType,Ta: nat] : time_TBOUND(B,heap_Time_Heap2(B,aTP_Lamp_nj(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Ta) ).

% TBOUND_empty
tff(fact_4503_raise__def,axiom,
    ! [B: $tType,S2: list(char)] : heap_Time_raise(B,S2) = heap_Time_Heap2(B,aTP_Lamp_nj(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))) ).

% raise_def
tff(fact_4504_the__heap__def,axiom,
    ! [B: $tType,Ma: heap_Time_Heap(B),H: heap_ext(product_unit)] : time_the_heap(B,Ma,H) = case_option(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),undefined(heap_ext(product_unit)),aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit))),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),heap_ext(product_unit)),product_case_prod(B,product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),aTP_Lamp_nl(B,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Ma),H)) ).

% the_heap_def
tff(fact_4505_successE,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( heap_Time_success(B,F,H)
     => ~ ! [R5: B,H5: 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,F),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)),R5),H5)) ) ).

% successE
tff(fact_4506_Heap_Osize_I2_J,axiom,
    ! [B: $tType,X: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(B),nat,size_size(heap_Time_Heap(B)),heap_Time_Heap2(B,X)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size(2)
tff(fact_4507_heap__def,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))] : heap_Time_heap(B,F) = heap_Time_Heap2(B,aa(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(B,product_prod(heap_ext(product_unit),nat)))),F)) ).

% heap_def
tff(fact_4508_tap__def,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),B)] : heap_Time_tap(B,F) = heap_Time_Heap2(B,aTP_Lamp_nm(fun(heap_ext(product_unit),B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),F)) ).

% tap_def
tff(fact_4509_Heap_Osize__gen,axiom,
    ! [B: $tType,Xa2: fun(B,nat),X: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] : heap_Time_size_Heap(B,Xa2,heap_Time_Heap2(B,X)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size_gen
tff(fact_4510_arg__min__list_Oelims,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [X: fun(B,C),Xa2: list(B),Y: B] :
          ( ( arg_min_list(B,C,X,Xa2) = Y )
         => ( ! [X2: B] :
                ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)) )
               => ( Y != X2 ) )
           => ( ! [X2: B,Y2: B,Zs2: list(B)] :
                  ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Zs2)) )
                 => ( Y != $let(
                        m2: B,
                        m2:= arg_min_list(B,C,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Zs2)),
                        $ite(aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,X,X2)),aa(B,C,X,m2)),X2,m2) ) ) )
             => ~ ( ( Xa2 = nil(B) )
                 => ( Y != undefined(B) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_4511_guard__def,axiom,
    ! [B: $tType,Pa: fun(heap_ext(product_unit),$o),F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(B,Pa,F) = heap_Time_Heap2(B,aa(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_nn(fun(heap_ext(product_unit),$o),fun(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Pa),F)) ).

% guard_def
tff(fact_4512_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_np(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% times_int.abs_eq
tff(fact_4513_int_Oabs__induct,axiom,
    ! [Pa: fun(int,$o),X: int] :
      ( ! [Y2: product_prod(nat,nat)] : aa(int,$o,Pa,aa(product_prod(nat,nat),int,abs_Integ,Y2))
     => aa(int,$o,Pa,X) ) ).

% int.abs_induct
tff(fact_4514_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y2: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y2)) ).

% eq_Abs_Integ
tff(fact_4515_success__guardI,axiom,
    ! [B: $tType,Pa: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))] :
      ( aa(heap_ext(product_unit),$o,Pa,H)
     => heap_Time_success(B,heap_Time_guard(B,Pa,F),H) ) ).

% success_guardI
tff(fact_4516_nat_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),X) ).

% nat.abs_eq
tff(fact_4517_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_4518_int__def,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),zero_zero(nat))) ).

% int_def
tff(fact_4519_arg__min__list_Osimps_I2_J,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),X: B,Y: B,Zs: list(B)] :
          arg_min_list(B,C,F,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),Zs))) = $let(
            m2: B,
            m2:= arg_min_list(B,C,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Zs)),
            $ite(aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,m2)),X,m2) ) ) ).

% arg_min_list.simps(2)
tff(fact_4520_execute__guard_I2_J,axiom,
    ! [B: $tType,Pa: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))] :
      ( aa(heap_ext(product_unit),$o,Pa,H)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_guard(B,Pa,F)),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(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H)) ) ) ).

% execute_guard(2)
tff(fact_4521_execute__guard_I1_J,axiom,
    ! [B: $tType,Pa: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))] :
      ( ~ aa(heap_ext(product_unit),$o,Pa,H)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_guard(B,Pa,F)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_guard(1)
tff(fact_4522_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_nq(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_4523_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_4524_of__int_Oabs__eq,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: product_prod(nat,nat)] : aa(int,B,ring_1_of_int(B),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),B,aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),aTP_Lamp_nr(nat,fun(nat,B))),X) ) ).

% of_int.abs_eq
tff(fact_4525_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa2),X) ) ).

% less_int.abs_eq
tff(fact_4526_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa2),X) ) ).

% less_eq_int.abs_eq
tff(fact_4527_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_nx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_4528_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_nz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_4529_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q2)) = 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))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_4530_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),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(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_4531_listrel1__iff__update,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),listrel1(B,R))
    <=> ? [Y5: B,N4: nat] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,nth(B,Xs),N4)),Y5)),R)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Xs))
          & ( Ys = list_update(B,Xs,N4,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_4532_sort__key__by__quicksort__code,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,C,F),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_oi(fun(B,C),fun(list(B),fun(B,fun(list(B),list(B)))),F),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_4533_sort__upto,axiom,
    ! [I: int,J: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_dk(int,int)),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_4534_length__sort,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),linorder_sort_key(B,C,F),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ).

% length_sort
tff(fact_4535_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_4536_sorted__list__of__multiset__mset,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : linord6283353356039996273ltiset(B,mset(B,Xs)) = aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs) ) ).

% sorted_list_of_multiset_mset
tff(fact_4537_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_4538_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_4539_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_4540_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_4541_list_Ocase__distrib,axiom,
    ! [C: $tType,B: $tType,D: $tType,H: fun(C,B),F1: C,F22: fun(D,fun(list(D),C)),List: list(D)] : aa(C,B,H,aa(list(D),C,case_list(C,D,F1,F22),List)) = aa(list(D),B,case_list(B,D,aa(C,B,H,F1),aa(fun(D,fun(list(D),C)),fun(D,fun(list(D),B)),aTP_Lamp_oj(fun(C,B),fun(fun(D,fun(list(D),C)),fun(D,fun(list(D),B))),H),F22)),List) ).

% list.case_distrib
tff(fact_4542_sort__key__const,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [C2: C,Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,C,aTP_Lamp_ok(C,fun(B,C),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_4543_diff__rat__def,axiom,
    ! [Q2: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q2),R) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q2),aa(rat,rat,uminus_uminus(rat),R)) ).

% diff_rat_def
tff(fact_4544_listrel1__eq__len,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),listrel1(B,R))
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_4545_sorted__sort__id,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_4546_sorted__sort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs)) ) ).

% sorted_sort
tff(fact_4547_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_4548_properties__for__sort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ys: list(B),Xs: list(B)] :
          ( ( mset(B,Ys) = mset(B,Xs) )
         => ( sorted_wrt(B,ord_less_eq(B),Ys)
           => ( aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs) = Ys ) ) ) ) ).

% properties_for_sort
tff(fact_4549_hd__def,axiom,
    ! [B: $tType,List: list(B)] : aa(list(B),B,hd(B),List) = aa(list(B),B,case_list(B,B,undefined(B),aTP_Lamp_ol(B,fun(list(B),B))),List) ).

% hd_def
tff(fact_4550_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = 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(P2)) ).

% rat_floor_code
tff(fact_4551_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = 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_om(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_4552_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P2)) = 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(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_4553_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_op(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_4554_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_or(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_4555_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),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_ot(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_4556_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),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_ov(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_4557_listrel1p__def,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),Xs: list(B),Ys: list(B)] :
      ( listrel1p(B,R,Xs,Ys)
    <=> aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Xs),Ys)),listrel1(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),R)))) ) ).

% listrel1p_def
tff(fact_4558_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),one_one(int)) ).

% quotient_of_int
tff(fact_4559_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = 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_ow(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_4560_sort__key__by__quicksort,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,C,F),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_sort_key(B,C,F),filter2(B,aa(list(B),fun(B,$o),aTP_Lamp_ox(fun(B,C),fun(list(B),fun(B,$o)),F),Xs),Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),filter2(B,aa(list(B),fun(B,$o),aTP_Lamp_oy(fun(B,C),fun(list(B),fun(B,$o)),F),Xs),Xs)),aa(list(B),list(B),linorder_sort_key(B,C,F),filter2(B,aa(list(B),fun(B,$o),aTP_Lamp_oz(fun(B,C),fun(list(B),fun(B,$o)),F),Xs),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_4561_filter__filter,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o),Xs: list(B)] : filter2(B,Pa,filter2(B,Q,Xs)) = filter2(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_pa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),Xs) ).

% filter_filter
tff(fact_4562_inverse__zero,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ( aa(B,B,inverse_inverse(B),zero_zero(B)) = zero_zero(B) ) ) ).

% inverse_zero
tff(fact_4563_inverse__nonzero__iff__nonzero,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( aa(B,B,inverse_inverse(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(B) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_4564_inverse__eq__1__iff,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [X: B] :
          ( ( aa(B,B,inverse_inverse(B),X) = one_one(B) )
        <=> ( X = one_one(B) ) ) ) ).

% inverse_eq_1_iff
tff(fact_4565_inverse__1,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ( aa(B,B,inverse_inverse(B),one_one(B)) = one_one(B) ) ) ).

% inverse_1
tff(fact_4566_set__filter,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : aa(list(B),set(B),set2(B),filter2(B,Pa,Xs)) = aa(fun(B,$o),set(B),collect(B),aa(list(B),fun(B,$o),aTP_Lamp_pb(fun(B,$o),fun(list(B),fun(B,$o)),Pa),Xs)) ).

% set_filter
tff(fact_4567_inverse__nonnegative__iff__nonnegative,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(B,B,inverse_inverse(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4568_inverse__nonpositive__iff__nonpositive,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),zero_zero(B)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4569_inverse__positive__iff__positive,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,inverse_inverse(B),A2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_4570_inverse__negative__iff__negative,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ).

% inverse_negative_iff_negative
tff(fact_4571_inverse__less__iff__less__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_4572_inverse__less__iff__less,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_4573_inverse__le__iff__le,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),B2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4574_inverse__le__iff__le__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
            <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4575_left__inverse,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),A2)),A2) = one_one(B) ) ) ) ).

% left_inverse
tff(fact_4576_right__inverse,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,inverse_inverse(B),A2)) = one_one(B) ) ) ) ).

% right_inverse
tff(fact_4577_inverse__eq__divide__numeral,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [W: num] : aa(B,B,inverse_inverse(B),aa(num,B,numeral_numeral(B),W)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_4578_power__int__minus1__right,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & monoid_mult(B) )
     => ! [Y: B] : power_int(B,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(B,B,inverse_inverse(B),Y) ) ).

% power_int_minus1_right
tff(fact_4579_inverse__eq__divide__neg__numeral,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [W: num] : aa(B,B,inverse_inverse(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),W))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_4580_sort__key__stable,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),K: C,Xs: list(B)] : filter2(B,aa(C,fun(B,$o),aTP_Lamp_pc(fun(B,C),fun(C,fun(B,$o)),F),K),aa(list(B),list(B),linorder_sort_key(B,C,F),Xs)) = filter2(B,aa(C,fun(B,$o),aTP_Lamp_pc(fun(B,C),fun(C,fun(B,$o)),F),K),Xs) ) ).

% sort_key_stable
tff(fact_4581_partition__in__shuffles,axiom,
    ! [B: $tType,Xs: list(B),Pa: fun(B,$o)] : aa(set(list(B)),$o,member(list(B),Xs),shuffles(B,filter2(B,Pa,Xs),filter2(B,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa),Xs))) ).

% partition_in_shuffles
tff(fact_4582_sgn__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,sgn_sgn(rat),A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_4583_nonzero__imp__inverse__nonzero,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,inverse_inverse(B),A2) != zero_zero(B) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_4584_nonzero__inverse__inverse__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,inverse_inverse(B),aa(B,B,inverse_inverse(B),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_4585_nonzero__inverse__eq__imp__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,inverse_inverse(B),A2) = aa(B,B,inverse_inverse(B),B2) )
         => ( ( A2 != zero_zero(B) )
           => ( ( B2 != zero_zero(B) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_4586_inverse__zero__imp__zero,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( aa(B,B,inverse_inverse(B),A2) = zero_zero(B) )
         => ( A2 = zero_zero(B) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_4587_field__class_Ofield__inverse__zero,axiom,
    ! [B: $tType] :
      ( field(B)
     => ( aa(B,B,inverse_inverse(B),zero_zero(B)) = zero_zero(B) ) ) ).

% field_class.field_inverse_zero
tff(fact_4588_power__int__inverse,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,N: int] : power_int(B,aa(B,B,inverse_inverse(B),X),N) = aa(B,B,inverse_inverse(B),power_int(B,X,N)) ) ).

% power_int_inverse
tff(fact_4589_length__filter__le,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),filter2(B,Pa,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_filter_le
tff(fact_4590_positive__imp__inverse__positive,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,inverse_inverse(B),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_4591_negative__imp__inverse__negative,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B))
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),zero_zero(B)) ) ) ).

% negative_imp_inverse_negative
tff(fact_4592_inverse__positive__imp__positive,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,inverse_inverse(B),A2))
         => ( ( A2 != zero_zero(B) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_4593_inverse__negative__imp__negative,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),zero_zero(B))
         => ( ( A2 != zero_zero(B) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),zero_zero(B)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_4594_less__imp__inverse__less__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),B2)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_4595_inverse__less__imp__less__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_4596_less__imp__inverse__less,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),B2)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% less_imp_inverse_less
tff(fact_4597_inverse__less__imp__less,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_4598_sorted__filter_H,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B),Pa: fun(B,$o)] :
          ( sorted_wrt(B,ord_less_eq(B),L)
         => sorted_wrt(B,ord_less_eq(B),filter2(B,Pa,L)) ) ) ).

% sorted_filter'
tff(fact_4599_nonzero__inverse__mult__distrib,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,inverse_inverse(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),B2)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_4600_nonzero__inverse__minus__eq,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,inverse_inverse(B),aa(B,B,uminus_uminus(B),A2)) = aa(B,B,uminus_uminus(B),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_4601_inverse__unique,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2) = one_one(B) )
         => ( aa(B,B,inverse_inverse(B),A2) = B2 ) ) ) ).

% inverse_unique
tff(fact_4602_inverse__numeral__1,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ( aa(B,B,inverse_inverse(B),aa(num,B,numeral_numeral(B),one2)) = aa(num,B,numeral_numeral(B),one2) ) ) ).

% inverse_numeral_1
tff(fact_4603_inverse__eq__divide,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] : aa(B,B,inverse_inverse(B),A2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2) ) ).

% inverse_eq_divide
tff(fact_4604_power__mult__inverse__distrib,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(B,B,inverse_inverse(B),X)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),X)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)) ) ).

% power_mult_inverse_distrib
tff(fact_4605_power__mult__power__inverse__commute,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: nat,N: nat] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,inverse_inverse(B),X)),N)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,inverse_inverse(B),X)),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),Ma)) ) ).

% power_mult_power_inverse_commute
tff(fact_4606_mult__inverse__of__nat__commute,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Xa2: nat,X: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(nat,B,semiring_1_of_nat(B),Xa2))),X) = aa(B,B,aa(B,fun(B,B),times_times(B),X),aa(B,B,inverse_inverse(B),aa(nat,B,semiring_1_of_nat(B),Xa2))) ) ).

% mult_inverse_of_nat_commute
tff(fact_4607_nonzero__abs__inverse,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,abs_abs(B),aa(B,B,inverse_inverse(B),A2)) = aa(B,B,inverse_inverse(B),aa(B,B,abs_abs(B),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_4608_mult__inverse__of__int__commute,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Xa2: int,X: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(int,B,ring_1_of_int(B),Xa2))),X) = aa(B,B,aa(B,fun(B,B),times_times(B),X),aa(B,B,inverse_inverse(B),aa(int,B,ring_1_of_int(B),Xa2))) ) ).

% mult_inverse_of_int_commute
tff(fact_4609_last__filter,axiom,
    ! [B: $tType,Xs: list(B),Pa: fun(B,$o)] :
      ( ( Xs != nil(B) )
     => ( aa(B,$o,Pa,last(B,Xs))
       => ( last(B,filter2(B,Pa,Xs)) = last(B,Xs) ) ) ) ).

% last_filter
tff(fact_4610_power__int__minus,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,N: int] : power_int(B,X,aa(int,int,uminus_uminus(int),N)) = aa(B,B,inverse_inverse(B),power_int(B,X,N)) ) ).

% power_int_minus
tff(fact_4611_list_Odisc__eq__case_I2_J,axiom,
    ! [B: $tType,List: list(B)] :
      ( ( List != nil(B) )
    <=> aa(list(B),$o,case_list($o,B,$false,aTP_Lamp_pd(B,fun(list(B),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_4612_list_Odisc__eq__case_I1_J,axiom,
    ! [B: $tType,List: list(B)] :
      ( ( List = nil(B) )
    <=> aa(list(B),$o,case_list($o,B,$true,aTP_Lamp_pe(B,fun(list(B),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_4613_sum__length__filter__compl,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),filter2(B,Pa,Xs))),aa(list(B),nat,size_size(list(B)),filter2(B,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa),Xs))) = aa(list(B),nat,size_size(list(B)),Xs) ).

% sum_length_filter_compl
tff(fact_4614_inter__set__filter,axiom,
    ! [B: $tType,A4: set(B),Xs: list(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(list(B),set(B),set2(B),Xs)) = aa(list(B),set(B),set2(B),filter2(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4),Xs)) ).

% inter_set_filter
tff(fact_4615_sorted__same,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [G: fun(list(B),B),Xs: list(B)] : sorted_wrt(B,ord_less_eq(B),filter2(B,aa(list(B),fun(B,$o),aTP_Lamp_pf(fun(list(B),B),fun(list(B),fun(B,$o)),G),Xs),Xs)) ) ).

% sorted_same
tff(fact_4616_mset__eq__length__filter,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Z: B] :
      ( ( mset(B,Xs) = mset(B,Ys) )
     => ( aa(list(B),nat,size_size(list(B)),filter2(B,aa(B,fun(B,$o),fequal(B),Z),Xs)) = aa(list(B),nat,size_size(list(B)),filter2(B,aa(B,fun(B,$o),fequal(B),Z),Ys)) ) ) ).

% mset_eq_length_filter
tff(fact_4617_replicate__length__filter,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : replicate(B,aa(list(B),nat,size_size(list(B)),filter2(B,aa(B,fun(B,$o),fequal(B),X),Xs)),X) = filter2(B,aa(B,fun(B,$o),fequal(B),X),Xs) ).

% replicate_length_filter
tff(fact_4618_length__filter__less,axiom,
    ! [B: $tType,X: B,Xs: list(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
     => ( ~ aa(B,$o,Pa,X)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),filter2(B,Pa,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ) ) ).

% length_filter_less
tff(fact_4619_inverse__le__imp__le,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_4620_le__imp__inverse__le,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),B2)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4621_inverse__le__imp__le__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4622_le__imp__inverse__le__neg,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),zero_zero(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),B2)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_4623_inverse__le__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),X)),one_one(B))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),X) ) ) ) ).

% inverse_le_1_iff
tff(fact_4624_one__less__inverse,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% one_less_inverse
tff(fact_4625_one__less__inverse__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(B,B,inverse_inverse(B),X))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),one_one(B)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4626_division__ring__inverse__add,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2))),aa(B,B,inverse_inverse(B),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_4627_inverse__add,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),aa(B,B,inverse_inverse(B),A2))),aa(B,B,inverse_inverse(B),B2)) ) ) ) ) ).

% inverse_add
tff(fact_4628_field__class_Ofield__inverse,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),A2)),A2) = one_one(B) ) ) ) ).

% field_class.field_inverse
tff(fact_4629_division__ring__inverse__diff,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),A2))),aa(B,B,inverse_inverse(B),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4630_nonzero__inverse__eq__divide,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [A2: B] :
          ( ( A2 != zero_zero(B) )
         => ( aa(B,B,inverse_inverse(B),A2) = aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4631_filter__eq__snocD,axiom,
    ! [B: $tType,Pa: fun(B,$o),L: list(B),L3: list(B),X: B] :
      ( ( filter2(B,Pa,L) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) )
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),L))
        & aa(B,$o,Pa,X) ) ) ).

% filter_eq_snocD
tff(fact_4632_inverse__le__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_4633_inverse__less__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),A2)),aa(B,B,inverse_inverse(B),B2))
        <=> ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) )
            & ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),zero_zero(B))
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_4634_one__le__inverse__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,inverse_inverse(B),X))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),one_one(B)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4635_inverse__less__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),X)),one_one(B))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),zero_zero(B))
            | aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),X) ) ) ) ).

% inverse_less_1_iff
tff(fact_4636_one__le__inverse,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),one_one(B))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(B,B,inverse_inverse(B),A2)) ) ) ) ).

% one_le_inverse
tff(fact_4637_filter__conv__foldr,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : filter2(B,Pa,Xs) = aa(list(B),list(B),foldr(B,list(B),aTP_Lamp_pg(fun(B,$o),fun(B,fun(list(B),list(B))),Pa),Xs),nil(B)) ).

% filter_conv_foldr
tff(fact_4638_reals__Archimedean,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => ? [N2: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,suc,N2)))),X) ) ) ).

% reals_Archimedean
tff(fact_4639_set__minus__filter__out,axiom,
    ! [B: $tType,Xs: list(B),Y: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(set(B),set(B),insert(B,Y),bot_bot(set(B)))) = aa(list(B),set(B),set2(B),filter2(B,aTP_Lamp_ph(B,fun(B,$o),Y),Xs)) ).

% set_minus_filter_out
tff(fact_4640_filter__shuffles__disjoint2_I1_J,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Zs: list(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)) = bot_bot(set(B)) )
     => ( aa(set(list(B)),$o,member(list(B),Zs),shuffles(B,Xs,Ys))
       => ( filter2(B,aTP_Lamp_pi(list(B),fun(B,$o),Ys),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4641_filter__shuffles__disjoint2_I2_J,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Zs: list(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)) = bot_bot(set(B)) )
     => ( aa(set(list(B)),$o,member(list(B),Zs),shuffles(B,Xs,Ys))
       => ( filter2(B,aTP_Lamp_pj(list(B),fun(B,$o),Ys),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4642_filter__shuffles__disjoint1_I1_J,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Zs: list(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)) = bot_bot(set(B)) )
     => ( aa(set(list(B)),$o,member(list(B),Zs),shuffles(B,Xs,Ys))
       => ( filter2(B,aTP_Lamp_pi(list(B),fun(B,$o),Xs),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4643_filter__shuffles__disjoint1_I2_J,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),Zs: list(B)] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)) = bot_bot(set(B)) )
     => ( aa(set(list(B)),$o,member(list(B),Zs),shuffles(B,Xs,Ys))
       => ( filter2(B,aTP_Lamp_pj(list(B),fun(B,$o),Xs),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4644_length__filter__conv__card,axiom,
    ! [B: $tType,P2: fun(B,$o),Xs: list(B)] : aa(list(B),nat,size_size(list(B)),filter2(B,P2,Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(B),fun(nat,$o),aTP_Lamp_pk(fun(B,$o),fun(list(B),fun(nat,$o)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_4645_filter__nth__ex__nth,axiom,
    ! [B: $tType,N: nat,Pa: fun(B,$o),Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),filter2(B,Pa,Xs)))
     => ? [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(list(B),nat,size_size(list(B)),Xs))
          & ( aa(nat,B,nth(B,filter2(B,Pa,Xs)),N) = aa(nat,B,nth(B,Xs),M2) )
          & ( filter2(B,Pa,take(B,M2,Xs)) = take(B,N,filter2(B,Pa,Xs)) ) ) ) ).

% filter_nth_ex_nth
tff(fact_4646_ex__inverse__of__nat__less,axiom,
    ! [B: $tType] :
      ( archim462609752435547400_field(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),X)
         => ? [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,inverse_inverse(B),aa(nat,B,semiring_1_of_nat(B),N2))),X) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4647_power__diff__conv__inverse,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [X: B,Ma: nat,N: nat] :
          ( ( X != zero_zero(B) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
           => ( aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,inverse_inverse(B),X)),Ma)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4648_distinct__length__filter,axiom,
    ! [B: $tType,Xs: list(B),Pa: fun(B,$o)] :
      ( distinct(B,Xs)
     => ( aa(list(B),nat,size_size(list(B)),filter2(B,Pa,Xs)) = aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(fun(B,$o),set(B),collect(B),Pa)),aa(list(B),set(B),set2(B),Xs))) ) ) ).

% distinct_length_filter
tff(fact_4649_part__def,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Pivot: C,Xs: list(B)] : linorder_part(B,C,F,Pivot,Xs) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),filter2(B,aa(C,fun(B,$o),aTP_Lamp_pl(fun(B,C),fun(C,fun(B,$o)),F),Pivot),Xs)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),filter2(B,aa(C,fun(B,$o),aTP_Lamp_pc(fun(B,C),fun(C,fun(B,$o)),F),Pivot),Xs)),filter2(B,aa(C,fun(B,$o),aTP_Lamp_pm(fun(B,C),fun(C,fun(B,$o)),F),Pivot),Xs))) ) ).

% part_def
tff(fact_4650_power__int__def,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & power(B) )
     => ! [X: B,N: int] :
          power_int(B,X,N) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N),aa(nat,B,aa(B,fun(nat,B),power_power(B),X),aa(int,nat,nat2,N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,inverse_inverse(B),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N)))) ) ).

% power_int_def
tff(fact_4651_transpose__aux__max,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Xss: list(list(C))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs))),aa(nat,nat,foldr(list(C),nat,aTP_Lamp_pn(list(C),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(nat,nat,foldr(list(C),nat,aTP_Lamp_po(list(C),fun(nat,nat)),filter2(list(C),aTP_Lamp_pp(list(C),$o),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_4652_sort__by__quicksort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),filter2(B,aTP_Lamp_pq(list(B),fun(B,$o),Xs),Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),filter2(B,aTP_Lamp_pr(list(B),fun(B,$o),Xs),Xs)),aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),filter2(B,aa(B,fun(B,$o),ord_less(B),aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_4653_power__int__numeral__neg__numeral,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Ma: num,N: num] : power_int(B,aa(num,B,numeral_numeral(B),Ma),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(B,B,inverse_inverse(B),aa(num,B,numeral_numeral(B),pow(Ma,N))) ) ).

% power_int_numeral_neg_numeral
tff(fact_4654_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_4655_remove__rev__alt__def,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : aa(list(B),list(B),remove_rev(B,X),Xs) = filter2(B,aTP_Lamp_ph(B,fun(B,$o),X),rev(B,Xs)) ).

% remove_rev_alt_def
tff(fact_4656_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_4657_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_4658_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_4659_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_4660_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_4661_transpose__max__length,axiom,
    ! [B: $tType,Xs: list(list(B))] : aa(nat,nat,foldr(list(B),nat,aTP_Lamp_ps(list(B),fun(nat,nat)),transpose(B,Xs)),zero_zero(nat)) = aa(list(list(B)),nat,size_size(list(list(B))),filter2(list(B),aTP_Lamp_pt(list(B),$o),Xs)) ).

% transpose_max_length
tff(fact_4662_Bleast__code,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Pa: fun(B,$o)] : bleast(B,aa(list(B),set(B),set2(B),Xs),Pa) = aa(list(B),B,case_list(B,B,abort_Bleast(B,aa(list(B),set(B),set2(B),Xs),Pa),aTP_Lamp_pu(B,fun(list(B),B))),filter2(B,Pa,aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs))) ) ).

% Bleast_code
tff(fact_4663_quicksort_Oelims,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: list(B),Y: list(B)] :
          ( ( aa(list(B),list(B),linorder_quicksort(B),X) = Y )
         => ( ( ( X = nil(B) )
             => ( Y != nil(B) ) )
           => ~ ! [X2: B,Xs2: list(B)] :
                  ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
                 => ( Y != aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_quicksort(B),filter2(B,aTP_Lamp_pv(B,fun(B,$o),X2),Xs2))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B))),aa(list(B),list(B),linorder_quicksort(B),filter2(B,aa(B,fun(B,$o),ord_less_eq(B),X2),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_4664_sort__quicksort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( linorder_sort_key(B,B,aTP_Lamp_hr(B,B)) = linorder_quicksort(B) ) ) ).

% sort_quicksort
tff(fact_4665_sorted__quicksort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),linorder_quicksort(B),Xs)) ) ).

% sorted_quicksort
tff(fact_4666_length__transpose,axiom,
    ! [B: $tType,Xs: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),transpose(B,Xs)) = aa(nat,nat,foldr(list(B),nat,aTP_Lamp_ps(list(B),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_4667_quicksort_Osimps_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Xs: list(B)] : aa(list(B),list(B),linorder_quicksort(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_quicksort(B),filter2(B,aTP_Lamp_pv(B,fun(B,$o),X),Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))),aa(list(B),list(B),linorder_quicksort(B),filter2(B,aa(B,fun(B,$o),ord_less_eq(B),X),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_4668_nth__nth__transpose__sorted,axiom,
    ! [B: $tType,Xs: list(list(B)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(B)),nat,size_size(list(list(B))),transpose(B,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(B)),nat,size_size(list(list(B))),filter2(list(B),aTP_Lamp_pw(nat,fun(list(B),$o),I),Xs)))
         => ( aa(nat,B,nth(B,aa(nat,list(B),nth(list(B),transpose(B,Xs)),I)),J) = aa(nat,B,nth(B,aa(nat,list(B),nth(list(B),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4669_distinct__concat_H,axiom,
    ! [B: $tType,Xs: list(list(B))] :
      ( distinct(list(B),filter2(list(B),aTP_Lamp_pt(list(B),$o),Xs))
     => ( ! [Ys3: list(B)] :
            ( aa(set(list(B)),$o,member(list(B),Ys3),aa(list(list(B)),set(list(B)),set2(list(B)),Xs))
           => distinct(B,Ys3) )
       => ( ! [Ys3: list(B),Zs2: list(B)] :
              ( aa(set(list(B)),$o,member(list(B),Ys3),aa(list(list(B)),set(list(B)),set2(list(B)),Xs))
             => ( aa(set(list(B)),$o,member(list(B),Zs2),aa(list(list(B)),set(list(B)),set2(list(B)),Xs))
               => ( ( Ys3 != Zs2 )
                 => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(list(B),set(B),set2(B),Ys3)),aa(list(B),set(B),set2(B),Zs2)) = bot_bot(set(B)) ) ) ) )
         => distinct(B,concat(B,Xs)) ) ) ) ).

% distinct_concat'
tff(fact_4670_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa2)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o)))),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_4671_map__ident,axiom,
    ! [B: $tType,X4: list(B)] : aa(list(B),list(B),map(B,B,aTP_Lamp_il(B,B)),X4) = X4 ).

% map_ident
tff(fact_4672_length__map,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Xs: list(C)] : aa(list(B),nat,size_size(list(B)),aa(list(C),list(B),map(C,B,F),Xs)) = aa(list(C),nat,size_size(list(C)),Xs) ).

% length_map
tff(fact_4673_length__concat__rev,axiom,
    ! [B: $tType,Xs: list(list(B))] : aa(list(B),nat,size_size(list(B)),concat(B,rev(list(B),Xs))) = aa(list(B),nat,size_size(list(B)),concat(B,Xs)) ).

% length_concat_rev
tff(fact_4674_nth__map,axiom,
    ! [C: $tType,B: $tType,N: nat,Xs: list(B),F: fun(B,C)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,C,nth(C,aa(list(B),list(C),map(B,C,F),Xs)),N) = aa(B,C,F,aa(nat,B,nth(B,Xs),N)) ) ) ).

% nth_map
tff(fact_4675_length__filter__map,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),F: fun(C,B),Xs: list(C)] : aa(list(B),nat,size_size(list(B)),filter2(B,Pa,aa(list(C),list(B),map(C,B,F),Xs))) = aa(list(C),nat,size_size(list(C)),filter2(C,aa(fun(C,B),fun(C,$o),aa(fun(B,$o),fun(fun(C,B),fun(C,$o)),comp(B,$o,C),Pa),F),Xs)) ).

% length_filter_map
tff(fact_4676_concat__map__singleton,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Xs: list(C)] : concat(B,aa(list(C),list(list(B)),map(C,list(B),aTP_Lamp_px(fun(C,B),fun(C,list(B)),F)),Xs)) = aa(list(C),list(B),map(C,B,F),Xs) ).

% concat_map_singleton
tff(fact_4677_map__eq__nth__eq,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),L: list(C),L3: list(C),I: nat] :
      ( ( aa(list(C),list(B),map(C,B,F),L) = aa(list(C),list(B),map(C,B,F),L3) )
     => ( aa(C,B,F,aa(nat,C,nth(C,L),I)) = aa(C,B,F,aa(nat,C,nth(C,L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_4678_sorted__wrt__map,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(B,$o)),F: fun(C,B),Xs: list(C)] :
      ( sorted_wrt(B,R2,aa(list(C),list(B),map(C,B,F),Xs))
    <=> sorted_wrt(C,aa(fun(C,B),fun(C,fun(C,$o)),aTP_Lamp_py(fun(B,fun(B,$o)),fun(fun(C,B),fun(C,fun(C,$o))),R2),F),Xs) ) ).

% sorted_wrt_map
tff(fact_4679_map__consI_I1_J,axiom,
    ! [B: $tType,C: $tType,W: list(B),F: fun(C,B),Ww: list(C),A2: C] :
      ( ( W = aa(list(C),list(B),map(C,B,F),Ww) )
     => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(C,B,F,A2)),W) = aa(list(C),list(B),map(C,B,F),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A2),Ww)) ) ) ).

% map_consI(1)
tff(fact_4680_map__eq__consE,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),Ls: list(C),Fa: B,Fl: list(B)] :
      ( ( aa(list(C),list(B),map(C,B,F),Ls) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Fa),Fl) )
     => ~ ! [A3: C,L2: list(C)] :
            ( ( Ls = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),L2) )
           => ( ( aa(C,B,F,A3) = Fa )
             => ( aa(list(C),list(B),map(C,B,F),L2) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_4681_list_Omap__ident,axiom,
    ! [B: $tType,Ta: list(B)] : aa(list(B),list(B),map(B,B,aTP_Lamp_il(B,B)),Ta) = Ta ).

% list.map_ident
tff(fact_4682_distinct__mapI,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),L: list(C)] :
      ( distinct(B,aa(list(C),list(B),map(C,B,F),L))
     => distinct(C,L) ) ).

% distinct_mapI
tff(fact_4683_foldl__map,axiom,
    ! [B: $tType,C: $tType,D: $tType,G: fun(B,fun(C,B)),A2: B,F: fun(D,C),Xs: list(D)] : aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,G),A2),aa(list(D),list(C),map(D,C,F),Xs)) = aa(list(D),B,aa(B,fun(list(D),B),foldl(B,D,aa(fun(D,C),fun(B,fun(D,B)),aTP_Lamp_pz(fun(B,fun(C,B)),fun(fun(D,C),fun(B,fun(D,B))),G),F)),A2),Xs) ).

% foldl_map
tff(fact_4684_append__eq__mapE,axiom,
    ! [C: $tType,B: $tType,Fl: list(B),Fl2: list(B),F: fun(C,B),Ls: list(C)] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Fl),Fl2) = aa(list(C),list(B),map(C,B,F),Ls) )
     => ~ ! [L2: list(C),L6: list(C)] :
            ( ( Ls = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L2),L6) )
           => ( ( aa(list(C),list(B),map(C,B,F),L2) = Fl )
             => ( aa(list(C),list(B),map(C,B,F),L6) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_4685_map__eq__appendE,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),Ls: list(C),Fl: list(B),Fl2: list(B)] :
      ( ( aa(list(C),list(B),map(C,B,F),Ls) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Fl),Fl2) )
     => ~ ! [L2: list(C),L6: list(C)] :
            ( ( Ls = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L2),L6) )
           => ( ( aa(list(C),list(B),map(C,B,F),L2) = Fl )
             => ( aa(list(C),list(B),map(C,B,F),L6) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_4686_Misc_Oappend__eq__map__conv,axiom,
    ! [C: $tType,B: $tType,Fl: list(B),Fl2: list(B),F: fun(C,B),Ls: list(C)] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Fl),Fl2) = aa(list(C),list(B),map(C,B,F),Ls) )
    <=> ? [L5: list(C),L7: list(C)] :
          ( ( Ls = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L5),L7) )
          & ( aa(list(C),list(B),map(C,B,F),L5) = Fl )
          & ( aa(list(C),list(B),map(C,B,F),L7) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_4687_Misc_Omap__eq__append__conv,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),Ls: list(C),Fl: list(B),Fl2: list(B)] :
      ( ( aa(list(C),list(B),map(C,B,F),Ls) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Fl),Fl2) )
    <=> ? [L5: list(C),L7: list(C)] :
          ( ( Ls = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L5),L7) )
          & ( aa(list(C),list(B),map(C,B,F),L5) = Fl )
          & ( aa(list(C),list(B),map(C,B,F),L7) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_4688_map__eq__imp__length__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(C,B),Xs: list(C),G: fun(D,B),Ys: list(D)] :
      ( ( aa(list(C),list(B),map(C,B,F),Xs) = aa(list(D),list(B),map(D,B,G),Ys) )
     => ( aa(list(C),nat,size_size(list(C)),Xs) = aa(list(D),nat,size_size(list(D)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4689_transpose__aux__filter__head,axiom,
    ! [B: $tType,Xss: list(list(B))] : concat(B,aa(list(list(B)),list(list(B)),map(list(B),list(B),case_list(list(B),B,nil(B),aTP_Lamp_qa(B,fun(list(B),list(B))))),Xss)) = aa(list(list(B)),list(B),map(list(B),B,hd(B)),filter2(list(B),aTP_Lamp_pt(list(B),$o),Xss)) ).

% transpose_aux_filter_head
tff(fact_4690_set__oo__map__alt,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),X4: list(B)] : aa(list(B),set(C),aa(fun(list(B),list(C)),fun(list(B),set(C)),aa(fun(list(C),set(C)),fun(fun(list(B),list(C)),fun(list(B),set(C))),comp(list(C),set(C),list(B)),set2(C)),map(B,C,F)),X4) = aa(set(B),set(C),image2(B,C,F),aa(list(B),set(B),set2(B),X4)) ).

% set_oo_map_alt
tff(fact_4691_map__consI_I2_J,axiom,
    ! [C: $tType,B: $tType,W: list(B),L: list(B),F: fun(C,B),Ww: list(C),A2: C] :
      ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),W),L) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(C),list(B),map(C,B,F),Ww)),L) )
     => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(C,B,F,A2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),W),L)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(C),list(B),map(C,B,F),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A2),Ww))),L) ) ) ).

% map_consI(2)
tff(fact_4692_distinct__map__eq,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),L: list(C),X: C,Y: C] :
      ( distinct(B,aa(list(C),list(B),map(C,B,F),L))
     => ( ( aa(C,B,F,X) = aa(C,B,F,Y) )
       => ( aa(set(C),$o,member(C,X),aa(list(C),set(C),set2(C),L))
         => ( aa(set(C),$o,member(C,Y),aa(list(C),set(C),set2(C),L))
           => ( X = Y ) ) ) ) ) ).

% distinct_map_eq
tff(fact_4693_sorted__map,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),Xs))
        <=> sorted_wrt(C,aTP_Lamp_qb(fun(C,B),fun(C,fun(C,$o)),F),Xs) ) ) ).

% sorted_map
tff(fact_4694_map__replicate__const,axiom,
    ! [C: $tType,B: $tType,K: B,Lst: list(C)] : aa(list(C),list(B),map(C,B,aa(B,fun(C,B),aTP_Lamp_it(B,fun(C,B)),K)),Lst) = replicate(B,aa(list(C),nat,size_size(list(C)),Lst),K) ).

% map_replicate_const
tff(fact_4695_sorted__filter,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C),Pa: fun(C,$o)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),Xs))
         => sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),filter2(C,Pa,Xs))) ) ) ).

% sorted_filter
tff(fact_4696_sorted__insort__key,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),X: C,Xs: list(C)] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),aa(list(C),list(C),aa(C,fun(list(C),list(C)),linorder_insort_key(C,B,F),X),Xs)))
        <=> sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),Xs)) ) ) ).

% sorted_insort_key
tff(fact_4697_sorted__sort__key,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C)] : sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),aa(list(C),list(C),linorder_sort_key(C,B,F),Xs))) ) ).

% sorted_sort_key
tff(fact_4698_concat__filter__neq__Nil,axiom,
    ! [B: $tType,Xs: list(list(B))] : concat(B,filter2(list(B),aTP_Lamp_pt(list(B),$o),Xs)) = concat(B,Xs) ).

% concat_filter_neq_Nil
tff(fact_4699_sorted__map__same,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),G: fun(list(C),B),Xs: list(C)] : sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),filter2(C,aa(list(C),fun(C,$o),aa(fun(list(C),B),fun(list(C),fun(C,$o)),aTP_Lamp_qc(fun(C,B),fun(fun(list(C),B),fun(list(C),fun(C,$o))),F),G),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_4700_foldl__foldl__conv__concat,axiom,
    ! [B: $tType,C: $tType,F: fun(B,fun(C,B)),A2: B,Xs: list(list(C))] : aa(list(list(C)),B,aa(B,fun(list(list(C)),B),foldl(B,list(C),foldl(B,C,F)),A2),Xs) = aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),A2),concat(C,Xs)) ).

% foldl_foldl_conv_concat
tff(fact_4701_distinct__idx,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),L: list(C),I: nat,J: nat] :
      ( distinct(B,aa(list(C),list(B),map(C,B,F),L))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(C),nat,size_size(list(C)),L))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(C),nat,size_size(list(C)),L))
         => ( ( aa(C,B,F,aa(nat,C,nth(C,L),I)) = aa(C,B,F,aa(nat,C,nth(C,L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_4702_map__upd__eq,axiom,
    ! [C: $tType,B: $tType,I: nat,L: list(B),F: fun(B,C),X: B] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
       => ( aa(B,C,F,aa(nat,B,nth(B,L),I)) = aa(B,C,F,X) ) )
     => ( aa(list(B),list(C),map(B,C,F),list_update(B,L,I,X)) = aa(list(B),list(C),map(B,C,F),L) ) ) ).

% map_upd_eq
tff(fact_4703_filter__insort,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C),Pa: fun(C,$o),X: C] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),Xs))
         => ( aa(C,$o,Pa,X)
           => ( filter2(C,Pa,aa(list(C),list(C),aa(C,fun(list(C),list(C)),linorder_insort_key(C,B,F),X),Xs)) = aa(list(C),list(C),aa(C,fun(list(C),list(C)),linorder_insort_key(C,B,F),X),filter2(C,Pa,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_4704_sorted__transpose,axiom,
    ! [B: $tType,Xs: list(list(B))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),transpose(B,Xs)))) ).

% sorted_transpose
tff(fact_4705_comm__append__is__replicate,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( Xs != nil(B) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Xs) )
         => ? [N2: nat,Zs2: list(B)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N2)
              & ( concat(B,replicate(list(B),N2,Zs2)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_4706_transpose__column,axiom,
    ! [B: $tType,Xs: list(list(B)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(B)),nat,size_size(list(list(B))),Xs))
       => ( aa(list(list(B)),list(B),map(list(B),B,aTP_Lamp_qd(nat,fun(list(B),B),I)),filter2(list(B),aTP_Lamp_pw(nat,fun(list(B),$o),I),transpose(B,Xs))) = aa(nat,list(B),nth(list(B),Xs),I) ) ) ) ).

% transpose_column
tff(fact_4707_map__by__foldl,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),L: list(C)] : aa(list(C),list(B),aa(list(B),fun(list(C),list(B)),foldl(list(B),C,aTP_Lamp_qe(fun(C,B),fun(list(B),fun(C,list(B))),F)),nil(B)),L) = aa(list(C),list(B),map(C,B,F),L) ).

% map_by_foldl
tff(fact_4708_nat_Orep__eq,axiom,
    ! [X: int] : aa(int,nat,nat2,X) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,X)) ).

% nat.rep_eq
tff(fact_4709_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_4710_properties__for__sort__key,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Ys: list(B),Xs: list(B),F: fun(B,C)] :
          ( ( mset(B,Ys) = mset(B,Xs) )
         => ( ! [K4: B] :
                ( aa(set(B),$o,member(B,K4),aa(list(B),set(B),set2(B),Ys))
               => ( filter2(B,aa(B,fun(B,$o),aTP_Lamp_qf(fun(B,C),fun(B,fun(B,$o)),F),K4),Ys) = filter2(B,aa(B,fun(B,$o),aTP_Lamp_qf(fun(B,C),fun(B,fun(B,$o)),F),K4),Xs) ) )
           => ( sorted_wrt(C,ord_less_eq(C),aa(list(B),list(C),map(B,C,F),Ys))
             => ( aa(list(B),list(B),linorder_sort_key(B,C,F),Xs) = Ys ) ) ) ) ) ).

% properties_for_sort_key
tff(fact_4711_of__int_Orep__eq,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [X: int] : aa(int,B,ring_1_of_int(B),X) = aa(product_prod(nat,nat),B,aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),aTP_Lamp_nr(nat,fun(nat,B))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_4712_nth__transpose,axiom,
    ! [B: $tType,I: nat,Xs: list(list(B))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(B)),nat,size_size(list(list(B))),transpose(B,Xs)))
     => ( aa(nat,list(B),nth(list(B),transpose(B,Xs)),I) = aa(list(list(B)),list(B),map(list(B),B,aTP_Lamp_qd(nat,fun(list(B),B),I)),filter2(list(B),aTP_Lamp_pw(nat,fun(list(B),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_4713_length__transpose__sorted,axiom,
    ! [B: $tType,Xs: list(list(B))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xs)))
     => ( aa(list(list(B)),nat,size_size(list(list(B))),transpose(B,Xs)) = $ite(Xs = nil(list(B)),zero_zero(nat),aa(list(B),nat,size_size(list(B)),aa(nat,list(B),nth(list(B),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_4714_transpose__column__length,axiom,
    ! [B: $tType,Xs: list(list(B)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(B)),nat,size_size(list(list(B))),Xs))
       => ( aa(list(list(B)),nat,size_size(list(list(B))),filter2(list(B),aTP_Lamp_pw(nat,fun(list(B),$o),I),transpose(B,Xs))) = aa(list(B),nat,size_size(list(B)),aa(nat,list(B),nth(list(B),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_4715_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Xa2)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2)) ) ).

% less_int.rep_eq
tff(fact_4716_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [B: $tType,C: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),S: set(C),F: fun(C,B),A4: set(C)] :
      ( folding_insort_key(B,C,Less_eq,Less,S,F)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),A4),S)
       => ( aa(set(C),$o,finite_finite2(C),A4)
         => ~ ! [L2: list(C)] :
                ( sorted_wrt(B,Less,aa(list(C),list(B),map(C,B,F),L2))
               => ( ( aa(list(C),set(C),set2(C),L2) = A4 )
                 => ( aa(list(C),nat,size_size(list(C)),L2) != aa(set(C),nat,finite_card(C),A4) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_4717_map__filter__map__filter,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Pa: fun(C,$o),Xs: list(C)] : aa(list(C),list(B),map(C,B,F),filter2(C,Pa,Xs)) = map_filter(C,B,aa(fun(C,$o),fun(C,option(B)),aTP_Lamp_qg(fun(C,B),fun(fun(C,$o),fun(C,option(B))),F),Pa),Xs) ).

% map_filter_map_filter
tff(fact_4718_insort__key__remove1,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [A2: B,Xs: list(B),F: fun(B,C)] :
          ( aa(set(B),$o,member(B,A2),aa(list(B),set(B),set2(B),Xs))
         => ( sorted_wrt(C,ord_less_eq(C),aa(list(B),list(C),map(B,C,F),Xs))
           => ( ( aa(list(B),B,hd(B),filter2(B,aa(fun(B,C),fun(B,$o),aTP_Lamp_qh(B,fun(fun(B,C),fun(B,$o)),A2),F),Xs)) = A2 )
             => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,C,F),A2),remove1(B,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_4719_product__concat__map,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C)] : product(B,C,Xs,Ys) = concat(product_prod(B,C),aa(list(B),list(list(product_prod(B,C))),map(B,list(product_prod(B,C)),aTP_Lamp_qi(list(C),fun(B,list(product_prod(B,C))),Ys)),Xs)) ).

% product_concat_map
tff(fact_4720_n__lists_Osimps_I2_J,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : n_lists(B,aa(nat,nat,suc,N),Xs) = concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),aTP_Lamp_qk(list(B),fun(list(B),list(list(B))),Xs)),n_lists(B,N,Xs))) ).

% n_lists.simps(2)
tff(fact_4721_product__lists_Osimps_I2_J,axiom,
    ! [B: $tType,Xs: list(B),Xss: list(list(B))] : product_lists(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Xs),Xss)) = concat(list(B),aa(list(B),list(list(list(B))),map(B,list(list(B)),aTP_Lamp_ql(list(list(B)),fun(B,list(list(B))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_4722_sorted__remove1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),A2: B] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),remove1(B,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_4723_subseqs_Osimps_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      subseqs(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = $let(
        xss: list(list(B)),
        xss:= subseqs(B,Xs),
        aa(list(list(B)),list(list(B)),aa(list(list(B)),fun(list(list(B)),list(list(B))),append(list(B)),aa(list(list(B)),list(list(B)),map(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X)),xss)),xss) ) ).

% subseqs.simps(2)
tff(fact_4724_map__filter__simps_I1_J,axiom,
    ! [B: $tType,C: $tType,F: fun(C,option(B)),X: C,Xs: list(C)] : map_filter(C,B,F,aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X),Xs)) = case_option(list(B),B,map_filter(C,B,F,Xs),aa(list(C),fun(B,list(B)),aTP_Lamp_qm(fun(C,option(B)),fun(list(C),fun(B,list(B))),F),Xs),aa(C,option(B),F,X)) ).

% map_filter_simps(1)
tff(fact_4725_transpose_Oelims,axiom,
    ! [B: $tType,X: list(list(B)),Y: list(list(B))] :
      ( ( transpose(B,X) = Y )
     => ( ( ( X = nil(list(B)) )
         => ( Y != nil(list(B)) ) )
       => ( ! [Xss2: list(list(B))] :
              ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),Xss2) )
             => ( Y != transpose(B,Xss2) ) )
         => ~ ! [X2: B,Xs2: list(B),Xss2: list(list(B))] :
                ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),Xss2) )
               => ( Y != aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),concat(B,aa(list(list(B)),list(list(B)),map(list(B),list(B),case_list(list(B),B,nil(B),aTP_Lamp_qa(B,fun(list(B),list(B))))),Xss2)))),transpose(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Xs2),concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),case_list(list(list(B)),B,nil(list(B)),aTP_Lamp_qn(B,fun(list(B),list(list(B)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_4726_transpose_Osimps_I3_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),Xss: list(list(B))] : transpose(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),Xss)) = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),concat(B,aa(list(list(B)),list(list(B)),map(list(B),list(B),case_list(list(B),B,nil(B),aTP_Lamp_qa(B,fun(list(B),list(B))))),Xss)))),transpose(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Xs),concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),case_list(list(list(B)),B,nil(list(B)),aTP_Lamp_qn(B,fun(list(B),list(list(B)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_4727_sorted__map__remove1,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C),X: C] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),Xs))
         => sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),remove1(C,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_4728_length__remove1,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      aa(list(B),nat,size_size(list(B)),remove1(B,X,Xs)) = $ite(aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)),aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_remove1
tff(fact_4729_insort__remove1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,Xs: list(B)] :
          ( aa(set(B),$o,member(B,A2),aa(list(B),set(B),set2(B),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),Xs)
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),A2),remove1(B,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_4730_execute__make,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,F: fun(nat,B),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(B),array_make(B,N,F)),H) = aa(product_prod(array(B),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(B),heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mu(nat,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(B,aa(list(nat),list(B),map(nat,B,F),upt(zero_zero(nat),N)),H))) ) ).

% execute_make
tff(fact_4731_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [B: $tType,C: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),S: set(C),F: fun(C,B),A4: set(C),L: list(C)] :
      ( folding_insort_key(B,C,Less_eq,Less,S,F)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),A4),S)
       => ( aa(set(C),$o,finite_finite2(C),A4)
         => ( ( sorted_wrt(B,Less,aa(list(C),list(B),map(C,B,F),L))
              & ( aa(list(C),set(C),set2(C),L) = A4 )
              & ( aa(list(C),nat,size_size(list(C)),L) = aa(set(C),nat,finite_card(C),A4) ) )
          <=> ( sorted8670434370408473282of_set(B,C,Less_eq,F,A4) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_4732_Array__Time_Omake__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,F: fun(nat,B)] : array_make(B,N,F) = heap_Time_heap(array(B),aa(fun(nat,B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_qo(nat,fun(fun(nat,B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),N),F)) ) ).

% Array_Time.make_def
tff(fact_4733_sort__upt,axiom,
    ! [Ma: nat,N: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_da(nat,nat)),upt(Ma,N)) = upt(Ma,N) ).

% sort_upt
tff(fact_4734_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero(nat),J) = nil(nat) )
    <=> ( J = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_4735_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K) )
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_4736_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_4737_drop__upt,axiom,
    ! [Ma: nat,I: nat,J: nat] : drop(nat,Ma,upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma),J) ).

% drop_upt
tff(fact_4738_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_4739_take__upt,axiom,
    ! [I: nat,Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma)),N)
     => ( take(nat,Ma,upt(I,N)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Ma)) ) ) ).

% take_upt
tff(fact_4740_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) ).

% length_upt
tff(fact_4741_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_4742_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_4743_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_qp(nat,fun(nat,nat),Ofs)),upt(A2,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_4744_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( upt(I,J) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys) )
      <=> ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
            & ( upt(I,K2) = Xs )
            & ( upt(K2,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_4745_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_4746_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I)),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_4747_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_4748_butlast__upt,axiom,
    ! [Ma: nat,N: nat] : aa(list(nat),list(nat),butlast(nat),upt(Ma,N)) = upt(Ma,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% butlast_upt
tff(fact_4749_atLeast__upt,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_4750_map__add__upt,axiom,
    ! [N: nat,Ma: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_qp(nat,fun(nat,nat),N)),upt(zero_zero(nat),Ma)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)) ).

% map_add_upt
tff(fact_4751_sorted__upt,axiom,
    ! [Ma: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(Ma,N)) ).

% sorted_upt
tff(fact_4752_map__replicate__trivial,axiom,
    ! [B: $tType,X: B,I: nat] : aa(list(nat),list(B),map(nat,B,aTP_Lamp_qq(B,fun(nat,B),X)),upt(zero_zero(nat),I)) = replicate(B,I,X) ).

% map_replicate_trivial
tff(fact_4753_filter__upt__take__conv,axiom,
    ! [B: $tType,Pa: fun(B,$o),Ma: nat,L: list(B),N: nat] : filter2(nat,aa(list(B),fun(nat,$o),aa(nat,fun(list(B),fun(nat,$o)),aTP_Lamp_qr(fun(B,$o),fun(nat,fun(list(B),fun(nat,$o))),Pa),Ma),L),upt(N,Ma)) = filter2(nat,aa(list(B),fun(nat,$o),aTP_Lamp_qs(fun(B,$o),fun(list(B),fun(nat,$o)),Pa),L),upt(N,Ma)) ).

% filter_upt_take_conv
tff(fact_4754_enumerate__map__upt,axiom,
    ! [B: $tType,N: nat,F: fun(nat,B),Ma: nat] : enumerate(B,N,aa(list(nat),list(B),map(nat,B,F),upt(N,Ma))) = aa(list(nat),list(product_prod(nat,B)),map(nat,product_prod(nat,B),aTP_Lamp_qt(fun(nat,B),fun(nat,product_prod(nat,B)),F)),upt(N,Ma)) ).

% enumerate_map_upt
tff(fact_4755_atMost__upto,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_4756_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_4757_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_4758_map__upt__Suc,axiom,
    ! [B: $tType,F: fun(nat,B),N: nat] : aa(list(nat),list(B),map(nat,B,F),upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(nat,B,F,zero_zero(nat))),aa(list(nat),list(B),map(nat,B,aTP_Lamp_qu(fun(nat,B),fun(nat,B),F)),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_4759_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,Pa: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U2)
     => ( ! [I2: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),U2) )
           => ~ aa(nat,$o,Pa,I2) )
       => ( filter2(nat,Pa,upt(zero_zero(nat),U)) = filter2(nat,Pa,upt(zero_zero(nat),U2)) ) ) ) ).

% upt_filter_extend
tff(fact_4760_map__decr__upt,axiom,
    ! [Ma: nat,N: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_kc(nat,nat)),upt(aa(nat,nat,suc,Ma),aa(nat,nat,suc,N))) = upt(Ma,N) ).

% map_decr_upt
tff(fact_4761_map__nth,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(nat),list(B),map(nat,B,nth(B,Xs)),upt(zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) = Xs ).

% map_nth
tff(fact_4762_nth__map__upt,axiom,
    ! [B: $tType,I: nat,N: nat,Ma: nat,F: fun(nat,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))
     => ( aa(nat,B,nth(B,aa(list(nat),list(B),map(nat,B,F),upt(Ma,N))),I) = aa(nat,B,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),I)) ) ) ).

% nth_map_upt
tff(fact_4763_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
      ( ( upt(L,H) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),I)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_4764_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_4765_enumerate__replicate__eq,axiom,
    ! [B: $tType,N: nat,Ma: nat,A2: B] : enumerate(B,N,replicate(B,Ma,A2)) = aa(list(nat),list(product_prod(nat,B)),map(nat,product_prod(nat,B),aTP_Lamp_qv(B,fun(nat,product_prod(nat,B)),A2)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Ma))) ).

% enumerate_replicate_eq
tff(fact_4766_map__upt__eqI,axiom,
    ! [B: $tType,Xs: list(B),N: nat,Ma: nat,F: fun(nat,B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Xs))
           => ( aa(nat,B,nth(B,Xs),I2) = aa(nat,B,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),I2)) ) )
       => ( aa(list(nat),list(B),map(nat,B,F),upt(Ma,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_4767_map__nth__upt__drop__take__conv,axiom,
    ! [B: $tType,N5: nat,L: list(B),M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),aa(list(B),nat,size_size(list(B)),L))
     => ( aa(list(nat),list(B),map(nat,B,nth(B,L)),upt(M,N5)) = drop(B,M,take(B,N5,L)) ) ) ).

% map_nth_upt_drop_take_conv
tff(fact_4768_filter__upt__last,axiom,
    ! [B: $tType,Pa: fun(B,$o),L: list(B),Js: list(nat),J: nat,I: nat] :
      ( ( filter2(nat,aa(list(B),fun(nat,$o),aTP_Lamp_qs(fun(B,$o),fun(list(B),fun(nat,$o)),Pa),L),upt(zero_zero(nat),aa(list(B),nat,size_size(list(B)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),I)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
         => ~ aa(B,$o,Pa,aa(nat,B,nth(B,L),I)) ) ) ) ).

% filter_upt_last
tff(fact_4769_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [B: $tType,C: $tType,Less_eq: fun(B,fun(B,$o)),Less: fun(B,fun(B,$o)),S: set(C),F: fun(C,B),A4: set(C)] :
      ( folding_insort_key(B,C,Less_eq,Less,S,F)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),A4),S)
       => ( aa(list(C),nat,size_size(list(C)),sorted8670434370408473282of_set(B,C,Less_eq,F,A4)) = aa(set(C),nat,finite_card(C),A4) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_4770_horner__sum__bit__eq__take__bit,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [A2: B,N: nat] : aa(list($o),B,aa(B,fun(list($o),B),aa(fun($o,B),fun(B,fun(list($o),B)),groups4207007520872428315er_sum($o,B),zero_neq_one_of_bool(B)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(list(nat),list($o),map(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2)),upt(zero_zero(nat),N))) = aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_4771_transpose__rectangle,axiom,
    ! [B: $tType,Xs: list(list(B)),N: nat] :
      ( ( ( Xs = nil(list(B)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(B)),nat,size_size(list(list(B))),Xs))
           => ( aa(list(B),nat,size_size(list(B)),aa(nat,list(B),nth(list(B),Xs),I2)) = N ) )
       => ( transpose(B,Xs) = aa(list(nat),list(list(B)),map(nat,list(B),aTP_Lamp_qx(list(list(B)),fun(nat,list(B)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_4772_product__code,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C)] : product_product(B,C,aa(list(B),set(B),set2(B),Xs),aa(list(C),set(C),set2(C),Ys)) = aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),concat(product_prod(B,C),aa(list(B),list(list(product_prod(B,C))),map(B,list(product_prod(B,C)),aTP_Lamp_qi(list(C),fun(B,list(product_prod(B,C))),Ys)),Xs))) ).

% product_code
tff(fact_4773_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F: fun(nat,B),Ns: list(nat)] :
          ( ! [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,X2)),aa(nat,B,F,Y2)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),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,F),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4774_transpose__aux__filter__tail,axiom,
    ! [B: $tType,Xss: list(list(B))] : concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),case_list(list(list(B)),B,nil(list(B)),aTP_Lamp_qn(B,fun(list(B),list(list(B)))))),Xss)) = aa(list(list(B)),list(list(B)),map(list(B),list(B),tl(B)),filter2(list(B),aTP_Lamp_pt(list(B),$o),Xss)) ).

% transpose_aux_filter_tail
tff(fact_4775_sum__list_ONil,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ( aa(list(B),B,groups8242544230860333062m_list(B),nil(B)) = zero_zero(B) ) ) ).

% sum_list.Nil
tff(fact_4776_sum__list_OCons,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ! [X: B,Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(list(B),B,groups8242544230860333062m_list(B),Xs)) ) ).

% sum_list.Cons
tff(fact_4777_sum__list__eq__0__iff,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [Ns: list(B)] :
          ( ( aa(list(B),B,groups8242544230860333062m_list(B),Ns) = zero_zero(B) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Ns))
             => ( X3 = zero_zero(B) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_4778_sum__list__append,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(B),Ys: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(list(B),B,groups8242544230860333062m_list(B),Xs)),aa(list(B),B,groups8242544230860333062m_list(B),Ys)) ) ).

% sum_list_append
tff(fact_4779_sum__list__0,axiom,
    ! [C: $tType,B: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aTP_Lamp_qy(C,B)),Xs)) = zero_zero(B) ) ).

% sum_list_0
tff(fact_4780_length__tl,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),tl(B),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_4781_in__hd__or__tl__conv,axiom,
    ! [B: $tType,L: list(B),X: B] :
      ( ( L != nil(B) )
     => ( ( ( X = aa(list(B),B,hd(B),L) )
          | aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(B),list(B),tl(B),L))) )
      <=> aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),L)) ) ) ).

% in_hd_or_tl_conv
tff(fact_4782_tl__replicate,axiom,
    ! [B: $tType,N: nat,X: B] : aa(list(B),list(B),tl(B),replicate(B,N,X)) = replicate(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ).

% tl_replicate
tff(fact_4783_sum__list__upt,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(Ma,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_da(nat,nat)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ) ).

% sum_list_upt
tff(fact_4784_rev__split__conv,axiom,
    ! [B: $tType,L: list(B)] :
      ( ( L != nil(B) )
     => ( aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),rev(B,aa(list(B),list(B),tl(B),L))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),L)),nil(B))) = rev(B,L) ) ) ).

% rev_split_conv
tff(fact_4785_in__set__tlD,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(B),list(B),tl(B),Xs)))
     => aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs)) ) ).

% in_set_tlD
tff(fact_4786_tl__obtain__elem,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( ( aa(list(B),list(B),tl(B),Xs) = nil(B) )
       => ~ ! [E4: B] : Xs != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E4),nil(B)) ) ) ).

% tl_obtain_elem
tff(fact_4787_sorted__tl,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),tl(B),Xs)) ) ) ).

% sorted_tl
tff(fact_4788_member__le__sum__list,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [X: B,Xs: list(B)] :
          ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(list(B),B,groups8242544230860333062m_list(B),Xs)) ) ) ).

% member_le_sum_list
tff(fact_4789_not__hd__in__tl,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      ( ( X != aa(list(B),B,hd(B),Xs) )
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
       => aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(B),list(B),tl(B),Xs))) ) ) ).

% not_hd_in_tl
tff(fact_4790_tl__last,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( aa(list(B),list(B),tl(B),Xs) != nil(B) )
     => ( last(B,Xs) = last(B,aa(list(B),list(B),tl(B),Xs)) ) ) ).

% tl_last
tff(fact_4791_rev__butlast__is__tl__rev,axiom,
    ! [B: $tType,L: list(B)] : rev(B,aa(list(B),list(B),butlast(B),L)) = aa(list(B),list(B),tl(B),rev(B,L)) ).

% rev_butlast_is_tl_rev
tff(fact_4792_sum__list__addf,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [F: fun(C,B),G: fun(C,B),Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bo(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,F),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,G),Xs))) ) ).

% sum_list_addf
tff(fact_4793_sum__list__mult__const,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_0(B)
     => ! [F: fun(C,B),C2: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aa(B,fun(C,B),aTP_Lamp_bq(fun(C,B),fun(B,fun(C,B)),F),C2)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,F),Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_4794_sum__list__const__mult,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [C2: B,F: fun(C,B),Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bp(B,fun(fun(C,B),fun(C,B)),C2),F)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,F),Xs))) ) ).

% sum_list_const_mult
tff(fact_4795_sum__list__subtractf,axiom,
    ! [B: $tType,C: $tType] :
      ( ab_group_add(B)
     => ! [F: fun(C,B),G: fun(C,B),Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bt(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),Xs)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,F),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,G),Xs))) ) ).

% sum_list_subtractf
tff(fact_4796_tl__def,axiom,
    ! [B: $tType,List: list(B)] : aa(list(B),list(B),tl(B),List) = aa(list(B),list(B),case_list(list(B),B,nil(B),aTP_Lamp_qz(B,fun(list(B),list(B)))),List) ).

% tl_def
tff(fact_4797_tl__append,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] : aa(list(B),list(B),tl(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(B),list(B),case_list(list(B),B,aa(list(B),list(B),tl(B),Ys),aTP_Lamp_ra(list(B),fun(B,fun(list(B),list(B))),Ys)),Xs) ).

% tl_append
tff(fact_4798_tl__subset,axiom,
    ! [B: $tType,Xs: list(B),A4: set(B)] :
      ( ( Xs != nil(B) )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),A4)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),aa(list(B),list(B),tl(B),Xs))),A4) ) ) ).

% tl_subset
tff(fact_4799_Misc_Onth__tl,axiom,
    ! [B: $tType,Xs: list(B),N: nat] :
      ( ( Xs != nil(B) )
     => ( aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Xs)),N) = aa(nat,B,nth(B,Xs),aa(nat,nat,suc,N)) ) ) ).

% Misc.nth_tl
tff(fact_4800_list__take__induct__tl2,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C),Pa: fun(C,fun(B,$o))] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(B),nat,size_size(list(B)),Xs))
           => aa(B,$o,aa(C,fun(B,$o),Pa,aa(nat,C,nth(C,Ys),N2)),aa(nat,B,nth(B,Xs),N2)) )
       => ! [N8: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N8),aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),tl(B),Xs)))
           => aa(B,$o,aa(C,fun(B,$o),Pa,aa(nat,C,nth(C,aa(list(C),list(C),tl(C),Ys)),N8)),aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Xs)),N8)) ) ) ) ).

% list_take_induct_tl2
tff(fact_4801_Groups__List_Osum__list__nonneg,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Xs: list(B)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(list(B),B,groups8242544230860333062m_list(B),Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_4802_sum__list__nonneg__eq__0__iff,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Xs: list(B)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),X2) )
         => ( ( aa(list(B),B,groups8242544230860333062m_list(B),Xs) = zero_zero(B) )
          <=> ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Xs))
               => ( X3 = zero_zero(B) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_4803_sum__list__nonpos,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Xs: list(B)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),Xs)),zero_zero(B)) ) ) ).

% sum_list_nonpos
tff(fact_4804_tl__take,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),list(B),tl(B),take(B,N,Xs)) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(list(B),list(B),tl(B),Xs)) ).

% tl_take
tff(fact_4805_distinct__hd__tl,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( distinct(B,Xs)
     => ( ( X = aa(list(B),B,hd(B),Xs) )
       => ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(B),list(B),tl(B),Xs))) ) ) ).

% distinct_hd_tl
tff(fact_4806_sum__list__abs,axiom,
    ! [B: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Xs: list(B)] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(list(B),B,groups8242544230860333062m_list(B),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(B),list(B),map(B,B,abs_abs(B)),Xs))) ) ).

% sum_list_abs
tff(fact_4807_butlast__rev__tl,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( aa(list(B),list(B),butlast(B),rev(B,Xs)) = rev(B,aa(list(B),list(B),tl(B),Xs)) ) ) ).

% butlast_rev_tl
tff(fact_4808_sum__list_Oeq__foldr,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(B,B,foldr(B,B,plus_plus(B),Xs),zero_zero(B)) ) ).

% sum_list.eq_foldr
tff(fact_4809_sum__list__replicate,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat,C2: B] : aa(list(B),B,groups8242544230860333062m_list(B),replicate(B,N,C2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),C2) ) ).

% sum_list_replicate
tff(fact_4810_remove1__tl,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( remove1(B,aa(list(B),B,hd(B),Xs),Xs) = aa(list(B),list(B),tl(B),Xs) ) ) ).

% remove1_tl
tff(fact_4811_sum__list__filter__le__nat,axiom,
    ! [B: $tType,F: fun(B,nat),Pa: fun(B,$o),Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F),filter2(B,Pa,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F),Xs))) ).

% sum_list_filter_le_nat
tff(fact_4812_sum__list__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( ( monoid_add(C)
        & ordere6658533253407199908up_add(C) )
     => ! [Xs: list(B),F: fun(B,C),G: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),map(B,C,F),Xs))),aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),map(B,C,G),Xs))) ) ) ).

% sum_list_mono
tff(fact_4813_length__concat,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(B),nat,size_size(list(B)),concat(B,Xss)) = aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xss)) ).

% length_concat
tff(fact_4814_sum__list__map__filter_H,axiom,
    ! [B: $tType,C: $tType] :
      ( monoid_add(B)
     => ! [F: fun(C,B),Pa: fun(C,$o),Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,F),filter2(C,Pa,Xs))) = aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aa(fun(C,$o),fun(C,B),aTP_Lamp_rb(fun(C,B),fun(fun(C,$o),fun(C,B)),F),Pa)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4815_distinct__sum__list__conv__Sum,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] :
          ( distinct(B,Xs)
         => ( aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),aTP_Lamp_rc(B,B)),aa(list(B),set(B),set2(B),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_4816_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [B: $tType,Xs: list(B)] :
      aa(list(B),nat,size_size(list(B)),Xs) = $ite(Xs = nil(B),zero_zero(nat),aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),tl(B),Xs)))) ).

% Nitpick.size_list_simp(2)
tff(fact_4817_List_Onth__tl,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),tl(B),Xs)))
     => ( aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Xs)),N) = aa(nat,B,nth(B,Xs),aa(nat,nat,suc,N)) ) ) ).

% List.nth_tl
tff(fact_4818_elem__le__sum__list,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [K: nat,Ns: list(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(B),nat,size_size(list(B)),Ns))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,nth(B,Ns),K)),aa(list(B),B,groups8242544230860333062m_list(B),Ns)) ) ) ).

% elem_le_sum_list
tff(fact_4819_sum__list__map__filter,axiom,
    ! [C: $tType,B: $tType] :
      ( monoid_add(C)
     => ! [Xs: list(B),Pa: fun(B,$o),F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => ( ~ aa(B,$o,Pa,X2)
               => ( aa(B,C,F,X2) = zero_zero(C) ) ) )
         => ( aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),map(B,C,F),filter2(B,Pa,Xs))) = aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),map(B,C,F),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_4820_sum__list__map__remove1,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [X: B,Xs: list(B),F: fun(B,C)] :
          ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
         => ( aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),map(B,C,F),Xs)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,F,X)),aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),map(B,C,F),remove1(B,X,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_4821_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aTP_Lamp_go(B,fun(C,B),R)),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))),R) ) ).

% sum_list_triv
tff(fact_4822_sum__list__Suc,axiom,
    ! [B: $tType,F: fun(B,nat),Xs: list(B)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,aTP_Lamp_gu(fun(B,nat),fun(B,nat),F)),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,F),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% sum_list_Suc
tff(fact_4823_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),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_4824_card__length__sum__list__rec,axiom,
    ! [Ma: nat,N5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Ma)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_rd(nat,fun(nat,fun(list(nat),$o)),Ma),N5))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_re(nat,fun(nat,fun(list(nat),$o)),Ma),N5)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_rf(nat,fun(nat,fun(list(nat),$o)),Ma),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_4825_card__length__sum__list,axiom,
    ! [Ma: nat,N5: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_rd(nat,fun(nat,fun(list(nat),$o)),Ma),N5))) = 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),N5),Ma)),one_one(nat))),N5) ).

% card_length_sum_list
tff(fact_4826_sum__list__map__eq__sum__count,axiom,
    ! [B: $tType,F: fun(B,nat),Xs: list(B)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F),Xs)) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(list(B),fun(B,nat),aTP_Lamp_rg(fun(B,nat),fun(list(B),fun(B,nat)),F),Xs)),aa(list(B),set(B),set2(B),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_4827_sum__list__update,axiom,
    ! [B: $tType] :
      ( ordere1170586879665033532d_diff(B)
     => ! [K: nat,Xs: list(B),X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(B),nat,size_size(list(B)),Xs))
         => ( aa(list(B),B,groups8242544230860333062m_list(B),list_update(B,Xs,K,X)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(list(B),B,groups8242544230860333062m_list(B),Xs)),X)),aa(nat,B,nth(B,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_4828_sum__list__map__eq__sum__count2,axiom,
    ! [B: $tType,Xs: list(B),X5: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),X5)
     => ( aa(set(B),$o,finite_finite2(B),X5)
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F),Xs)) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(fun(B,nat),fun(B,nat),aTP_Lamp_rh(list(B),fun(fun(B,nat),fun(B,nat)),Xs),F)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4829_merge__list__correct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ls: list(list(B)),As: list(list(B))] :
          ( ! [L2: list(B)] :
              ( aa(set(list(B)),$o,member(list(B),L2),aa(list(list(B)),set(list(B)),set2(list(B)),Ls))
             => ( distinct(B,L2)
                & sorted_wrt(B,ord_less_eq(B),L2) ) )
         => ( ! [L2: list(B)] :
                ( aa(set(list(B)),$o,member(list(B),L2),aa(list(list(B)),set(list(B)),set2(list(B)),As))
               => ( distinct(B,L2)
                  & sorted_wrt(B,ord_less_eq(B),L2) ) )
           => ( distinct(B,merge_list(B,As,Ls))
              & sorted_wrt(B,ord_less_eq(B),merge_list(B,As,Ls))
              & ( aa(list(B),set(B),set2(B),merge_list(B,As,Ls)) = aa(list(B),set(B),set2(B),concat(B,aa(list(list(B)),list(list(B)),aa(list(list(B)),fun(list(list(B)),list(list(B))),append(list(B)),As),Ls))) ) ) ) ) ) ).

% merge_list_correct
tff(fact_4830_transpose__transpose,axiom,
    ! [B: $tType,Xs: list(list(B))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xs)))
     => ( transpose(B,transpose(B,Xs)) = takeWhile(list(B),aTP_Lamp_pt(list(B),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_4831_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [B: $tType,F: fun(B,nat),Xs: list(B)] :
      size_list(B,F,Xs) = $ite(Xs = nil(B),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,F,aa(list(B),B,hd(B),Xs))),size_list(B,F,aa(list(B),list(B),tl(B),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_4832_size__list__append,axiom,
    ! [B: $tType,F: fun(B,nat),Xs: list(B),Ys: list(B)] : size_list(B,F,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(B,F,Xs)),size_list(B,F,Ys)) ).

% size_list_append
tff(fact_4833_length__takeWhile__le,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_takeWhile_le
tff(fact_4834_sorted__takeWhile,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Pa: fun(B,$o)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),takeWhile(B,Pa,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_4835_takeWhile__eq__take,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : takeWhile(B,Pa,Xs) = take(B,aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_4836_merge__list_Osimps_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( merge_list(B,nil(list(B)),nil(list(B))) = nil(B) ) ) ).

% merge_list.simps(1)
tff(fact_4837_merge__list_Osimps_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B)] : merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),nil(list(B)))) = L ) ).

% merge_list.simps(2)
tff(fact_4838_merge__list_Osimps_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [La2: list(B),Acc22: list(list(B))] : merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22),nil(list(B))) = merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22)) ) ).

% merge_list.simps(3)
tff(fact_4839_merge__list_Osimps_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [La2: list(B),Acc22: list(list(B)),L: list(B)] : merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),nil(list(B)))) = merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22))) ) ).

% merge_list.simps(4)
tff(fact_4840_list_Osize__gen_I1_J,axiom,
    ! [B: $tType,X: fun(B,nat)] : size_list(B,X,nil(B)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_4841_size__list__pointwise,axiom,
    ! [B: $tType,Xs: list(B),F: fun(B,nat),G: fun(B,nat)] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,X2)),aa(B,nat,G,X2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(B,F,Xs)),size_list(B,G,Xs)) ) ).

% size_list_pointwise
tff(fact_4842_size__list__estimation_H,axiom,
    ! [B: $tType,X: B,Xs: list(B),Y: nat,F: fun(B,nat)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(B,nat,F,X))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(B,F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_4843_takeWhile__nth,axiom,
    ! [B: $tType,J: nat,Pa: fun(B,$o),Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs)))
     => ( aa(nat,B,nth(B,takeWhile(B,Pa,Xs)),J) = aa(nat,B,nth(B,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_4844_nth__length__takeWhile,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs))),aa(list(B),nat,size_size(list(B)),Xs))
     => ~ aa(B,$o,Pa,aa(nat,B,nth(B,Xs),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_4845_drop__takeWhile,axiom,
    ! [B: $tType,I: nat,Pa: fun(B,$o),L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,L)))
     => ( drop(B,I,takeWhile(B,Pa,L)) = takeWhile(B,Pa,drop(B,I,L)) ) ) ).

% drop_takeWhile
tff(fact_4846_takeWhile__not__last,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( distinct(B,Xs)
     => ( takeWhile(B,aTP_Lamp_ri(list(B),fun(B,$o),Xs),Xs) = aa(list(B),list(B),butlast(B),Xs) ) ) ).

% takeWhile_not_last
tff(fact_4847_less__length__takeWhile__conv,axiom,
    ! [B: $tType,I: nat,Pa: fun(B,$o),L: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,L)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
        & ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J3),I)
           => aa(B,$o,Pa,aa(nat,B,nth(B,L),J3)) ) ) ) ).

% less_length_takeWhile_conv
tff(fact_4848_eq__len__takeWhile__conv,axiom,
    ! [B: $tType,I: nat,Pa: fun(B,$o),L: list(B)] :
      ( ( I = aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,L)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(list(B),nat,size_size(list(B)),L))
        & ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I)
           => aa(B,$o,Pa,aa(nat,B,nth(B,L),J3)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
         => ~ aa(B,$o,Pa,aa(nat,B,nth(B,L),I)) ) ) ) ).

% eq_len_takeWhile_conv
tff(fact_4849_length__takeWhile__less__P__nth,axiom,
    ! [B: $tType,J: nat,Pa: fun(B,$o),Xs: list(B)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(B,$o,Pa,aa(nat,B,nth(B,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_4850_takeWhile__eq__take__P__nth,axiom,
    ! [B: $tType,N: nat,Xs: list(B),Pa: fun(B,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Xs))
           => aa(B,$o,Pa,aa(nat,B,nth(B,Xs),I2)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
         => ~ aa(B,$o,Pa,aa(nat,B,nth(B,Xs),N)) )
       => ( takeWhile(B,Pa,Xs) = take(B,N,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_4851_size__list__conv__sum__list,axiom,
    ! [B: $tType,F: fun(B,nat),Xs: list(B)] : size_list(B,F,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,F),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_4852_list_Osize__gen_I2_J,axiom,
    ! [B: $tType,X: fun(B,nat),X21: B,X222: list(B)] : size_list(B,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),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(B,nat,X,X21)),size_list(B,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_4853_filter__equals__takeWhile__sorted__rev,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C),Ta: B] :
          ( sorted_wrt(B,ord_less_eq(B),rev(B,aa(list(C),list(B),map(C,B,F),Xs)))
         => ( filter2(C,aa(B,fun(C,$o),aTP_Lamp_rj(fun(C,B),fun(B,fun(C,$o)),F),Ta),Xs) = takeWhile(C,aa(B,fun(C,$o),aTP_Lamp_rj(fun(C,B),fun(B,fun(C,$o)),F),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4854_mergesort__remdups__def,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : aa(list(B),list(B),mergesort_remdups(B),Xs) = merge_list(B,nil(list(B)),aa(list(B),list(list(B)),map(B,list(B),aTP_Lamp_rk(B,list(B))),Xs)) ) ).

% mergesort_remdups_def
tff(fact_4855_extract__def,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : extract(B,Pa,Xs) = aa(list(B),option(product_prod(list(B),product_prod(B,list(B)))),case_list(option(product_prod(list(B),product_prod(B,list(B)))),B,none(product_prod(list(B),product_prod(B,list(B)))),aa(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B)))))),aTP_Lamp_rl(fun(B,$o),fun(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B))))))),Pa),Xs)),dropWhile(B,aa(fun(B,$o),fun(B,$o),aa(fun($o,$o),fun(fun(B,$o),fun(B,$o)),comp($o,$o,B),fNot),Pa),Xs)) ).

% extract_def
tff(fact_4856_transpose_Opelims,axiom,
    ! [B: $tType,X: list(list(B)),Y: list(list(B))] :
      ( ( transpose(B,X) = Y )
     => ( accp(list(list(B)),transpose_rel(B),X)
       => ( ( ( X = nil(list(B)) )
           => ( ( Y = nil(list(B)) )
             => ~ accp(list(list(B)),transpose_rel(B),nil(list(B))) ) )
         => ( ! [Xss2: list(list(B))] :
                ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),Xss2) )
               => ( ( Y = transpose(B,Xss2) )
                 => ~ accp(list(list(B)),transpose_rel(B),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),Xss2)) ) )
           => ~ ! [X2: B,Xs2: list(B),Xss2: list(list(B))] :
                  ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),concat(B,aa(list(list(B)),list(list(B)),map(list(B),list(B),case_list(list(B),B,nil(B),aTP_Lamp_qa(B,fun(list(B),list(B))))),Xss2)))),transpose(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Xs2),concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),case_list(list(list(B)),B,nil(list(B)),aTP_Lamp_qn(B,fun(list(B),list(list(B)))))),Xss2))))) )
                   => ~ accp(list(list(B)),transpose_rel(B),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_4857_transpose_Opsimps_I3_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),Xss: list(list(B))] :
      ( accp(list(list(B)),transpose_rel(B),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),Xss))
     => ( transpose(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),Xss)) = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),concat(B,aa(list(list(B)),list(list(B)),map(list(B),list(B),case_list(list(B),B,nil(B),aTP_Lamp_qa(B,fun(list(B),list(B))))),Xss)))),transpose(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Xs),concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),case_list(list(list(B)),B,nil(list(B)),aTP_Lamp_qn(B,fun(list(B),list(list(B)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_4858_length__dropWhile__le,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),dropWhile(B,Pa,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_dropWhile_le
tff(fact_4859_sorted__dropWhile,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Pa: fun(B,$o)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),dropWhile(B,Pa,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_4860_dropWhile__eq__drop,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : dropWhile(B,Pa,Xs) = drop(B,aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_4861_length__dropWhile__takeWhile,axiom,
    ! [B: $tType,X: nat,Pa: fun(B,$o),Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(B),nat,size_size(list(B)),dropWhile(B,Pa,Xs)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs)))),aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_dropWhile_takeWhile
tff(fact_4862_dropWhile__nth,axiom,
    ! [B: $tType,J: nat,Pa: fun(B,$o),Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),dropWhile(B,Pa,Xs)))
     => ( aa(nat,B,nth(B,dropWhile(B,Pa,Xs)),J) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(B),nat,size_size(list(B)),takeWhile(B,Pa,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_4863_transpose_Opinduct,axiom,
    ! [B: $tType,A0: list(list(B)),Pa: fun(list(list(B)),$o)] :
      ( accp(list(list(B)),transpose_rel(B),A0)
     => ( ( accp(list(list(B)),transpose_rel(B),nil(list(B)))
         => aa(list(list(B)),$o,Pa,nil(list(B))) )
       => ( ! [Xss2: list(list(B))] :
              ( accp(list(list(B)),transpose_rel(B),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),Xss2))
             => ( aa(list(list(B)),$o,Pa,Xss2)
               => aa(list(list(B)),$o,Pa,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),nil(B)),Xss2)) ) )
         => ( ! [X2: B,Xs2: list(B),Xss2: list(list(B))] :
                ( accp(list(list(B)),transpose_rel(B),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),Xss2))
               => ( aa(list(list(B)),$o,Pa,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Xs2),concat(list(B),aa(list(list(B)),list(list(list(B))),map(list(B),list(list(B)),case_list(list(list(B)),B,nil(list(B)),aTP_Lamp_qn(B,fun(list(B),list(list(B)))))),Xss2))))
                 => aa(list(list(B)),$o,Pa,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)),Xss2)) ) )
           => aa(list(list(B)),$o,Pa,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_4864_dropWhile__neq__rev,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( distinct(B,Xs)
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
       => ( dropWhile(B,aTP_Lamp_ph(B,fun(B,$o),X),rev(B,Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),rev(B,takeWhile(B,aTP_Lamp_ph(B,fun(B,$o),X),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_4865_takeWhile__neq__rev,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( distinct(B,Xs)
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
       => ( takeWhile(B,aTP_Lamp_ph(B,fun(B,$o),X),rev(B,Xs)) = rev(B,aa(list(B),list(B),tl(B),dropWhile(B,aTP_Lamp_ph(B,fun(B,$o),X),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_4866_quicksort_Opelims,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: list(B),Y: list(B)] :
          ( ( aa(list(B),list(B),linorder_quicksort(B),X) = Y )
         => ( accp(list(B),linord6200660962353139674rt_rel(B),X)
           => ( ( ( X = nil(B) )
               => ( ( Y = nil(B) )
                 => ~ accp(list(B),linord6200660962353139674rt_rel(B),nil(B)) ) )
             => ~ ! [X2: B,Xs2: list(B)] :
                    ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
                   => ( ( Y = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_quicksort(B),filter2(B,aTP_Lamp_pv(B,fun(B,$o),X2),Xs2))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B))),aa(list(B),list(B),linorder_quicksort(B),filter2(B,aa(B,fun(B,$o),ord_less_eq(B),X2),Xs2)))) )
                     => ~ accp(list(B),linord6200660962353139674rt_rel(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_4867_remove__rev__def,axiom,
    ! [B: $tType,X: B] : remove_rev(B,X) = aa(fun(B,$o),fun(list(B),list(B)),filter_rev(B),aa(fun(B,$o),fun(B,$o),aa(fun($o,$o),fun(fun(B,$o),fun(B,$o)),comp($o,$o,B),fNot),aa(B,fun(B,$o),fequal(B),X))) ).

% remove_rev_def
tff(fact_4868_find__dropWhile,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : find(B,Pa,Xs) = aa(list(B),option(B),case_list(option(B),B,none(B),aTP_Lamp_rm(B,fun(list(B),option(B)))),dropWhile(B,aa(fun(B,$o),fun(B,$o),aa(fun($o,$o),fun(fun(B,$o),fun(B,$o)),comp($o,$o,B),fNot),Pa),Xs)) ).

% find_dropWhile
tff(fact_4869_find__SomeD_I1_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B),X: B] :
      ( ( find(B,Pa,Xs) = aa(B,option(B),some(B),X) )
     => aa(B,$o,Pa,X) ) ).

% find_SomeD(1)
tff(fact_4870_find__SomeD_I2_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B),X: B] :
      ( ( find(B,Pa,Xs) = aa(B,option(B),some(B),X) )
     => aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs)) ) ).

% find_SomeD(2)
tff(fact_4871_find__Some__iff,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B),X: B] :
      ( ( find(B,Pa,Xs) = aa(B,option(B),some(B),X) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
          & aa(B,$o,Pa,aa(nat,B,nth(B,Xs),I3))
          & ( X = aa(nat,B,nth(B,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(B,$o,Pa,aa(nat,B,nth(B,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_4872_find__Some__iff2,axiom,
    ! [B: $tType,X: B,Pa: fun(B,$o),Xs: list(B)] :
      ( ( aa(B,option(B),some(B),X) = find(B,Pa,Xs) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
          & aa(B,$o,Pa,aa(nat,B,nth(B,Xs),I3))
          & ( X = aa(nat,B,nth(B,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(B,$o,Pa,aa(nat,B,nth(B,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_4873_filter__rev__alt,axiom,
    ! [B: $tType,Pa: fun(B,$o),L: list(B)] : aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter_rev(B),Pa),L) = filter2(B,Pa,rev(B,L)) ).

% filter_rev_alt
tff(fact_4874_partition__rev__filter__conv,axiom,
    ! [B: $tType,Pa: fun(B,$o),Yes2: list(B),No2: list(B),Xs: list(B)] : partition_rev(B,Pa,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)),Yes2),No2),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)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),rev(B,filter2(B,Pa,Xs))),Yes2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),rev(B,filter2(B,aa(fun(B,$o),fun(B,$o),aa(fun($o,$o),fun(fun(B,$o),fun(B,$o)),comp($o,$o,B),fNot),Pa),Xs))),No2)) ).

% partition_rev_filter_conv
tff(fact_4875_effect__makeI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),H4: heap_ext(product_unit),F: fun(nat,B),N: nat,H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit))),product_Pair(array(B),heap_ext(product_unit)),A2),H4) = array_alloc(B,aa(list(nat),list(B),map(nat,B,F),upt(zero_zero(nat),N)),H) )
         => heap_Time_effect(array(B),array_make(B,N,F),H,H4,A2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_makeI
tff(fact_4876_map__distinct__upd__conv,axiom,
    ! [C: $tType,B: $tType,I: nat,L: list(B),F: fun(B,C),X: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
     => ( distinct(B,L)
       => ( list_update(C,aa(list(B),list(C),map(B,C,F),L),I,X) = aa(list(B),list(C),map(B,C,fun_upd(B,C,F,aa(nat,B,nth(B,L),I),X)),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4877_empty__upd__none,axiom,
    ! [B: $tType,C: $tType,X: B,X4: B] : aa(B,option(C),fun_upd(B,option(C),aTP_Lamp_jv(B,option(C)),X,none(C)),X4) = none(C) ).

% empty_upd_none
tff(fact_4878_image__update,axiom,
    ! [C: $tType,B: $tType,X: B,A4: set(B),F: fun(B,C),N: C] :
      ( ~ aa(set(B),$o,member(B,X),A4)
     => ( aa(set(B),set(C),image2(B,C,fun_upd(B,C,F,X,N)),A4) = aa(set(B),set(C),image2(B,C,F),A4) ) ) ).

% image_update
tff(fact_4879_success__effectE,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),H: heap_ext(product_unit)] :
      ( heap_Time_success(B,C2,H)
     => ~ ! [R5: B,H5: heap_ext(product_unit),N2: nat] : ~ heap_Time_effect(B,C2,H,H5,R5,N2) ) ).

% success_effectE
tff(fact_4880_effect__success,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,C2,H,H4,R,N)
     => heap_Time_success(B,C2,H) ) ).

% effect_success
tff(fact_4881_effect__deterministic_I3_J,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),A2: B,N: nat,H6: heap_ext(product_unit),B2: B,N3: nat] :
      ( heap_Time_effect(B,F,H,H4,A2,N)
     => ( heap_Time_effect(B,F,H,H6,B2,N3)
       => ( N = N3 ) ) ) ).

% effect_deterministic(3)
tff(fact_4882_effect__deterministic_I2_J,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),A2: B,N: nat,H6: heap_ext(product_unit),B2: B,N3: nat] :
      ( heap_Time_effect(B,F,H,H4,A2,N)
     => ( heap_Time_effect(B,F,H,H6,B2,N3)
       => ( H4 = H6 ) ) ) ).

% effect_deterministic(2)
tff(fact_4883_effect__deterministic_I1_J,axiom,
    ! [B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),A2: B,N: nat,H6: heap_ext(product_unit),B2: B,N3: nat] :
      ( heap_Time_effect(B,F,H,H4,A2,N)
     => ( heap_Time_effect(B,F,H,H6,B2,N3)
       => ( A2 = B2 ) ) ) ).

% effect_deterministic(1)
tff(fact_4884_effect__ifE,axiom,
    ! [B: $tType,C2: $o,Ta: heap_Time_Heap(B),E2: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,
          $ite((C2),Ta,E2),
          H,H4,R,N)
     => ( ( (C2)
         => ~ heap_Time_effect(B,Ta,H,H4,R,N) )
       => ~ ( ~ (C2)
           => ~ heap_Time_effect(B,E2,H,H4,R,N) ) ) ) ).

% effect_ifE
tff(fact_4885_effect__ifI,axiom,
    ! [B: $tType,C2: $o,Ta: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat,E2: heap_Time_Heap(B)] :
      ( ( (C2)
       => heap_Time_effect(B,Ta,H,H4,R,N) )
     => ( ( ~ (C2)
         => heap_Time_effect(B,E2,H,H4,R,N) )
       => heap_Time_effect(B,
            $ite((C2),Ta,E2),
            H,H4,R,N) ) ) ).

% effect_ifI
tff(fact_4886_effect__LetI,axiom,
    ! [C: $tType,B: $tType,X: B,Ta: B,F: fun(B,heap_Time_Heap(C)),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: C,N: nat] :
      ( ( X = Ta )
     => ( heap_Time_effect(C,aa(B,heap_Time_Heap(C),F,X),H,H4,R,N)
       => heap_Time_effect(C,aa(B,heap_Time_Heap(C),F,Ta),H,H4,R,N) ) ) ).

% effect_LetI
tff(fact_4887_effect__raiseE,axiom,
    ! [B: $tType,X: list(char),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] : ~ heap_Time_effect(B,heap_Time_raise(B,X),H,H4,R,N) ).

% effect_raiseE
tff(fact_4888_finite__update__induct,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),C2: C,Pa: fun(fun(B,C),$o)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aTP_Lamp_rn(fun(B,C),fun(C,fun(B,$o)),F),C2)))
     => ( aa(fun(B,C),$o,Pa,aTP_Lamp_iu(C,fun(B,C),C2))
       => ( ! [A3: B,B3: C,F4: fun(B,C)] :
              ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,C),fun(B,$o),aTP_Lamp_ro(C,fun(fun(B,C),fun(B,$o)),C2),F4)))
             => ( ( aa(B,C,F4,A3) = C2 )
               => ( ( B3 != C2 )
                 => ( aa(fun(B,C),$o,Pa,F4)
                   => aa(fun(B,C),$o,Pa,fun_upd(B,C,F4,A3,B3)) ) ) ) )
         => aa(fun(B,C),$o,Pa,F) ) ) ) ).

% finite_update_induct
tff(fact_4889_effect__bindI,axiom,
    ! [C: $tType,B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat,G: fun(B,heap_Time_Heap(C)),H6: heap_ext(product_unit),R3: C,N3: nat] :
      ( heap_Time_effect(B,F,H,H4,R,N)
     => ( heap_Time_effect(C,aa(B,heap_Time_Heap(C),G,R),H4,H6,R3,N3)
       => heap_Time_effect(C,heap_Time_bind(B,C,F,G),H,H6,R3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N3)) ) ) ).

% effect_bindI
tff(fact_4890_effect__bindE,axiom,
    ! [C: $tType,B: $tType,F: heap_Time_Heap(C),G: fun(C,heap_Time_Heap(B)),H: heap_ext(product_unit),H6: heap_ext(product_unit),R3: B,N: nat] :
      ( heap_Time_effect(B,heap_Time_bind(C,B,F,G),H,H6,R3,N)
     => ~ ! [H5: heap_ext(product_unit),R5: C,N12: nat] :
            ( heap_Time_effect(C,F,H,H5,R5,N12)
           => ! [N23: nat] :
                ( heap_Time_effect(B,aa(C,heap_Time_Heap(B),G,R5),H5,H6,R3,N23)
               => ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N12),N23) ) ) ) ) ).

% effect_bindE
tff(fact_4891_effect__returnI,axiom,
    ! [B: $tType,H: heap_ext(product_unit),H4: heap_ext(product_unit),X: B] :
      ( ( H = H4 )
     => heap_Time_effect(B,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),H,H4,X,one_one(nat)) ) ).

% effect_returnI
tff(fact_4892_effect__returnE,axiom,
    ! [B: $tType,X: B,H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),H,H4,R,N)
     => ~ ( ( R = X )
         => ( ( H4 = H )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_returnE
tff(fact_4893_effect__ureturnE,axiom,
    ! [B: $tType,X: B,H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X),H,H4,R,N)
     => ~ ( ( R = X )
         => ( ( H4 = H )
           => ( N != zero_zero(nat) ) ) ) ) ).

% effect_ureturnE
tff(fact_4894_effect__ureturnI,axiom,
    ! [B: $tType,H: heap_ext(product_unit),H4: heap_ext(product_unit),X: B] :
      ( ( H = H4 )
     => heap_Time_effect(B,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X),H,H4,X,zero_zero(nat)) ) ).

% effect_ureturnI
tff(fact_4895_success__bind__effectI,axiom,
    ! [B: $tType,C: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),X: B,N: nat,G: fun(B,heap_Time_Heap(C))] :
      ( heap_Time_effect(B,F,H,H4,X,N)
     => ( heap_Time_success(C,aa(B,heap_Time_Heap(C),G,X),H4)
       => heap_Time_success(C,heap_Time_bind(B,C,F,G),H) ) ) ).

% success_bind_effectI
tff(fact_4896_effect__assertI,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,H4: heap_ext(product_unit),H: heap_ext(product_unit),R: B,N: nat] :
      ( aa(B,$o,Pa,X)
     => ( ( H4 = H )
       => ( ( R = X )
         => ( ( N = one_one(nat) )
           => heap_Time_effect(B,heap_Time_assert(B,Pa,X),H,H4,R,N) ) ) ) ) ).

% effect_assertI
tff(fact_4897_effect__assertE,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B,H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,heap_Time_assert(B,Pa,X),H,H4,R,N)
     => ~ ( aa(B,$o,Pa,X)
         => ( ( R = X )
           => ( ( H4 = H )
             => ( N != one_one(nat) ) ) ) ) ) ).

% effect_assertE
tff(fact_4898_effect__tapI,axiom,
    ! [B: $tType,H4: heap_ext(product_unit),H: heap_ext(product_unit),R: B,F: fun(heap_ext(product_unit),B)] :
      ( ( H4 = H )
     => ( ( R = aa(heap_ext(product_unit),B,F,H) )
       => heap_Time_effect(B,heap_Time_tap(B,F),H,H4,R,one_one(nat)) ) ) ).

% effect_tapI
tff(fact_4899_effect__tapE,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,heap_Time_tap(B,F),H,H4,R,N)
     => ~ ( ( H4 = H )
         => ( ( R = aa(heap_ext(product_unit),B,F,H) )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_tapE
tff(fact_4900_partition__rev_Osimps_I2_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),Yes2: list(B),No2: list(B),X: B,Xs: list(B)] :
      partition_rev(B,Pa,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)),Yes2),No2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = partition_rev(B,Pa,
        $ite(aa(B,$o,Pa,X),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),X),Yes2)),No2),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)),Yes2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),No2))),
        Xs) ).

% partition_rev.simps(2)
tff(fact_4901_partition__rev_Osimps_I1_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),Yes2: list(B),No2: list(B)] : partition_rev(B,Pa,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)),Yes2),No2),nil(B)) = 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)),Yes2),No2) ).

% partition_rev.simps(1)
tff(fact_4902_effect__lookupE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [R: ref(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),X: B,N: nat] :
          ( heap_Time_effect(B,ref_lookup(B,R),H,H4,X,N)
         => ~ ( ( H4 = H )
             => ( ( X = ref_get(B,H,R) )
               => ( N != one_one(nat) ) ) ) ) ) ).

% effect_lookupE
tff(fact_4903_effect__lookupI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H4: heap_ext(product_unit),H: heap_ext(product_unit),X: B,R: ref(B),N: nat] :
          ( ( H4 = H )
         => ( ( X = ref_get(B,H,R) )
           => ( ( N = one_one(nat) )
             => heap_Time_effect(B,ref_lookup(B,R),H,H4,X,N) ) ) ) ) ).

% effect_lookupI
tff(fact_4904_partition__rev_Oelims,axiom,
    ! [B: $tType,X: fun(B,$o),Xa2: product_prod(list(B),list(B)),Xb: list(B),Y: product_prod(list(B),list(B))] :
      ( ( partition_rev(B,X,Xa2,Xb) = Y )
     => ( ! [Yes: list(B),No: list(B)] :
            ( ( Xa2 = 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)),Yes),No) )
           => ( ( Xb = nil(B) )
             => ( Y != 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)),Yes),No) ) ) )
       => ~ ! [Yes: list(B),No: list(B)] :
              ( ( Xa2 = 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)),Yes),No) )
             => ! [X2: B,Xs2: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
                 => ( Y != partition_rev(B,X,
                        $ite(aa(B,$o,X,X2),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),X2),Yes)),No),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)),Yes),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),No))),
                        Xs2) ) ) ) ) ) ).

% partition_rev.elims
tff(fact_4905_effectI,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),H: heap_ext(product_unit),R: B,H4: heap_ext(product_unit),N: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,C2),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)),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),N))) )
     => heap_Time_effect(B,C2,H,H4,R,N) ) ).

% effectI
tff(fact_4906_effect__def,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,C2,H,H4,R,N)
    <=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,C2),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)),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),N))) ) ) ).

% effect_def
tff(fact_4907_effect__of__listI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),H4: heap_ext(product_unit),Xs: list(B),H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit))),product_Pair(array(B),heap_ext(product_unit)),A2),H4) = array_alloc(B,Xs,H) )
         => heap_Time_effect(array(B),array_of_list(B,Xs),H,H4,A2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(B),nat,size_size(list(B)),Xs))) ) ) ).

% effect_of_listI
tff(fact_4908_effect__newI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),H4: heap_ext(product_unit),N: nat,X: B,H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit))),product_Pair(array(B),heap_ext(product_unit)),A2),H4) = array_alloc(B,replicate(B,N,X),H) )
         => heap_Time_effect(array(B),array_new(B,N,X),H,H4,A2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_newI
tff(fact_4909_map__upds__append1,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C),Ma: fun(B,option(C)),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(C),nat,size_size(list(C)),Ys))
     => ( map_upds(B,C,Ma,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))),Ys) = fun_upd(B,option(C),map_upds(B,C,Ma,Xs,Ys),X,aa(C,option(C),some(C),aa(nat,C,nth(C,Ys),aa(list(B),nat,size_size(list(B)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_4910_quicksort__by__rel_Oelims,axiom,
    ! [B: $tType,X: fun(B,fun(B,$o)),Xa2: list(B),Xb: list(B),Y: list(B)] :
      ( ( aa(list(B),list(B),quicksort_by_rel(B,X,Xa2),Xb) = Y )
     => ( ( ( Xb = nil(B) )
         => ( Y != Xa2 ) )
       => ~ ! [X2: B,Xs2: list(B)] :
              ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
             => ( Y != aa(product_prod(list(B),list(B)),list(B),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(B,fun(list(B),fun(list(B),list(B))),aa(list(B),fun(B,fun(list(B),fun(list(B),list(B)))),aTP_Lamp_rp(fun(B,fun(B,$o)),fun(list(B),fun(B,fun(list(B),fun(list(B),list(B))))),X),Xa2),X2)),partition_rev(B,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),X),X2),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)),nil(B)),nil(B)),Xs2)) ) ) ) ) ).

% quicksort_by_rel.elims
tff(fact_4911_quicksort__by__rel_Osimps_I2_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B),X: B,Xs: list(B)] : aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(product_prod(list(B),list(B)),list(B),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(B,fun(list(B),fun(list(B),list(B))),aa(list(B),fun(B,fun(list(B),fun(list(B),list(B)))),aTP_Lamp_rp(fun(B,fun(B,$o)),fun(list(B),fun(B,fun(list(B),fun(list(B),list(B))))),R2),Sl2),X)),partition_rev(B,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),R2),X),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)),nil(B)),nil(B)),Xs)) ).

% quicksort_by_rel.simps(2)
tff(fact_4912_fun__upds__append2__drop,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Ma: fun(B,option(C)),Zs: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( map_upds(B,C,Ma,Xs,aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys),Zs)) = map_upds(B,C,Ma,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_4913_fun__upds__append__drop,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Ma: fun(B,option(C)),Zs: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( map_upds(B,C,Ma,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Zs),Ys) = map_upds(B,C,Ma,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_4914_set__quicksort__by__rel,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B),Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),Xs)) = aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Sl2)) ).

% set_quicksort_by_rel
tff(fact_4915_quicksort__by__rel__permutes,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B),Xs: list(B)] : mset(B,aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),Xs)) = mset(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Sl2)) ).

% quicksort_by_rel_permutes
tff(fact_4916_map__upds__list__update2__drop,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),I: nat,Ma: fun(B,option(C)),Ys: list(C),Y: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Xs)),I)
     => ( map_upds(B,C,Ma,Xs,list_update(C,Ys,I,Y)) = map_upds(B,C,Ma,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_4917_quicksort__by__rel_Osimps_I1_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B)] : aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),nil(B)) = Sl2 ).

% quicksort_by_rel.simps(1)
tff(fact_4918_quicksort__by__rel__remove__acc__guared,axiom,
    ! [B: $tType,Sl2: list(B),R2: fun(B,fun(B,$o)),Xs: list(B)] :
      ( ( Sl2 != nil(B) )
     => ( aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),quicksort_by_rel(B,R2,nil(B)),Xs)),Sl2) ) ) ).

% quicksort_by_rel_remove_acc_guared
tff(fact_4919_quicksort__by__rel__remove__acc,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B),Xs: list(B)] : aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),quicksort_by_rel(B,R2,nil(B)),Xs)),Sl2) ).

% quicksort_by_rel_remove_acc
tff(fact_4920_sorted__wrt__quicksort__by__rel,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] :
      ( ! [X2: B,Y2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),R2,X2),Y2)
          | aa(B,$o,aa(B,fun(B,$o),R2,Y2),X2) )
     => ( ! [X2: B,Y2: B,Z2: B] :
            ( aa(B,$o,aa(B,fun(B,$o),R2,X2),Y2)
           => ( aa(B,$o,aa(B,fun(B,$o),R2,Y2),Z2)
             => aa(B,$o,aa(B,fun(B,$o),R2,X2),Z2) ) )
       => sorted_wrt(B,R2,aa(list(B),list(B),quicksort_by_rel(B,R2,nil(B)),Xs)) ) ) ).

% sorted_wrt_quicksort_by_rel
tff(fact_4921_sorted__quicksort__by__rel,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),quicksort_by_rel(B,ord_less_eq(B),nil(B)),Xs)) ) ).

% sorted_quicksort_by_rel
tff(fact_4922_sort__quicksort__by__rel,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( linorder_sort_key(B,B,aTP_Lamp_hr(B,B)) = quicksort_by_rel(B,ord_less_eq(B),nil(B)) ) ) ).

% sort_quicksort_by_rel
tff(fact_4923_map__upd__upds__conv__if,axiom,
    ! [B: $tType,C: $tType,F: fun(B,option(C)),X: B,Y: C,Xs: list(B),Ys: list(C)] :
      map_upds(B,C,fun_upd(B,option(C),F,X,aa(C,option(C),some(C),Y)),Xs,Ys) = $ite(aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),take(B,aa(list(C),nat,size_size(list(C)),Ys),Xs))),map_upds(B,C,F,Xs,Ys),fun_upd(B,option(C),map_upds(B,C,F,Xs,Ys),X,aa(C,option(C),some(C),Y))) ).

% map_upd_upds_conv_if
tff(fact_4924_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_rq(rat,fun(int,$o),X)) ).

% floor_rat_def
tff(fact_4925_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_nq(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4926_map__filter__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,option(B)),Xs: list(C)] : map_filter(C,B,F,Xs) = aa(list(C),list(B),map(C,B,aa(fun(C,option(B)),fun(C,B),aa(fun(option(B),B),fun(fun(C,option(B)),fun(C,B)),comp(option(B),B,C),the2(B)),F)),filter2(C,aTP_Lamp_rr(fun(C,option(B)),fun(C,$o),F),Xs)) ).

% map_filter_def
tff(fact_4927_option_Othe__def,axiom,
    ! [B: $tType,Option: option(B)] : aa(option(B),B,the2(B),Option) = case_option(B,B,undefined(B),aTP_Lamp_il(B,B),Option) ).

% option.the_def
tff(fact_4928_old_Orec__nat__def,axiom,
    ! [B: $tType,X4: B,Xa: fun(nat,fun(B,B)),Xb2: nat] : aa(nat,B,aa(fun(nat,fun(B,B)),fun(nat,B),aa(B,fun(fun(nat,fun(B,B)),fun(nat,B)),rec_nat(B),X4),Xa),Xb2) = the(B,rec_set_nat(B,X4,Xa,Xb2)) ).

% old.rec_nat_def
tff(fact_4929_the__sym__eq__trivial,axiom,
    ! [B: $tType,X: B] : the(B,aa(B,fun(B,$o),fequal(B),X)) = X ).

% the_sym_eq_trivial
tff(fact_4930_the__eq__trivial,axiom,
    ! [B: $tType,A2: B] : the(B,aTP_Lamp_af(B,fun(B,$o),A2)) = A2 ).

% the_eq_trivial
tff(fact_4931_the__equality,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B] :
      ( aa(B,$o,Pa,A2)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => ( X2 = A2 ) )
       => ( the(B,Pa) = A2 ) ) ) ).

% the_equality
tff(fact_4932_The__split__eq,axiom,
    ! [B: $tType,C: $tType,X: B,Y: C] : the(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(C,fun(B,fun(C,$o)),aTP_Lamp_lo(B,fun(C,fun(B,fun(C,$o))),X),Y))) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) ).

% The_split_eq
tff(fact_4933_theI,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B] :
      ( aa(B,$o,Pa,A2)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => ( X2 = A2 ) )
       => aa(B,$o,Pa,the(B,Pa)) ) ) ).

% theI
tff(fact_4934_theI_H,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ? [X4: B] :
          ( aa(B,$o,Pa,X4)
          & ! [Y2: B] :
              ( aa(B,$o,Pa,Y2)
             => ( Y2 = X4 ) ) )
     => aa(B,$o,Pa,the(B,Pa)) ) ).

% theI'
tff(fact_4935_theI2,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B,Q: fun(B,$o)] :
      ( aa(B,$o,Pa,A2)
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => ( X2 = A2 ) )
       => ( ! [X2: B] :
              ( aa(B,$o,Pa,X2)
             => aa(B,$o,Q,X2) )
         => aa(B,$o,Q,the(B,Pa)) ) ) ) ).

% theI2
tff(fact_4936_If__def,axiom,
    ! [B: $tType,Pa: $o,X: B,Y: B] :
      $ite((Pa),X,Y) = the(B,aa(B,fun(B,$o),aa(B,fun(B,fun(B,$o)),aTP_Lamp_rs($o,fun(B,fun(B,fun(B,$o))),(Pa)),X),Y)) ).

% If_def
tff(fact_4937_the1I2,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] :
      ( ? [X4: B] :
          ( aa(B,$o,Pa,X4)
          & ! [Y2: B] :
              ( aa(B,$o,Pa,Y2)
             => ( Y2 = X4 ) ) )
     => ( ! [X2: B] :
            ( aa(B,$o,Pa,X2)
           => aa(B,$o,Q,X2) )
       => aa(B,$o,Q,the(B,Pa)) ) ) ).

% the1I2
tff(fact_4938_the1__equality,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B] :
      ( ? [X4: B] :
          ( aa(B,$o,Pa,X4)
          & ! [Y2: B] :
              ( aa(B,$o,Pa,Y2)
             => ( Y2 = X4 ) ) )
     => ( aa(B,$o,Pa,A2)
       => ( the(B,Pa) = A2 ) ) ) ).

% the1_equality
tff(fact_4939_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_np(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4940_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_nz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4941_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_nx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4942_the__elem__def,axiom,
    ! [B: $tType,X5: set(B)] : the_elem(B,X5) = the(B,aTP_Lamp_rt(set(B),fun(B,$o),X5)) ).

% the_elem_def
tff(fact_4943_sort__mergesort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( linorder_sort_key(B,B,aTP_Lamp_hr(B,B)) = mergesort(B) ) ) ).

% sort_mergesort
tff(fact_4944_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_ru(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_4945_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_1
tff(fact_4946_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_2
tff(fact_4947_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Ma: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),Ma)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Ma) ).

% prod_encode_prod_decode_aux
tff(fact_4948_list__encode_Oelims,axiom,
    ! [X: list(nat),Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X2: nat,Xs2: list(nat)] :
              ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs2) )
             => ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_4949_list__encode_Opelims,axiom,
    ! [X: list(nat),Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( accp(list(nat),nat_list_encode_rel,X)
       => ( ( ( X = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X2: nat,Xs2: list(nat)] :
                ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs2) )
               => ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),nat_list_encode(Xs2)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_4950_merge__list_Opsimps_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [La2: list(B),Acc22: list(list(B)),L: list(B)] :
          ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),nil(list(B)))))
         => ( merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),nil(list(B)))) = merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22))) ) ) ) ).

% merge_list.psimps(4)
tff(fact_4951_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_4952_merge__list_Opsimps_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),nil(list(B))))
       => ( merge_list(B,nil(list(B)),nil(list(B))) = nil(B) ) ) ) ).

% merge_list.psimps(1)
tff(fact_4953_merge__list_Opsimps_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: list(B)] :
          ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),nil(list(B)))))
         => ( merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L),nil(list(B)))) = L ) ) ) ).

% merge_list.psimps(2)
tff(fact_4954_merge__list_Opsimps_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [La2: list(B),Acc22: list(list(B))] :
          ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22)),nil(list(B))))
         => ( merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22),nil(list(B))) = merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La2),Acc22)) ) ) ) ).

% merge_list.psimps(3)
tff(fact_4955_merge__list_Opelims,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: list(list(B)),Xa2: list(list(B)),Y: list(B)] :
          ( ( merge_list(B,X,Xa2) = Y )
         => ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),X),Xa2))
           => ( ( ( X = nil(list(B)) )
               => ( ( Xa2 = nil(list(B)) )
                 => ( ( Y = nil(B) )
                   => ~ accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),nil(list(B)))) ) ) )
             => ( ( ( X = nil(list(B)) )
                 => ! [L2: list(B)] :
                      ( ( Xa2 = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))) )
                     => ( ( Y = L2 )
                       => ~ accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))))) ) ) )
               => ( ! [La: list(B),Acc2: list(list(B))] :
                      ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2) )
                     => ( ( Xa2 = nil(list(B)) )
                       => ( ( Y = merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)) )
                         => ~ accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),nil(list(B)))) ) ) )
                 => ( ! [La: list(B),Acc2: list(list(B))] :
                        ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2) )
                       => ! [L2: list(B)] :
                            ( ( Xa2 = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))) )
                           => ( ( Y = merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2))) )
                             => ~ accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))))) ) ) )
                   => ~ ! [L12: list(B),L23: list(B),Ls2: list(list(B))] :
                          ( ( Xa2 = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L12),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L23),Ls2)) )
                         => ( ( Y = merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),merge(B,L12,L23)),X),Ls2) )
                           => ~ accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),X),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L12),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L23),Ls2)))) ) ) ) ) ) ) ) ) ) ).

% merge_list.pelims
tff(fact_4956_merge__list_Opsimps_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Acc22: list(list(B)),L1: list(B),L22: list(B),Ls: list(list(B))] :
          ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),Acc22),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L1),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L22),Ls))))
         => ( merge_list(B,Acc22,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L1),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L22),Ls))) = merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),merge(B,L1,L22)),Acc22),Ls) ) ) ) ).

% merge_list.psimps(5)
tff(fact_4957_merge__list_Opinduct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A0: list(list(B)),A12: list(list(B)),Pa: fun(list(list(B)),fun(list(list(B)),$o))] :
          ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),A0),A12))
         => ( ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),nil(list(B))))
             => aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,nil(list(B))),nil(list(B))) )
           => ( ! [L2: list(B)] :
                  ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B)))))
                 => aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B)))) )
             => ( ! [La: list(B),Acc2: list(list(B))] :
                    ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),nil(list(B))))
                   => ( aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2))
                     => aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),nil(list(B))) ) )
               => ( ! [La: list(B),Acc2: list(list(B)),L2: list(B)] :
                      ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B)))))
                     => ( aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,nil(list(B))),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)))
                       => aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B)))) ) )
                 => ( ! [Acc2: list(list(B)),L12: list(B),L23: list(B),Ls2: list(list(B))] :
                        ( accp(product_prod(list(list(B)),list(list(B))),merge_list_rel(B),aa(list(list(B)),product_prod(list(list(B)),list(list(B))),aa(list(list(B)),fun(list(list(B)),product_prod(list(list(B)),list(list(B)))),product_Pair(list(list(B)),list(list(B))),Acc2),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L12),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L23),Ls2))))
                       => ( aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),merge(B,L12,L23)),Acc2)),Ls2)
                         => aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,Acc2),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L12),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L23),Ls2))) ) )
                   => aa(list(list(B)),$o,aa(list(list(B)),fun(list(list(B)),$o),Pa,A0),A12) ) ) ) ) ) ) ) ).

% merge_list.pinduct
tff(fact_4958_merge_Osimps_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L22: list(B)] : merge(B,nil(B),L22) = L22 ) ).

% merge.simps(1)
tff(fact_4959_merge_Osimps_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X12: B,L1: list(B),X22: B,L22: list(B)] :
          merge(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X12),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),L22)) = $ite(
            aa(B,$o,aa(B,fun(B,$o),ord_less(B),X12),X22),
            aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X12),merge(B,L1,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),L22))),
            $ite(X12 = X22,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X12),merge(B,L1,L22)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),merge(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X12),L1),L22))) ) ) ).

% merge.simps(3)
tff(fact_4960_merge_Osimps_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [V2: B,Va2: list(B)] : merge(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2),nil(B)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2) ) ).

% merge.simps(2)
tff(fact_4961_merge__list_Osimps_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Acc22: list(list(B)),L1: list(B),L22: list(B),Ls: list(list(B))] : merge_list(B,Acc22,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L1),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L22),Ls))) = merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),merge(B,L1,L22)),Acc22),Ls) ) ).

% merge_list.simps(5)
tff(fact_4962_merge_Oelims,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: list(B),Xa2: list(B),Y: list(B)] :
          ( ( merge(B,X,Xa2) = Y )
         => ( ( ( X = nil(B) )
             => ( Y != Xa2 ) )
           => ( ! [V4: B,Va: list(B)] :
                  ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
                 => ( ( Xa2 = nil(B) )
                   => ( Y != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) ) ) )
             => ~ ! [X1: B,L12: list(B)] :
                    ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),L12) )
                   => ! [X23: B,L23: list(B)] :
                        ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),L23) )
                       => ( Y != $ite(
                              aa(B,$o,aa(B,fun(B,$o),ord_less(B),X1),X23),
                              aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),merge(B,L12,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),L23))),
                              $ite(X1 = X23,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),merge(B,L12,L23)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),merge(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),L12),L23))) ) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_4963_merge__list_Oelims,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: list(list(B)),Xa2: list(list(B)),Y: list(B)] :
          ( ( merge_list(B,X,Xa2) = Y )
         => ( ( ( X = nil(list(B)) )
             => ( ( Xa2 = nil(list(B)) )
               => ( Y != nil(B) ) ) )
           => ( ( ( X = nil(list(B)) )
               => ! [L2: list(B)] :
                    ( ( Xa2 = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))) )
                   => ( Y != L2 ) ) )
             => ( ! [La: list(B),Acc2: list(list(B))] :
                    ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2) )
                   => ( ( Xa2 = nil(list(B)) )
                     => ( Y != merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2)) ) ) )
               => ( ! [La: list(B),Acc2: list(list(B))] :
                      ( ( X = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2) )
                     => ! [L2: list(B)] :
                          ( ( Xa2 = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),nil(list(B))) )
                         => ( Y != merge_list(B,nil(list(B)),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L2),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),La),Acc2))) ) ) )
                 => ~ ! [L12: list(B),L23: list(B),Ls2: list(list(B))] :
                        ( ( Xa2 = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L12),aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),L23),Ls2)) )
                       => ( Y != merge_list(B,aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),merge(B,L12,L23)),X),Ls2) ) ) ) ) ) ) ) ) ).

% merge_list.elims
tff(fact_4964_merge_Opelims,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: list(B),Xa2: list(B),Y: list(B)] :
          ( ( merge(B,X,Xa2) = Y )
         => ( accp(product_prod(list(B),list(B)),merge_rel(B),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)),X),Xa2))
           => ( ( ( X = nil(B) )
               => ( ( Y = Xa2 )
                 => ~ accp(product_prod(list(B),list(B)),merge_rel(B),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)),nil(B)),Xa2)) ) )
             => ( ! [V4: B,Va: list(B)] :
                    ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
                   => ( ( Xa2 = nil(B) )
                     => ( ( Y = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
                       => ~ accp(product_prod(list(B),list(B)),merge_rel(B),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),V4),Va)),nil(B))) ) ) )
               => ~ ! [X1: B,L12: list(B)] :
                      ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),L12) )
                     => ! [X23: B,L23: list(B)] :
                          ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),L23) )
                         => ( ( Y = $ite(
                                  aa(B,$o,aa(B,fun(B,$o),ord_less(B),X1),X23),
                                  aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),merge(B,L12,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),L23))),
                                  $ite(X1 = X23,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),merge(B,L12,L23)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),merge(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),L12),L23))) ) )
                           => ~ accp(product_prod(list(B),list(B)),merge_rel(B),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),X1),L12)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),L23))) ) ) ) ) ) ) ) ) ).

% merge.pelims
tff(fact_4965_filter__rev__aux__alt,axiom,
    ! [B: $tType,A2: list(B),Pa: fun(B,$o),L: list(B)] : aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter_rev_aux(B,A2),Pa),L) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),filter2(B,Pa,rev(B,L))),A2) ).

% filter_rev_aux_alt
tff(fact_4966_rat__floor__lemma,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),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),A2),B2)),one_one(int)))) ) ).

% rat_floor_lemma
tff(fact_4967_less__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2))
        <=> aa(int,$o,aa(int,fun(int,$o),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),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))) ) ) ) ).

% less_rat
tff(fact_4968_le__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2))
        <=> aa(int,$o,aa(int,fun(int,$o),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),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))) ) ) ) ).

% le_rat
tff(fact_4969_add__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = 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),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)) ) ) ) ).

% add_rat
tff(fact_4970_diff__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)) ) ) ) ).

% diff_rat
tff(fact_4971_eq__rat_I3_J,axiom,
    ! [A2: int,C2: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),A2) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),C2) ).

% eq_rat(3)
tff(fact_4972_Rat__induct__pos,axiom,
    ! [Pa: fun(rat,$o),Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => aa(rat,$o,Pa,aa(int,rat,aa(int,fun(int,rat),fract,A3),B3)) )
     => aa(rat,$o,Pa,Q2) ) ).

% Rat_induct_pos
tff(fact_4973_eq__rat_I2_J,axiom,
    ! [A2: int] : aa(int,rat,aa(int,fun(int,rat),fract,A2),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% eq_rat(2)
tff(fact_4974_mult__rat__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),times_times(int),C2),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ) ) ).

% mult_rat_cancel
tff(fact_4975_eq__rat_I1_J,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( ( aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) = aa(int,rat,aa(int,fun(int,rat),fract,C2),D2) )
        <=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A2),D2) = aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_4976_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K) ).

% Fract_of_nat_eq
tff(fact_4977_rat__number__collapse_I1_J,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),K) = zero_zero(rat) ).

% rat_number_collapse(1)
tff(fact_4978_rat__number__collapse_I6_J,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),zero_zero(int)) = zero_zero(rat) ).

% rat_number_collapse(6)
tff(fact_4979_One__rat__def,axiom,
    one_one(rat) = aa(int,rat,aa(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).

% One_rat_def
tff(fact_4980_Fract__of__int__eq,axiom,
    ! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),one_one(int)) = aa(int,rat,ring_1_of_int(rat),K) ).

% Fract_of_int_eq
tff(fact_4981_filter__rev__aux_Osimps_I2_J,axiom,
    ! [B: $tType,A2: list(B),Pa: fun(B,$o),X: B,Xs: list(B)] :
      aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter_rev_aux(B,A2),Pa),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = $ite(aa(B,$o,Pa,X),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter_rev_aux(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),A2)),Pa),Xs),aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter_rev_aux(B,A2),Pa),Xs)) ).

% filter_rev_aux.simps(2)
tff(fact_4982_filter__rev__aux_Osimps_I1_J,axiom,
    ! [B: $tType,A2: list(B),Pa: fun(B,$o)] : aa(list(B),list(B),aa(fun(B,$o),fun(list(B),list(B)),filter_rev_aux(B,A2),Pa),nil(B)) = A2 ).

% filter_rev_aux.simps(1)
tff(fact_4983_Zero__rat__def,axiom,
    zero_zero(rat) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).

% Zero_rat_def
tff(fact_4984_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa(num,rat,numeral_numeral(rat),K) = aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% rat_number_expand(3)
tff(fact_4985_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),W)),one_one(int)) = aa(num,rat,numeral_numeral(rat),W) ).

% rat_number_collapse(3)
tff(fact_4986_zero__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% zero_less_Fract_iff
tff(fact_4987_Fract__less__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% Fract_less_zero_iff
tff(fact_4988_Fract__less__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),B2) ) ) ).

% Fract_less_one_iff
tff(fact_4989_one__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),A2) ) ) ).

% one_less_Fract_iff
tff(fact_4990_rat__number__collapse_I5_J,axiom,
    aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).

% rat_number_collapse(5)
tff(fact_4991_Fract__add__one,axiom,
    ! [N: int,Ma: 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),Ma),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,Ma),N)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_4992_Fract__le__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% Fract_le_zero_iff
tff(fact_4993_zero__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% zero_le_Fract_iff
tff(fact_4994_Fract__le__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),B2) ) ) ).

% Fract_le_one_iff
tff(fact_4995_one__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2) ) ) ).

% one_le_Fract_iff
tff(fact_4996_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% rat_number_expand(5)
tff(fact_4997_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W)) ).

% rat_number_collapse(4)
tff(fact_4998_filter__rev__def,axiom,
    ! [B: $tType] : filter_rev(B) = filter_rev_aux(B,nil(B)) ).

% filter_rev_def
tff(fact_4999_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : pow(X,aa(num,num,bit1,Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y))),X) ).

% pow.simps(3)
tff(fact_5000_restrict__map__upds,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),D3: set(B),Ma: fun(B,option(C))] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),D3)
       => ( restrict_map(B,C,map_upds(B,C,Ma,Xs,Ys),D3) = map_upds(B,C,restrict_map(B,C,Ma,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),D3),aa(list(B),set(B),set2(B),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_5001_ring__1__class_Oof__int__def,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ( ring_1_of_int(B) = aa(fun(product_prod(nat,nat),B),fun(int,B),map_fun(int,product_prod(nat,nat),B,B,rep_Integ,id(B)),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),aTP_Lamp_nr(nat,fun(nat,B)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_5002_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_5003_id__funpow,axiom,
    ! [B: $tType,N: nat] : aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),id(B)) = id(B) ).

% id_funpow
tff(fact_5004_rotate0,axiom,
    ! [B: $tType] : rotate(B,zero_zero(nat)) = id(list(B)) ).

% rotate0
tff(fact_5005_restrict__map__empty,axiom,
    ! [B: $tType,C: $tType,D3: set(B),X4: B] : aa(B,option(C),restrict_map(B,C,aTP_Lamp_jv(B,option(C)),D3),X4) = none(C) ).

% restrict_map_empty
tff(fact_5006_push__bit__0__id,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ( aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),zero_zero(nat)) = id(B) ) ) ).

% push_bit_0_id
tff(fact_5007_drop__bit__0,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ( aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),zero_zero(nat)) = id(B) ) ) ).

% drop_bit_0
tff(fact_5008_map__fun_Oidentity,axiom,
    ! [C: $tType,B: $tType] : map_fun(B,B,C,C,aTP_Lamp_il(B,B),aTP_Lamp_rv(C,C)) = id(fun(B,C)) ).

% map_fun.identity
tff(fact_5009_pointfree__idE,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),G: fun(B,C),X: B] :
      ( ( aa(fun(B,C),fun(B,B),aa(fun(C,B),fun(fun(B,C),fun(B,B)),comp(C,B,B),F),G) = id(B) )
     => ( aa(C,B,F,aa(B,C,G,X)) = X ) ) ).

% pointfree_idE
tff(fact_5010_restrict__map__subset__eq,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),R2: set(B),M3: fun(B,option(C)),R7: set(B)] :
      ( ( restrict_map(B,C,Ma,R2) = M3 )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),R7),R2)
       => ( restrict_map(B,C,Ma,R7) = restrict_map(B,C,M3,R7) ) ) ) ).

% restrict_map_subset_eq
tff(fact_5011_le__map__restrict,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [Ma: fun(B,option(C)),X5: set(B)] : aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),restrict_map(B,C,Ma,X5)),Ma) ) ).

% le_map_restrict
tff(fact_5012_restrict__map__eq_I2_J,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B)),A4: set(C),K: C,V2: B] :
      ( ( aa(C,option(B),restrict_map(C,B,Ma,A4),K) = aa(B,option(B),some(B),V2) )
    <=> ( ( aa(C,option(B),Ma,K) = aa(B,option(B),some(B),V2) )
        & aa(set(C),$o,member(C,K),A4) ) ) ).

% restrict_map_eq(2)
tff(fact_5013_List_Omap_Oidentity,axiom,
    ! [B: $tType] : map(B,B,aTP_Lamp_il(B,B)) = id(list(B)) ).

% List.map.identity
tff(fact_5014_map__option_Oidentity,axiom,
    ! [B: $tType] : map_option(B,B,aTP_Lamp_il(B,B)) = id(option(B)) ).

% map_option.identity
tff(fact_5015_funpow__simps__right_I1_J,axiom,
    ! [B: $tType,F: fun(B,B)] : aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),zero_zero(nat)),F) = id(B) ).

% funpow_simps_right(1)
tff(fact_5016_sqr_Osimps_I2_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,bit0,sqr(N))) ).

% sqr.simps(2)
tff(fact_5017_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5018_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_5019_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_5020_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = aa(num,num,aa(num,fun(num,num),times_times(num),X),X) ).

% sqr_conv_mult
tff(fact_5021_foldr__filter,axiom,
    ! [B: $tType,C: $tType,F: fun(C,fun(B,B)),Pa: fun(C,$o),Xs: list(C)] : foldr(C,B,F,filter2(C,Pa,Xs)) = foldr(C,B,aa(fun(C,$o),fun(C,fun(B,B)),aTP_Lamp_rw(fun(C,fun(B,B)),fun(fun(C,$o),fun(C,fun(B,B))),F),Pa),Xs) ).

% foldr_filter
tff(fact_5022_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_5023_map__restrict__insert__none__simp,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B)),X: C,S2: set(C)] :
      ( ( aa(C,option(B),Ma,X) = none(B) )
     => ( restrict_map(C,B,Ma,aa(set(C),set(C),uminus_uminus(set(C)),aa(set(C),set(C),insert(C,X),S2))) = restrict_map(C,B,Ma,aa(set(C),set(C),uminus_uminus(set(C)),S2)) ) ) ).

% map_restrict_insert_none_simp
tff(fact_5024_restrict__map__upd,axiom,
    ! [C: $tType,B: $tType,F: fun(B,option(C)),S: set(B),K: B,V2: C] : fun_upd(B,option(C),restrict_map(B,C,F,S),K,aa(C,option(C),some(C),V2)) = restrict_map(B,C,fun_upd(B,option(C),F,K,aa(C,option(C),some(C),V2)),aa(set(B),set(B),insert(B,K),S)) ).

% restrict_map_upd
tff(fact_5025_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : pow(X,aa(num,num,bit0,Y)) = sqr(pow(X,Y)) ).

% pow.simps(2)
tff(fact_5026_map__upd__eq__restrict,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),X: B] : fun_upd(B,option(C),Ma,X,none(C)) = restrict_map(B,C,Ma,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) ).

% map_upd_eq_restrict
tff(fact_5027_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_5028_map__of__map__restrict,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),Ks: list(B)] : map_of(B,C,aa(list(B),list(product_prod(B,C)),map(B,product_prod(B,C),aTP_Lamp_rx(fun(B,C),fun(B,product_prod(B,C)),F)),Ks)) = restrict_map(B,C,aa(fun(B,C),fun(B,option(C)),aa(fun(C,option(C)),fun(fun(B,C),fun(B,option(C))),comp(C,option(C),B),some(C)),F),aa(list(B),set(B),set2(B),Ks)) ).

% map_of_map_restrict
tff(fact_5029_nths__Cons,axiom,
    ! [B: $tType,X: B,L: list(B),A4: set(nat)] :
      nths(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L),A4) = aa(list(B),list(B),
        aa(list(B),fun(list(B),list(B)),append(B),
          $ite(aa(set(nat),$o,member(nat,zero_zero(nat)),A4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)),nil(B))),
        nths(B,L,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ry(set(nat),fun(nat,$o),A4)))) ).

% nths_Cons
tff(fact_5030_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,positive,aa(int,rat,aa(int,fun(int,rat),fract,A2),B2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% positive_rat
tff(fact_5031_empty__eq__map__of__iff,axiom,
    ! [C: $tType,B: $tType,Xys: list(product_prod(B,C))] :
      ( ( aTP_Lamp_jv(B,option(C)) = map_of(B,C,Xys) )
    <=> ( Xys = nil(product_prod(B,C)) ) ) ).

% empty_eq_map_of_iff
tff(fact_5032_nths__singleton,axiom,
    ! [B: $tType,X: B,A4: set(nat)] :
      nths(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)),A4) = $ite(aa(set(nat),$o,member(nat,zero_zero(nat)),A4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)),nil(B)) ).

% nths_singleton
tff(fact_5033_less__eq__integer__def,axiom,
    ord_less_eq(code_integer) = aa(fun(int,fun(int,$o)),fun(code_integer,fun(code_integer,$o)),map_fun(code_integer,int,fun(int,$o),fun(code_integer,$o),code_int_of_integer,map_fun(code_integer,int,$o,$o,code_int_of_integer,id($o))),ord_less_eq(int)) ).

% less_eq_integer_def
tff(fact_5034_less__integer__def,axiom,
    ord_less(code_integer) = aa(fun(int,fun(int,$o)),fun(code_integer,fun(code_integer,$o)),map_fun(code_integer,int,fun(int,$o),fun(code_integer,$o),code_int_of_integer,map_fun(code_integer,int,$o,$o,code_int_of_integer,id($o))),ord_less(int)) ).

% less_integer_def
tff(fact_5035_map__of__None__filterD,axiom,
    ! [C: $tType,B: $tType,Xs: list(product_prod(C,B)),X: C,Pa: fun(product_prod(C,B),$o)] :
      ( ( aa(C,option(B),map_of(C,B,Xs),X) = none(B) )
     => ( aa(C,option(B),map_of(C,B,filter2(product_prod(C,B),Pa,Xs)),X) = none(B) ) ) ).

% map_of_None_filterD
tff(fact_5036_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_5037_times__integer__def,axiom,
    times_times(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)),times_times(int)) ).

% times_integer_def
tff(fact_5038_minus__integer__def,axiom,
    minus_minus(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)),minus_minus(int)) ).

% minus_integer_def
tff(fact_5039_divide__integer__def,axiom,
    divide_divide(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)),divide_divide(int)) ).

% divide_integer_def
tff(fact_5040_modulo__integer__def,axiom,
    modulo_modulo(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)),modulo_modulo(int)) ).

% modulo_integer_def
tff(fact_5041_positive__add,axiom,
    ! [X: rat,Y: rat] :
      ( aa(rat,$o,positive,X)
     => ( aa(rat,$o,positive,Y)
       => aa(rat,$o,positive,aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),X),Y)) ) ) ).

% positive_add
tff(fact_5042_and__integer__def,axiom,
    bit_se5824344872417868541ns_and(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)),bit_se5824344872417868541ns_and(int)) ).

% and_integer_def
tff(fact_5043_or__integer__def,axiom,
    bit_se1065995026697491101ons_or(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)),bit_se1065995026697491101ons_or(int)) ).

% or_integer_def
tff(fact_5044_take__bit__integer__def,axiom,
    bit_se2584673776208193580ke_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se2584673776208193580ke_bit(int)) ).

% take_bit_integer_def
tff(fact_5045_xor__integer__def,axiom,
    bit_se5824344971392196577ns_xor(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)),bit_se5824344971392196577ns_xor(int)) ).

% xor_integer_def
tff(fact_5046_drop__bit__integer__def,axiom,
    bit_se4197421643247451524op_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se4197421643247451524op_bit(int)) ).

% drop_bit_integer_def
tff(fact_5047_push__bit__integer__def,axiom,
    bit_se4730199178511100633sh_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se4730199178511100633sh_bit(int)) ).

% push_bit_integer_def
tff(fact_5048_nths__all,axiom,
    ! [B: $tType,Xs: list(B),I5: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Xs))
         => aa(set(nat),$o,member(nat,I2),I5) )
     => ( nths(B,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5049_sorted__nths,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),I5: set(nat)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),nths(B,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_5050_unset__bit__integer__def,axiom,
    bit_se2638667681897837118et_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se2638667681897837118et_bit(int)) ).

% unset_bit_integer_def
tff(fact_5051_set__bit__integer__def,axiom,
    bit_se5668285175392031749et_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se5668285175392031749et_bit(int)) ).

% set_bit_integer_def
tff(fact_5052_flip__bit__integer__def,axiom,
    bit_se8732182000553998342ip_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se8732182000553998342ip_bit(int)) ).

% flip_bit_integer_def
tff(fact_5053_drop__eq__nths,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : drop(B,N,Xs) = nths(B,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_5054_nat__of__integer__def,axiom,
    code_nat_of_integer = aa(fun(int,nat),fun(code_integer,nat),map_fun(code_integer,int,nat,nat,code_int_of_integer,id(nat)),nat2) ).

% nat_of_integer_def
tff(fact_5055_map__of__Some__split,axiom,
    ! [C: $tType,B: $tType,Xs: list(product_prod(C,B)),K: C,V2: B] :
      ( ( aa(C,option(B),map_of(C,B,Xs),K) = aa(B,option(B),some(B),V2) )
     => ? [Ys3: list(product_prod(C,B)),Zs2: list(product_prod(C,B))] :
          ( ( Xs = aa(list(product_prod(C,B)),list(product_prod(C,B)),aa(list(product_prod(C,B)),fun(list(product_prod(C,B)),list(product_prod(C,B))),append(product_prod(C,B)),Ys3),aa(list(product_prod(C,B)),list(product_prod(C,B)),aa(product_prod(C,B),fun(list(product_prod(C,B)),list(product_prod(C,B))),cons(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),K),V2)),Zs2)) )
          & ( aa(C,option(B),map_of(C,B,Ys3),K) = none(B) ) ) ) ).

% map_of_Some_split
tff(fact_5056_nths__append,axiom,
    ! [B: $tType,L: list(B),L3: list(B),A4: set(nat)] : nths(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L3),A4) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),nths(B,L,A4)),nths(B,L3,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_rz(list(B),fun(set(nat),fun(nat,$o)),L),A4)))) ).

% nths_append
tff(fact_5057_nths__drop,axiom,
    ! [B: $tType,N: nat,Xs: list(B),I5: set(nat)] : nths(B,drop(B,N,Xs),I5) = nths(B,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I5)) ).

% nths_drop
tff(fact_5058_filter__in__nths,axiom,
    ! [B: $tType,Xs: list(B),S2: set(nat)] :
      ( distinct(B,Xs)
     => ( filter2(B,aa(set(nat),fun(B,$o),aTP_Lamp_sa(list(B),fun(set(nat),fun(B,$o)),Xs),S2),Xs) = nths(B,Xs,S2) ) ) ).

% filter_in_nths
tff(fact_5059_bit__integer__def,axiom,
    bit_se5641148757651400278ts_bit(code_integer) = aa(fun(int,fun(nat,$o)),fun(code_integer,fun(nat,$o)),map_fun(code_integer,int,fun(nat,$o),fun(nat,$o),code_int_of_integer,id(fun(nat,$o))),bit_se5641148757651400278ts_bit(int)) ).

% bit_integer_def
tff(fact_5060_length__nths,axiom,
    ! [B: $tType,Xs: list(B),I5: set(nat)] : aa(list(B),nat,size_size(list(B)),nths(B,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_sb(list(B),fun(set(nat),fun(nat,$o)),Xs),I5))) ).

% length_nths
tff(fact_5061_uminus__integer__def,axiom,
    uminus_uminus(code_integer) = 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),uminus_uminus(int)) ).

% uminus_integer_def
tff(fact_5062_abs__integer__def,axiom,
    abs_abs(code_integer) = 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),abs_abs(int)) ).

% abs_integer_def
tff(fact_5063_map__of__map,axiom,
    ! [C: $tType,D: $tType,B: $tType,F: fun(D,C),Xs: list(product_prod(B,D))] : map_of(B,C,aa(list(product_prod(B,D)),list(product_prod(B,C)),map(product_prod(B,D),product_prod(B,C),aa(fun(B,fun(D,product_prod(B,C))),fun(product_prod(B,D),product_prod(B,C)),product_case_prod(B,D,product_prod(B,C)),aTP_Lamp_sc(fun(D,C),fun(B,fun(D,product_prod(B,C))),F))),Xs)) = aa(fun(B,option(D)),fun(B,option(C)),aa(fun(option(D),option(C)),fun(fun(B,option(D)),fun(B,option(C))),comp(option(D),option(C),B),map_option(D,C,F)),map_of(B,D,Xs)) ).

% map_of_map
tff(fact_5064_mask__integer__def,axiom,
    bit_se2239418461657761734s_mask(code_integer) = aa(fun(nat,int),fun(nat,code_integer),map_fun(nat,nat,int,code_integer,id(nat),code_integer_of_int),bit_se2239418461657761734s_mask(int)) ).

% mask_integer_def
tff(fact_5065_sgn__integer__def,axiom,
    sgn_sgn(code_integer) = 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),sgn_sgn(int)) ).

% sgn_integer_def
tff(fact_5066_not__integer__def,axiom,
    bit_ri4277139882892585799ns_not(code_integer) = 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),bit_ri4277139882892585799ns_not(int)) ).

% not_integer_def
tff(fact_5067_filter__eq__nths,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : filter2(B,Pa,Xs) = nths(B,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(list(B),fun(nat,$o),aTP_Lamp_pk(fun(B,$o),fun(list(B),fun(nat,$o)),Pa),Xs))) ).

% filter_eq_nths
tff(fact_5068_num__of__integer__def,axiom,
    code_num_of_integer = aa(fun(int,num),fun(code_integer,num),map_fun(code_integer,int,num,num,code_int_of_integer,id(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)) ).

% num_of_integer_def
tff(fact_5069_mset__zip__take__Cons__drop__twice,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),J: nat,X: B,Y: C] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(B),nat,size_size(list(B)),Xs))
       => ( mset(product_prod(B,C),zip(B,C,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J,Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),drop(B,J,Xs))),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),take(C,J,Ys)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y),drop(C,J,Ys))))) = aa(multiset(product_prod(B,C)),multiset(product_prod(B,C)),aa(product_prod(B,C),fun(multiset(product_prod(B,C)),multiset(product_prod(B,C))),add_mset(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y)),mset(product_prod(B,C),zip(B,C,Xs,Ys))) ) ) ) ).

% mset_zip_take_Cons_drop_twice
tff(fact_5070_dual__min,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( min(B,aTP_Lamp_kh(B,fun(B,$o))) = ord_max(B) ) ) ).

% dual_min
tff(fact_5071_mask__mod__exp,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat,Ma: nat] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)),one_one(B))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),N))),one_one(B)) ) ).

% mask_mod_exp
tff(fact_5072_subset__mset_Omin__arg__le_I1_J,axiom,
    ! [B: $tType,N: multiset(B),Ma: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),N),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),Ma),N))
    <=> ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),Ma),N) = N ) ) ).

% subset_mset.min_arg_le(1)
tff(fact_5073_subset__mset_Omin__arg__le_I2_J,axiom,
    ! [B: $tType,Ma: multiset(B),N: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Ma),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),Ma),N))
    <=> ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),Ma),N) = Ma ) ) ).

% subset_mset.min_arg_le(2)
tff(fact_5074_min_Obounded__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),C2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% min.bounded_iff
tff(fact_5075_min_Oabsorb2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5076_min_Oabsorb1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_5077_min__arg__le_I2_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Ma: B,N: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N))
        <=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N) = Ma ) ) ) ).

% min_arg_le(2)
tff(fact_5078_min__arg__le_I1_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [N: B,Ma: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N),aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N))
        <=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N) = N ) ) ) ).

% min_arg_le(1)
tff(fact_5079_min__eq__arg_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ma: B,N: B] :
          ( ( aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N) = N )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N),Ma) ) ) ).

% min_eq_arg(2)
tff(fact_5080_min__eq__arg_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ma: B,N: B] :
          ( ( aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N) = Ma )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),N) ) ) ).

% min_eq_arg(1)
tff(fact_5081_min__arg__not__ge_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ma: B,N: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N)),Ma)
        <=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N) = Ma ) ) ) ).

% min_arg_not_ge(1)
tff(fact_5082_min__arg__not__ge_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Ma: B,N: B] :
          ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N)),N)
        <=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),Ma),N) = N ) ) ) ).

% min_arg_not_ge(2)
tff(fact_5083_min__less__self__conv_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ).

% min_less_self_conv(1)
tff(fact_5084_min__less__self__conv_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2) ) ) ).

% min_less_self_conv(2)
tff(fact_5085_min__simps_I1_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = A2 ) ) ) ).

% min_simps(1)
tff(fact_5086_min__simps_I2_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_5087_min__bot,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_min(B),bot_bot(B)),X) = bot_bot(B) ) ).

% min_bot
tff(fact_5088_min__bot2,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_min(B),X),bot_bot(B)) = bot_bot(B) ) ).

% min_bot2
tff(fact_5089_zip__eq__zip__same__len,axiom,
    ! [B: $tType,C: $tType,A2: list(B),B2: list(C),A5: list(B),B5: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),A2) = aa(list(C),nat,size_size(list(C)),B2) )
     => ( ( aa(list(B),nat,size_size(list(B)),A5) = aa(list(C),nat,size_size(list(C)),B5) )
       => ( ( zip(B,C,A2,B2) = zip(B,C,A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_5090_max__min__same_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),X),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)) = X ) ).

% max_min_same(1)
tff(fact_5091_max__min__same_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),X) = X ) ).

% max_min_same(2)
tff(fact_5092_max__min__same_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_5093_max__min__same_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Y: B,X: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),Y),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)) = Y ) ).

% max_min_same(4)
tff(fact_5094_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_5095_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_5096_min__Suc__Suc,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Ma)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),N)) ).

% min_Suc_Suc
tff(fact_5097_min__number__of_I1_J,axiom,
    ! [B: $tType] :
      ( ( numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_min(B),aa(num,B,numeral_numeral(B),U)),aa(num,B,numeral_numeral(B),V2)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),U)),aa(num,B,numeral_numeral(B),V2)),aa(num,B,numeral_numeral(B),U),aa(num,B,numeral_numeral(B),V2)) ) ).

% min_number_of(1)
tff(fact_5098_min__0__1_I4_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_min(B),aa(num,B,numeral_numeral(B),X)),zero_zero(B)) = zero_zero(B) ) ).

% min_0_1(4)
tff(fact_5099_min__0__1_I3_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_min(B),zero_zero(B)),aa(num,B,numeral_numeral(B),X)) = zero_zero(B) ) ).

% min_0_1(3)
tff(fact_5100_min__0__1_I1_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ( aa(B,B,aa(B,fun(B,B),ord_min(B),zero_zero(B)),one_one(B)) = zero_zero(B) ) ) ).

% min_0_1(1)
tff(fact_5101_min__0__1_I2_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ( aa(B,B,aa(B,fun(B,B),ord_min(B),one_one(B)),zero_zero(B)) = zero_zero(B) ) ) ).

% min_0_1(2)
tff(fact_5102_min__0__1_I6_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_min(B),aa(num,B,numeral_numeral(B),X)),one_one(B)) = one_one(B) ) ).

% min_0_1(6)
tff(fact_5103_min__0__1_I5_J,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: num] : aa(B,B,aa(B,fun(B,B),ord_min(B),one_one(B)),aa(num,B,numeral_numeral(B),X)) = one_one(B) ) ).

% min_0_1(5)
tff(fact_5104_Int__atMost,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atMost(B),A2)),aa(B,set(B),set_ord_atMost(B),B2)) = aa(B,set(B),set_ord_atMost(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)) ) ).

% Int_atMost
tff(fact_5105_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A2)) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_5106_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A2)),B2) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_5107_length__take,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),take(B,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),N) ).

% length_take
tff(fact_5108_min__number__of_I4_J,axiom,
    ! [B: $tType] :
      ( ( uminus(B)
        & numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) ) ).

% min_number_of(4)
tff(fact_5109_min__number__of_I3_J,axiom,
    ! [B: $tType] :
      ( ( uminus(B)
        & numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(num,B,numeral_numeral(B),V2)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U))),aa(num,B,numeral_numeral(B),V2)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),U)),aa(num,B,numeral_numeral(B),V2)) ) ).

% min_number_of(3)
tff(fact_5110_min__number__of_I2_J,axiom,
    ! [B: $tType] :
      ( ( uminus(B)
        & numeral(B)
        & ord(B) )
     => ! [U: num,V2: num] :
          aa(B,B,aa(B,fun(B,B),ord_min(B),aa(num,B,numeral_numeral(B),U)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),U)),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))),aa(num,B,numeral_numeral(B),U),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),V2))) ) ).

% min_number_of(2)
tff(fact_5111_Int__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),set_or1337092689740270186AtMost(B,C2,D2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),ord_max(B),A2),C2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),D2)) ) ).

% Int_atLeastAtMost
tff(fact_5112_Int__atLeastLessThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or7035219750837199246ssThan(B,A2,B2)),set_or7035219750837199246ssThan(B,C2,D2)) = set_or7035219750837199246ssThan(B,aa(B,B,aa(B,fun(B,B),ord_max(B),A2),C2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),D2)) ) ).

% Int_atLeastLessThan
tff(fact_5113_Int__atLeastAtMostR1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,C2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atMost(B),B2)),set_or1337092689740270186AtMost(B,C2,D2)) = set_or1337092689740270186AtMost(B,C2,aa(B,B,aa(B,fun(B,B),ord_min(B),B2),D2)) ) ).

% Int_atLeastAtMostR1
tff(fact_5114_Int__atLeastAtMostL1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),aa(B,set(B),set_ord_atMost(B),D2)) = set_or1337092689740270186AtMost(B,A2,aa(B,B,aa(B,fun(B,B),ord_min(B),B2),D2)) ) ).

% Int_atLeastAtMostL1
tff(fact_5115_Int__greaterThanAtMost,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or3652927894154168847AtMost(B,A2,B2)),set_or3652927894154168847AtMost(B,C2,D2)) = set_or3652927894154168847AtMost(B,aa(B,B,aa(B,fun(B,B),ord_max(B),A2),C2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),D2)) ) ).

% Int_greaterThanAtMost
tff(fact_5116_Int__greaterThanLessThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or5935395276787703475ssThan(B,A2,B2)),set_or5935395276787703475ssThan(B,C2,D2)) = set_or5935395276787703475ssThan(B,aa(B,B,aa(B,fun(B,B),ord_max(B),A2),C2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),D2)) ) ).

% Int_greaterThanLessThan
tff(fact_5117_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),N)) ).

% min_numeral_Suc
tff(fact_5118_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_5119_zip__append,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Us: list(C),Ys: list(B),Vs: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Us) )
     => ( zip(B,C,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Us),Vs)) = aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),zip(B,C,Xs,Us)),zip(B,C,Ys,Vs)) ) ) ).

% zip_append
tff(fact_5120_length__zip,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] : aa(list(product_prod(B,C)),nat,size_size(list(product_prod(B,C))),zip(B,C,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(C),nat,size_size(list(C)),Ys)) ).

% length_zip
tff(fact_5121_map__of__zip__is__None,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C),X: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( ( aa(B,option(C),map_of(B,C,zip(B,C,Xs,Ys)),X) = none(C) )
      <=> ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs)) ) ) ).

% map_of_zip_is_None
tff(fact_5122_nth__zip,axiom,
    ! [B: $tType,C: $tType,I: nat,Xs: list(B),Ys: list(C)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(C),nat,size_size(list(C)),Ys))
       => ( aa(nat,product_prod(B,C),nth(product_prod(B,C),zip(B,C,Xs,Ys)),I) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(nat,B,nth(B,Xs),I)),aa(nat,C,nth(C,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5123_min__def__raw,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X4: B,Xa: B] :
          aa(B,B,aa(B,fun(B,B),ord_min(B),X4),Xa) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Xa),X4,Xa) ) ).

% min_def_raw
tff(fact_5124_min__le__iff__disj,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),Z)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_5125_min_OcoboundedI2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_5126_min_OcoboundedI1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_5127_min_Oabsorb__iff2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5128_min_Oabsorb__iff1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_5129_min_Ocobounded2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),B2) ) ).

% min.cobounded2
tff(fact_5130_min_Ocobounded1,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),A2) ) ).

% min.cobounded1
tff(fact_5131_min_Oorder__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( A2 = aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_5132_min_OboundedI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),C2)) ) ) ) ).

% min.boundedI
tff(fact_5133_min_OboundedE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),ord_min(B),B2),C2))
         => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2) ) ) ) ).

% min.boundedE
tff(fact_5134_min_OorderI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2) ) ) ).

% min.orderI
tff(fact_5135_min_OorderE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( A2 = aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) ) ) ) ).

% min.orderE
tff(fact_5136_min_Omono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)),aa(B,B,aa(B,fun(B,B),ord_min(B),C2),D2)) ) ) ) ).

% min.mono
tff(fact_5137_min__absorb2,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_5138_min__absorb1,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_5139_min__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B] :
          aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2),A2,B2) ) ).

% min_def
tff(fact_5140_of__int__min,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: int,Y: int] : aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),ord_min(int),X),Y)) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(int,B,ring_1_of_int(B),X)),aa(int,B,ring_1_of_int(B),Y)) ) ).

% of_int_min
tff(fact_5141_zip__obtain__same__length,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Pa: fun(list(product_prod(B,C)),$o)] :
      ( ! [Zs2: list(B),Ws2: list(C),N2: nat] :
          ( ( aa(list(B),nat,size_size(list(B)),Zs2) = aa(list(C),nat,size_size(list(C)),Ws2) )
         => ( ( N2 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(C),nat,size_size(list(C)),Ys)) )
           => ( ( Zs2 = take(B,N2,Xs) )
             => ( ( Ws2 = take(C,N2,Ys) )
               => aa(list(product_prod(B,C)),$o,Pa,zip(B,C,Zs2,Ws2)) ) ) ) )
     => aa(list(product_prod(B,C)),$o,Pa,zip(B,C,Xs,Ys)) ) ).

% zip_obtain_same_length
tff(fact_5142_nat__mult__min__left,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_min_left
tff(fact_5143_nat__mult__min__right,axiom,
    ! [Ma: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ma),Q2)) ).

% nat_mult_min_right
tff(fact_5144_ord_Omin_Ocong,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o))] : min(B,Less_eq) = min(B,Less_eq) ).

% ord.min.cong
tff(fact_5145_ord_Omin__def,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),A2: B,B2: B] :
      aa(B,B,aa(B,fun(B,B),min(B,Less_eq),A2),B2) = $ite(aa(B,$o,aa(B,fun(B,$o),Less_eq,A2),B2),A2,B2) ).

% ord.min_def
tff(fact_5146_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5147_zip__inj,axiom,
    ! [B: $tType,C: $tType,A2: list(B),B2: list(C),A5: list(B),B5: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),A2) = aa(list(C),nat,size_size(list(C)),B2) )
     => ( ( aa(list(B),nat,size_size(list(B)),A5) = aa(list(C),nat,size_size(list(C)),B5) )
       => ( ( zip(B,C,A2,B2) = zip(B,C,A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_5148_min__add__distrib__left,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [X: B,Y: B,Z: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),Z) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Z)),aa(B,B,aa(B,fun(B,B),plus_plus(B),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_5149_min__add__distrib__right,axiom,
    ! [B: $tType] :
      ( ordere2412721322843649153imp_le(B)
     => ! [X: B,Y: B,Z: B] : aa(B,B,aa(B,fun(B,B),plus_plus(B),X),aa(B,B,aa(B,fun(B,B),ord_min(B),Y),Z)) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)),aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Z)) ) ).

% min_add_distrib_right
tff(fact_5150_min__diff__distrib__left,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B,Y: B,Z: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),Z) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X),Z)),aa(B,B,aa(B,fun(B,B),minus_minus(B),Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_5151_min__diff,axiom,
    ! [Ma: nat,I: 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),Ma),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),N)),I) ).

% min_diff
tff(fact_5152_zip__assoc,axiom,
    ! [C: $tType,B: $tType,D: $tType,Xs: list(B),Ys: list(C),Zs: list(D)] : zip(B,product_prod(C,D),Xs,zip(C,D,Ys,Zs)) = aa(list(product_prod(product_prod(B,C),D)),list(product_prod(B,product_prod(C,D))),map(product_prod(product_prod(B,C),D),product_prod(B,product_prod(C,D)),aa(fun(product_prod(B,C),fun(D,product_prod(B,product_prod(C,D)))),fun(product_prod(product_prod(B,C),D),product_prod(B,product_prod(C,D))),product_case_prod(product_prod(B,C),D,product_prod(B,product_prod(C,D))),aa(fun(B,fun(C,fun(D,product_prod(B,product_prod(C,D))))),fun(product_prod(B,C),fun(D,product_prod(B,product_prod(C,D)))),product_case_prod(B,C,fun(D,product_prod(B,product_prod(C,D)))),aTP_Lamp_sd(B,fun(C,fun(D,product_prod(B,product_prod(C,D)))))))),zip(product_prod(B,C),D,zip(B,C,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_5153_of__nat__min,axiom,
    ! [B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [X: nat,Y: nat] : aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(nat,B,semiring_1_of_nat(B),X)),aa(nat,B,semiring_1_of_nat(B),Y)) ) ).

% of_nat_min
tff(fact_5154_minus__max__eq__min,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [X: B,Y: B] : aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,uminus_uminus(B),X)),aa(B,B,uminus_uminus(B),Y)) ) ).

% minus_max_eq_min
tff(fact_5155_minus__min__eq__max,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [X: B,Y: B] : aa(B,B,uminus_uminus(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,uminus_uminus(B),X)),aa(B,B,uminus_uminus(B),Y)) ) ).

% minus_min_eq_max
tff(fact_5156_ex__mset__zip__left,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Xs5: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( ( mset(B,Xs5) = mset(B,Xs) )
       => ? [Ys5: list(C)] :
            ( ( aa(list(C),nat,size_size(list(C)),Ys5) = aa(list(B),nat,size_size(list(B)),Xs5) )
            & ( mset(product_prod(B,C),zip(B,C,Xs5,Ys5)) = mset(product_prod(B,C),zip(B,C,Xs,Ys)) ) ) ) ) ).

% ex_mset_zip_left
tff(fact_5157_zip__rev,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( zip(B,C,rev(B,Xs),rev(C,Ys)) = rev(product_prod(B,C),zip(B,C,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_5158_greaterThan__Int__greaterThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_lessThan(B),A2)),aa(B,set(B),set_ord_lessThan(B),B2)) = aa(B,set(B),set_ord_lessThan(B),aa(B,B,aa(B,fun(B,B),ord_min(B),A2),B2)) ) ).

% greaterThan_Int_greaterThan
tff(fact_5159_zip__Cons1,axiom,
    ! [B: $tType,C: $tType,X: B,Xs: list(B),Ys: list(C)] : zip(B,C,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),Ys) = aa(list(C),list(product_prod(B,C)),case_list(list(product_prod(B,C)),C,nil(product_prod(B,C)),aa(list(B),fun(C,fun(list(C),list(product_prod(B,C)))),aTP_Lamp_se(B,fun(list(B),fun(C,fun(list(C),list(product_prod(B,C))))),X),Xs)),Ys) ).

% zip_Cons1
tff(fact_5160_zip__Cons,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Y: C,Ys: list(C)] : zip(B,C,Xs,aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y),Ys)) = aa(list(B),list(product_prod(B,C)),case_list(list(product_prod(B,C)),B,nil(product_prod(B,C)),aa(list(C),fun(B,fun(list(B),list(product_prod(B,C)))),aTP_Lamp_sf(C,fun(list(C),fun(B,fun(list(B),list(product_prod(B,C))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_5161_zip__left__commute,axiom,
    ! [C: $tType,B: $tType,D: $tType,Xs: list(B),Ys: list(C),Zs: list(D)] : zip(B,product_prod(C,D),Xs,zip(C,D,Ys,Zs)) = aa(list(product_prod(C,product_prod(B,D))),list(product_prod(B,product_prod(C,D))),map(product_prod(C,product_prod(B,D)),product_prod(B,product_prod(C,D)),aa(fun(C,fun(product_prod(B,D),product_prod(B,product_prod(C,D)))),fun(product_prod(C,product_prod(B,D)),product_prod(B,product_prod(C,D))),product_case_prod(C,product_prod(B,D),product_prod(B,product_prod(C,D))),aTP_Lamp_sh(C,fun(product_prod(B,D),product_prod(B,product_prod(C,D)))))),zip(C,product_prod(B,D),Ys,zip(B,D,Xs,Zs))) ).

% zip_left_commute
tff(fact_5162_map2__map__map,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,H: fun(C,fun(D,B)),F: fun(E,C),Xs: list(E),G: fun(E,D)] : aa(list(product_prod(C,D)),list(B),map(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),H)),zip(C,D,aa(list(E),list(C),map(E,C,F),Xs),aa(list(E),list(D),map(E,D,G),Xs))) = aa(list(E),list(B),map(E,B,aa(fun(E,D),fun(E,B),aa(fun(E,C),fun(fun(E,D),fun(E,B)),aTP_Lamp_si(fun(C,fun(D,B)),fun(fun(E,C),fun(fun(E,D),fun(E,B))),H),F),G)),Xs) ).

% map2_map_map
tff(fact_5163_zip__commute,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C)] : zip(B,C,Xs,Ys) = aa(list(product_prod(C,B)),list(product_prod(B,C)),map(product_prod(C,B),product_prod(B,C),aa(fun(C,fun(B,product_prod(B,C))),fun(product_prod(C,B),product_prod(B,C)),product_case_prod(C,B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))))),zip(C,B,Ys,Xs)) ).

% zip_commute
tff(fact_5164_zip__same__conv__map,axiom,
    ! [B: $tType,Xs: list(B)] : zip(B,B,Xs,Xs) = aa(list(B),list(product_prod(B,B)),map(B,product_prod(B,B),aTP_Lamp_sk(B,product_prod(B,B))),Xs) ).

% zip_same_conv_map
tff(fact_5165_pair__list__split,axiom,
    ! [B: $tType,C: $tType,L: list(product_prod(B,C))] :
      ~ ! [L12: list(B),L23: list(C)] :
          ( ( L = zip(B,C,L12,L23) )
         => ( ( aa(list(B),nat,size_size(list(B)),L12) = aa(list(C),nat,size_size(list(C)),L23) )
           => ( aa(list(product_prod(B,C)),nat,size_size(list(product_prod(B,C))),L) != aa(list(C),nat,size_size(list(C)),L23) ) ) ) ).

% pair_list_split
tff(fact_5166_map__of__zip__inject,axiom,
    ! [C: $tType,B: $tType,Ys: list(B),Xs: list(C),Zs: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(C),nat,size_size(list(C)),Xs) )
       => ( distinct(C,Xs)
         => ( ( map_of(C,B,zip(C,B,Xs,Ys)) = map_of(C,B,zip(C,B,Xs,Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_5167_map__zip__map2,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,F: fun(product_prod(C,D),B),Xs: list(C),G: fun(E,D),Ys: list(E)] : aa(list(product_prod(C,D)),list(B),map(product_prod(C,D),B,F),zip(C,D,Xs,aa(list(E),list(D),map(E,D,G),Ys))) = aa(list(product_prod(C,E)),list(B),map(product_prod(C,E),B,aa(fun(C,fun(E,B)),fun(product_prod(C,E),B),product_case_prod(C,E,B),aa(fun(E,D),fun(C,fun(E,B)),aTP_Lamp_sl(fun(product_prod(C,D),B),fun(fun(E,D),fun(C,fun(E,B))),F),G))),zip(C,E,Xs,Ys)) ).

% map_zip_map2
tff(fact_5168_map__zip__map,axiom,
    ! [C: $tType,B: $tType,E: $tType,D: $tType,F: fun(product_prod(C,D),B),G: fun(E,C),Xs: list(E),Ys: list(D)] : aa(list(product_prod(C,D)),list(B),map(product_prod(C,D),B,F),zip(C,D,aa(list(E),list(C),map(E,C,G),Xs),Ys)) = aa(list(product_prod(E,D)),list(B),map(product_prod(E,D),B,aa(fun(E,fun(D,B)),fun(product_prod(E,D),B),product_case_prod(E,D,B),aa(fun(E,C),fun(E,fun(D,B)),aTP_Lamp_sm(fun(product_prod(C,D),B),fun(fun(E,C),fun(E,fun(D,B))),F),G))),zip(E,D,Xs,Ys)) ).

% map_zip_map
tff(fact_5169_subset__mset_Omin__add__distrib__right,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B),Z: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),Y),Z)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Y)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Z)) ).

% subset_mset.min_add_distrib_right
tff(fact_5170_subset__mset_Omin__add__distrib__left,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B),Z: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),X),Y)),Z) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),min(multiset(B),subseteq_mset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Z)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Y),Z)) ).

% subset_mset.min_add_distrib_left
tff(fact_5171_map__prod__fun__zip,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,F: fun(D,B),G: fun(E,C),Xs: list(D),Ys: list(E)] : aa(list(product_prod(D,E)),list(product_prod(B,C)),map(product_prod(D,E),product_prod(B,C),aa(fun(D,fun(E,product_prod(B,C))),fun(product_prod(D,E),product_prod(B,C)),product_case_prod(D,E,product_prod(B,C)),aa(fun(E,C),fun(D,fun(E,product_prod(B,C))),aTP_Lamp_sn(fun(D,B),fun(fun(E,C),fun(D,fun(E,product_prod(B,C)))),F),G))),zip(D,E,Xs,Ys)) = zip(B,C,aa(list(D),list(B),map(D,B,F),Xs),aa(list(E),list(C),map(E,C,G),Ys)) ).

% map_prod_fun_zip
tff(fact_5172_zip__map2,axiom,
    ! [C: $tType,B: $tType,D: $tType,Xs: list(B),F: fun(D,C),Ys: list(D)] : zip(B,C,Xs,aa(list(D),list(C),map(D,C,F),Ys)) = aa(list(product_prod(B,D)),list(product_prod(B,C)),map(product_prod(B,D),product_prod(B,C),aa(fun(B,fun(D,product_prod(B,C))),fun(product_prod(B,D),product_prod(B,C)),product_case_prod(B,D,product_prod(B,C)),aTP_Lamp_sc(fun(D,C),fun(B,fun(D,product_prod(B,C))),F))),zip(B,D,Xs,Ys)) ).

% zip_map2
tff(fact_5173_zip__map1,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(D,B),Xs: list(D),Ys: list(C)] : zip(B,C,aa(list(D),list(B),map(D,B,F),Xs),Ys) = aa(list(product_prod(D,C)),list(product_prod(B,C)),map(product_prod(D,C),product_prod(B,C),aa(fun(D,fun(C,product_prod(B,C))),fun(product_prod(D,C),product_prod(B,C)),product_case_prod(D,C,product_prod(B,C)),aTP_Lamp_so(fun(D,B),fun(D,fun(C,product_prod(B,C))),F))),zip(D,C,Xs,Ys)) ).

% zip_map1
tff(fact_5174_max__mult__distrib__left,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [P2: B,X: B,Y: B] :
          aa(B,B,aa(B,fun(B,B),times_times(B),P2),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),P2),aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),times_times(B),P2),X)),aa(B,B,aa(B,fun(B,B),times_times(B),P2),Y)),aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),times_times(B),P2),X)),aa(B,B,aa(B,fun(B,B),times_times(B),P2),Y))) ) ).

% max_mult_distrib_left
tff(fact_5175_min__mult__distrib__left,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [P2: B,X: B,Y: B] :
          aa(B,B,aa(B,fun(B,B),times_times(B),P2),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),P2),aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),times_times(B),P2),X)),aa(B,B,aa(B,fun(B,B),times_times(B),P2),Y)),aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),times_times(B),P2),X)),aa(B,B,aa(B,fun(B,B),times_times(B),P2),Y))) ) ).

% min_mult_distrib_left
tff(fact_5176_max__mult__distrib__right,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B,P2: B] :
          aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),P2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),P2),aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),P2)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),P2)),aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),P2)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),P2))) ) ).

% max_mult_distrib_right
tff(fact_5177_min__mult__distrib__right,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [X: B,Y: B,P2: B] :
          aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),P2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),P2),aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),P2)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),P2)),aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),times_times(B),X),P2)),aa(B,B,aa(B,fun(B,B),times_times(B),Y),P2))) ) ).

% min_mult_distrib_right
tff(fact_5178_max__divide__distrib__right,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B,P2: B] :
          aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),P2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),P2),aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),P2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),P2)),aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),P2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),P2))) ) ).

% max_divide_distrib_right
tff(fact_5179_min__divide__distrib__right,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [X: B,Y: B,P2: B] :
          aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),ord_min(B),X),Y)),P2) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),P2),aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),P2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),P2)),aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),X),P2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),Y),P2))) ) ).

% min_divide_distrib_right
tff(fact_5180_min__Suc1,axiom,
    ! [N: nat,Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),Ma) = case_nat(nat,zero_zero(nat),aTP_Lamp_sp(nat,fun(nat,nat),N),Ma) ).

% min_Suc1
tff(fact_5181_min__Suc2,axiom,
    ! [Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ma),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_sq(nat,fun(nat,nat),N),Ma) ).

% min_Suc2
tff(fact_5182_last__zip,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] :
      ( ( Xs != nil(B) )
     => ( ( Ys != nil(C) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
         => ( last(product_prod(B,C),zip(B,C,Xs,Ys)) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),last(B,Xs)),last(C,Ys)) ) ) ) ) ).

% last_zip
tff(fact_5183_in__set__impl__in__set__zip2,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Y: C] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(set(C),$o,member(C,Y),aa(list(C),set(C),set2(C),Ys))
       => ~ ! [X2: B] : ~ aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y)),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_5184_in__set__impl__in__set__zip1,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),X: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
       => ~ ! [Y2: C] : ~ aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y2)),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_5185_map__of__zip__is__Some,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),X: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
      <=> ? [Y5: C] : aa(B,option(C),map_of(B,C,zip(B,C,Xs,Ys)),X) = aa(C,option(C),some(C),Y5) ) ) ).

% map_of_zip_is_Some
tff(fact_5186_map__upds__fold__map__upd,axiom,
    ! [B: $tType,C: $tType,Ma: fun(B,option(C)),Ks: list(B),Vs: list(C)] : map_upds(B,C,Ma,Ks,Vs) = aa(list(product_prod(B,C)),fun(B,option(C)),aa(fun(B,option(C)),fun(list(product_prod(B,C)),fun(B,option(C))),foldl(fun(B,option(C)),product_prod(B,C),aTP_Lamp_ss(fun(B,option(C)),fun(product_prod(B,C),fun(B,option(C))))),Ma),zip(B,C,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_5187_enumerate__eq__zip,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : enumerate(B,N,Xs) = zip(nat,B,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(B),nat,size_size(list(B)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_5188_foldl__snd__zip,axiom,
    ! [C: $tType,D: $tType,B: $tType,Ys: list(B),Xs: list(C),F: fun(D,fun(B,D)),B2: D] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Ys)),aa(list(C),nat,size_size(list(C)),Xs))
     => ( aa(list(product_prod(C,B)),D,aa(D,fun(list(product_prod(C,B)),D),foldl(D,product_prod(C,B),aTP_Lamp_su(fun(D,fun(B,D)),fun(D,fun(product_prod(C,B),D)),F)),B2),zip(C,B,Xs,Ys)) = aa(list(B),D,aa(D,fun(list(B),D),foldl(D,B,F),B2),Ys) ) ) ).

% foldl_snd_zip
tff(fact_5189_map__zip2,axiom,
    ! [B: $tType,C: $tType,K: B,L: list(C)] : aa(list(C),list(product_prod(B,C)),map(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),K)),L) = zip(B,C,replicate(B,aa(list(C),nat,size_size(list(C)),L),K),L) ).

% map_zip2
tff(fact_5190_map__zip1,axiom,
    ! [B: $tType,C: $tType,K: C,L: list(B)] : aa(list(B),list(product_prod(B,C)),map(B,product_prod(B,C),aa(C,fun(B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))),K)),L) = zip(B,C,L,replicate(C,aa(list(B),nat,size_size(list(B)),L),K)) ).

% map_zip1
tff(fact_5191_foldr__snd__zip,axiom,
    ! [C: $tType,B: $tType,D: $tType,Ys: list(B),Xs: list(C),F: fun(B,fun(D,D)),B2: D] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Ys)),aa(list(C),nat,size_size(list(C)),Xs))
     => ( aa(D,D,foldr(product_prod(C,B),D,aa(fun(C,fun(B,fun(D,D))),fun(product_prod(C,B),fun(D,D)),product_case_prod(C,B,fun(D,D)),aTP_Lamp_sv(fun(B,fun(D,D)),fun(C,fun(B,fun(D,D))),F)),zip(C,B,Xs,Ys)),B2) = aa(D,D,foldr(B,D,F,Ys),B2) ) ) ).

% foldr_snd_zip
tff(fact_5192_zip__replicate2,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),N: nat,Y: C] : zip(B,C,Xs,replicate(C,N,Y)) = aa(list(B),list(product_prod(B,C)),map(B,product_prod(B,C),aa(C,fun(B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))),Y)),take(B,N,Xs)) ).

% zip_replicate2
tff(fact_5193_map__of__zip__upd,axiom,
    ! [C: $tType,B: $tType,Ys: list(B),Xs: list(C),Zs: list(B),X: C,Y: B,Z: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(C),nat,size_size(list(C)),Xs) )
       => ( ~ aa(set(C),$o,member(C,X),aa(list(C),set(C),set2(C),Xs))
         => ( ( fun_upd(C,option(B),map_of(C,B,zip(C,B,Xs,Ys)),X,aa(B,option(B),some(B),Y)) = fun_upd(C,option(B),map_of(C,B,zip(C,B,Xs,Zs)),X,aa(B,option(B),some(B),Z)) )
           => ( map_of(C,B,zip(C,B,Xs,Ys)) = map_of(C,B,zip(C,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_5194_ran__map__of__zip,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( distinct(B,Xs)
       => ( ran(B,C,map_of(B,C,zip(B,C,Xs,Ys))) = aa(list(C),set(C),set2(C),Ys) ) ) ) ).

% ran_map_of_zip
tff(fact_5195_zip__append1,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(B),Zs: list(C)] : zip(B,C,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys),Zs) = aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),zip(B,C,Xs,take(C,aa(list(B),nat,size_size(list(B)),Xs),Zs))),zip(B,C,Ys,drop(C,aa(list(B),nat,size_size(list(B)),Xs),Zs))) ).

% zip_append1
tff(fact_5196_zip__append2,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Zs: list(C)] : zip(B,C,Xs,aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys),Zs)) = aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),zip(B,C,take(B,aa(list(C),nat,size_size(list(C)),Ys),Xs),Ys)),zip(B,C,drop(B,aa(list(C),nat,size_size(list(C)),Ys),Xs),Zs)) ).

% zip_append2
tff(fact_5197_map__of__zip__nth,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),I: nat] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( distinct(B,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(C),nat,size_size(list(C)),Ys))
         => ( aa(B,option(C),map_of(B,C,zip(B,C,Xs,Ys)),aa(nat,B,nth(B,Xs),I)) = aa(C,option(C),some(C),aa(nat,C,nth(C,Ys),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_5198_min__list_Oelims,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X: list(B),Y: B] :
          ( ( min_list(B,X) = Y )
         => ( ! [X2: B,Xs2: list(B)] :
                ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
               => ( Y != aa(list(B),B,case_list(B,B,X2,aa(list(B),fun(B,fun(list(B),B)),aTP_Lamp_sw(B,fun(list(B),fun(B,fun(list(B),B))),X2),Xs2)),Xs2) ) )
           => ~ ( ( X = nil(B) )
               => ( Y != undefined(B) ) ) ) ) ) ).

% min_list.elims
tff(fact_5199_integer__of__nat__def,axiom,
    code_integer_of_nat = aa(fun(nat,int),fun(nat,code_integer),map_fun(nat,nat,int,code_integer,id(nat),code_integer_of_int),semiring_1_of_nat(int)) ).

% integer_of_nat_def
tff(fact_5200_map__of__distinct__upd4,axiom,
    ! [B: $tType,C: $tType,X: B,Xs: list(product_prod(B,C)),Ys: list(product_prod(B,C)),Y: C] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs)))
     => ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Ys)))
       => ( map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),Ys)) = fun_upd(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),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),X),Y)),Ys))),X,none(C)) ) ) ) ).

% map_of_distinct_upd4
tff(fact_5201_nat__of__integer__integer__of__nat,axiom,
    ! [N: nat] : aa(code_integer,nat,code_nat_of_integer,aa(nat,code_integer,code_integer_of_nat,N)) = N ).

% nat_of_integer_integer_of_nat
tff(fact_5202_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L) ).

% fst_divmod_integer
tff(fact_5203_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_min(code_integer),K),L)) = aa(int,int,aa(int,fun(int,int),ord_min(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ).

% int_of_integer_min
tff(fact_5204_img__fst,axiom,
    ! [C: $tType,B: $tType,A2: B,B2: C,S: set(product_prod(B,C))] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),S)
     => aa(set(B),$o,member(B,A2),aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),S)) ) ).

% img_fst
tff(fact_5205_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,N)) = aa(nat,int,semiring_1_of_nat(int),N) ).

% int_of_integer_integer_of_nat
tff(fact_5206_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,X)) = aa(nat,int,semiring_1_of_nat(int),X) ).

% integer_of_nat.rep_eq
tff(fact_5207_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% fst_divmod_abs
tff(fact_5208_map__fst__zip,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),zip(B,C,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5209_map__fst__enumerate,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : aa(list(product_prod(nat,B)),list(nat),map(product_prod(nat,B),nat,product_fst(nat,B)),enumerate(B,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(B),nat,size_size(list(B)),Xs))) ).

% map_fst_enumerate
tff(fact_5210_sorted__wrt__map__linord,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(B)
     => ! [L: list(product_prod(B,C))] :
          ( sorted_wrt(product_prod(B,C),aTP_Lamp_sx(product_prod(B,C),fun(product_prod(B,C),$o)),L)
        <=> sorted_wrt(B,ord_less_eq(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_5211_map__fst__mk__snd,axiom,
    ! [C: $tType,B: $tType,K: C,L: list(B)] : aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),aa(list(B),list(product_prod(B,C)),map(B,product_prod(B,C),aa(C,fun(B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))),K)),L)) = L ).

% map_fst_mk_snd
tff(fact_5212_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_5213_sorted__wrt__map__rev__linord,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(B)
     => ! [L: list(product_prod(B,C))] :
          ( sorted_wrt(product_prod(B,C),aTP_Lamp_sy(product_prod(B,C),fun(product_prod(B,C),$o)),L)
        <=> sorted_wrt(B,ord_less_eq(B),rev(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),L))) ) ) ).

% sorted_wrt_map_rev_linord
tff(fact_5214_one__div__numeral,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),divide_divide(B),one_one(B)),aa(num,B,numeral_numeral(B),N)) = aa(product_prod(B,B),B,product_fst(B,B),unique8689654367752047608divmod(B,one2,N)) ) ).

% one_div_numeral
tff(fact_5215_map__of__rev__distinct,axiom,
    ! [C: $tType,B: $tType,Ma: list(product_prod(B,C))] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Ma))
     => ( map_of(B,C,rev(product_prod(B,C),Ma)) = map_of(B,C,Ma) ) ) ).

% map_of_rev_distinct
tff(fact_5216_map__fst__mk__fst,axiom,
    ! [C: $tType,B: $tType,K: B,L: list(C)] : aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),aa(list(C),list(product_prod(B,C)),map(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),K)),L)) = replicate(B,aa(list(C),nat,size_size(list(C)),L),K) ).

% map_fst_mk_fst
tff(fact_5217_fst__diag__fst,axiom,
    ! [C: $tType,B: $tType] : aa(fun(product_prod(B,C),product_prod(B,B)),fun(product_prod(B,C),B),aa(fun(product_prod(B,B),B),fun(fun(product_prod(B,C),product_prod(B,B)),fun(product_prod(B,C),B)),comp(product_prod(B,B),B,product_prod(B,C)),product_fst(B,B)),aa(fun(product_prod(B,C),B),fun(product_prod(B,C),product_prod(B,B)),aa(fun(B,product_prod(B,B)),fun(fun(product_prod(B,C),B),fun(product_prod(B,C),product_prod(B,B))),comp(B,product_prod(B,B),product_prod(B,C)),aTP_Lamp_sk(B,product_prod(B,B))),product_fst(B,C))) = product_fst(B,C) ).

% fst_diag_fst
tff(fact_5218_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_5219_measure__fst,axiom,
    ! [C: $tType,B: $tType,F: fun(B,nat)] :
      ( fun_is_measure(B,F)
     => fun_is_measure(product_prod(B,C),aTP_Lamp_sz(fun(B,nat),fun(product_prod(B,C),nat),F)) ) ).

% measure_fst
tff(fact_5220_fst__def,axiom,
    ! [C: $tType,B: $tType,Prod: product_prod(B,C)] : aa(product_prod(B,C),B,product_fst(B,C),Prod) = aa(product_prod(B,C),B,aa(fun(B,fun(C,B)),fun(product_prod(B,C),B),product_case_prod(B,C,B),aTP_Lamp_it(B,fun(C,B))),Prod) ).

% fst_def
tff(fact_5221_fn__fst__conv,axiom,
    ! [C: $tType,D: $tType,B: $tType,F: fun(B,D)] : aTP_Lamp_ta(fun(B,D),fun(product_prod(B,C),D),F) = aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),aTP_Lamp_tb(fun(B,D),fun(B,fun(C,D)),F)) ).

% fn_fst_conv
tff(fact_5222_fstE,axiom,
    ! [C: $tType,B: $tType,X: product_prod(B,C),A2: B,B2: C,Pa: fun(B,$o)] :
      ( ( X = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2) )
     => ( aa(B,$o,Pa,aa(product_prod(B,C),B,product_fst(B,C),X))
       => aa(B,$o,Pa,A2) ) ) ).

% fstE
tff(fact_5223_in__fst__imageE,axiom,
    ! [C: $tType,B: $tType,X: B,S: set(product_prod(B,C))] :
      ( aa(set(B),$o,member(B,X),aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),S))
     => ~ ! [Y2: C] : ~ aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y2)),S) ) ).

% in_fst_imageE
tff(fact_5224_fst__image__mp,axiom,
    ! [C: $tType,B: $tType,A4: set(product_prod(B,C)),B4: set(B),X: B,Y: C] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),A4)),B4)
     => ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y)),A4)
       => aa(set(B),$o,member(B,X),B4) ) ) ).

% fst_image_mp
tff(fact_5225_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiring_1_of_nat(code_integer) ).

% integer_of_nat_eq_of_nat
tff(fact_5226_fst__diag__id,axiom,
    ! [B: $tType,Z: B] : aa(B,B,aa(fun(B,product_prod(B,B)),fun(B,B),aa(fun(product_prod(B,B),B),fun(fun(B,product_prod(B,B)),fun(B,B)),comp(product_prod(B,B),B,B),product_fst(B,B)),aTP_Lamp_sk(B,product_prod(B,B))),Z) = aa(B,B,id(B),Z) ).

% fst_diag_id
tff(fact_5227_distinct__map__fst__filterI,axiom,
    ! [C: $tType,B: $tType,Xs: list(product_prod(B,C)),Pa: fun(product_prod(B,C),$o)] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs))
     => distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),filter2(product_prod(B,C),Pa,Xs))) ) ).

% distinct_map_fst_filterI
tff(fact_5228_sorted__enumerate,axiom,
    ! [B: $tType,N: nat,Xs: list(B)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,B)),list(nat),map(product_prod(nat,B),nat,product_fst(nat,B)),enumerate(B,N,Xs))) ).

% sorted_enumerate
tff(fact_5229_fst__foldl,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(B,fun(D,B)),G: fun(B,fun(C,fun(D,C))),A2: B,B2: C,Xs: list(D)] : aa(product_prod(B,C),B,product_fst(B,C),aa(list(D),product_prod(B,C),aa(product_prod(B,C),fun(list(D),product_prod(B,C)),foldl(product_prod(B,C),D,aa(fun(B,fun(C,fun(D,product_prod(B,C)))),fun(product_prod(B,C),fun(D,product_prod(B,C))),product_case_prod(B,C,fun(D,product_prod(B,C))),aa(fun(B,fun(C,fun(D,C))),fun(B,fun(C,fun(D,product_prod(B,C)))),aTP_Lamp_tc(fun(B,fun(D,B)),fun(fun(B,fun(C,fun(D,C))),fun(B,fun(C,fun(D,product_prod(B,C))))),F),G))),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),Xs)) = aa(list(D),B,aa(B,fun(list(D),B),foldl(B,D,F),A2),Xs) ).

% fst_foldl
tff(fact_5230_integer__of__nat__0,axiom,
    aa(nat,code_integer,code_integer_of_nat,zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_5231_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : aa(nat,code_integer,code_integer_of_nat,X) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).

% integer_of_nat.abs_eq
tff(fact_5232_distinct__map__fstD,axiom,
    ! [B: $tType,C: $tType,Xs: list(product_prod(B,C)),X: B,Y: C,Z: C] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs))
     => ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y)),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),Xs))
       => ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Z)),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),Xs))
         => ( Y = Z ) ) ) ) ).

% distinct_map_fstD
tff(fact_5233_integer__of__nat__numeral,axiom,
    ! [N: num] : aa(nat,code_integer,code_integer_of_nat,aa(num,nat,numeral_numeral(nat),N)) = aa(num,code_integer,numeral_numeral(code_integer),N) ).

% integer_of_nat_numeral
tff(fact_5234_integer__of__nat__1,axiom,
    aa(nat,code_integer,code_integer_of_nat,one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_5235_map__fst__zip__take,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C)] : aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),zip(B,C,Xs,Ys)) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(C),nat,size_size(list(C)),Ys)),Xs) ).

% map_fst_zip_take
tff(fact_5236_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_5237_set__map__of__compr,axiom,
    ! [C: $tType,B: $tType,Xs: list(product_prod(B,C))] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs))
     => ( aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),Xs) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_td(list(product_prod(B,C)),fun(B,fun(C,$o)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_5238_map__of__Some__filter__not__in,axiom,
    ! [C: $tType,B: $tType,Xs: list(product_prod(C,B)),K: C,V2: B,Pa: fun(product_prod(C,B),$o)] :
      ( ( aa(C,option(B),map_of(C,B,Xs),K) = aa(B,option(B),some(B),V2) )
     => ( ~ aa(product_prod(C,B),$o,Pa,aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),K),V2))
       => ( distinct(C,aa(list(product_prod(C,B)),list(C),map(product_prod(C,B),C,product_fst(C,B)),Xs))
         => ( aa(C,option(B),map_of(C,B,filter2(product_prod(C,B),Pa,Xs)),K) = none(B) ) ) ) ) ).

% map_of_Some_filter_not_in
tff(fact_5239_map__of__distinct__lookup,axiom,
    ! [B: $tType,C: $tType,X: B,Xs: list(product_prod(B,C)),Ys: list(product_prod(B,C)),Y: C] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs)))
     => ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Ys)))
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),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),X),Y)),Ys))),X) = aa(C,option(C),some(C),Y) ) ) ) ).

% map_of_distinct_lookup
tff(fact_5240_min__list_Osimps,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X: B,Xs: list(B)] : min_list(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),B,case_list(B,B,X,aa(list(B),fun(B,fun(list(B),B)),aTP_Lamp_sw(B,fun(list(B),fun(B,fun(list(B),B))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_5241_map__of__distinct__upd3,axiom,
    ! [B: $tType,C: $tType,X: B,Xs: list(product_prod(B,C)),Ys: list(product_prod(B,C)),Y: C,Y6: C] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs)))
     => ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Ys)))
       => ( map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),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),X),Y)),Ys))) = fun_upd(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),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),X),Y6)),Ys))),X,aa(C,option(C),some(C),Y)) ) ) ) ).

% map_of_distinct_upd3
tff(fact_5242_map__of__distinct__upd2,axiom,
    ! [B: $tType,C: $tType,X: B,Xs: list(product_prod(B,C)),Ys: list(product_prod(B,C)),Y: C] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs)))
     => ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Ys)))
       => ( map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),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),X),Y)),Ys))) = fun_upd(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(list(product_prod(B,C)),fun(list(product_prod(B,C)),list(product_prod(B,C))),append(product_prod(B,C)),Xs),Ys)),X,aa(C,option(C),some(C),Y)) ) ) ) ).

% map_of_distinct_upd2
tff(fact_5243_min__list_Opelims,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [X: list(B),Y: B] :
          ( ( min_list(B,X) = Y )
         => ( accp(list(B),min_list_rel(B),X)
           => ( ! [X2: B,Xs2: list(B)] :
                  ( ( X = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
                 => ( ( Y = aa(list(B),B,case_list(B,B,X2,aa(list(B),fun(B,fun(list(B),B)),aTP_Lamp_sw(B,fun(list(B),fun(B,fun(list(B),B))),X2),Xs2)),Xs2) )
                   => ~ accp(list(B),min_list_rel(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)) ) )
             => ~ ( ( X = nil(B) )
                 => ( ( Y = undefined(B) )
                   => ~ accp(list(B),min_list_rel(B),nil(B)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_5244_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( Y = $ite(Xa2 = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa2)))))) ) ) ).

% bezw.elims
tff(fact_5245_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),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,aa(nat,nat,aa(nat,fun(nat,nat),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,aa(nat,nat,aa(nat,fun(nat,nat),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.simps
tff(fact_5246_img__snd,axiom,
    ! [C: $tType,B: $tType,A2: B,B2: C,S: set(product_prod(B,C))] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),S)
     => aa(set(C),$o,member(C,B2),aa(set(product_prod(B,C)),set(C),image2(product_prod(B,C),C,product_snd(B,C)),S)) ) ).

% img_snd
tff(fact_5247_map__snd__zip,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(list(product_prod(B,C)),list(C),map(product_prod(B,C),C,product_snd(B,C)),zip(B,C,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5248_map__snd__mk__fst,axiom,
    ! [C: $tType,B: $tType,K: C,L: list(B)] : aa(list(product_prod(C,B)),list(B),map(product_prod(C,B),B,product_snd(C,B)),aa(list(B),list(product_prod(C,B)),map(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),K)),L)) = L ).

% map_snd_mk_fst
tff(fact_5249_one__mod__numeral,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [N: num] : aa(B,B,aa(B,fun(B,B),modulo_modulo(B),one_one(B)),aa(num,B,numeral_numeral(B),N)) = aa(product_prod(B,B),B,product_snd(B,B),unique8689654367752047608divmod(B,one2,N)) ) ).

% one_mod_numeral
tff(fact_5250_map__snd__mk__snd,axiom,
    ! [C: $tType,B: $tType,K: B,L: list(C)] : aa(list(product_prod(C,B)),list(B),map(product_prod(C,B),B,product_snd(C,B)),aa(list(C),list(product_prod(C,B)),map(C,product_prod(C,B),aa(B,fun(C,product_prod(C,B)),aTP_Lamp_te(B,fun(C,product_prod(C,B))),K)),L)) = replicate(B,aa(list(C),nat,size_size(list(C)),L),K) ).

% map_snd_mk_snd
tff(fact_5251_snd__diag__snd,axiom,
    ! [C: $tType,B: $tType] : aa(fun(product_prod(B,C),product_prod(C,C)),fun(product_prod(B,C),C),aa(fun(product_prod(C,C),C),fun(fun(product_prod(B,C),product_prod(C,C)),fun(product_prod(B,C),C)),comp(product_prod(C,C),C,product_prod(B,C)),product_snd(C,C)),aa(fun(product_prod(B,C),C),fun(product_prod(B,C),product_prod(C,C)),aa(fun(C,product_prod(C,C)),fun(fun(product_prod(B,C),C),fun(product_prod(B,C),product_prod(C,C))),comp(C,product_prod(C,C),product_prod(B,C)),aTP_Lamp_tf(C,product_prod(C,C))),product_snd(B,C))) = product_snd(B,C) ).

% snd_diag_snd
tff(fact_5252_snd__def,axiom,
    ! [B: $tType,C: $tType,Prod: product_prod(C,B)] : aa(product_prod(C,B),B,product_snd(C,B),Prod) = aa(product_prod(C,B),B,aa(fun(C,fun(B,B)),fun(product_prod(C,B),B),product_case_prod(C,B,B),aTP_Lamp_tg(C,fun(B,B))),Prod) ).

% snd_def
tff(fact_5253_fn__snd__conv,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(C,D)] : aTP_Lamp_th(fun(C,D),fun(product_prod(B,C),D),F) = aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),aTP_Lamp_ti(fun(C,D),fun(B,fun(C,D)),F)) ).

% fn_snd_conv
tff(fact_5254_Ex__prod__contract,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o))] :
      ( ? [A6: B,X_12: C] : aa(C,$o,aa(B,fun(C,$o),Pa,A6),X_12)
    <=> ? [Z5: product_prod(B,C)] : aa(C,$o,aa(B,fun(C,$o),Pa,aa(product_prod(B,C),B,product_fst(B,C),Z5)),aa(product_prod(B,C),C,product_snd(B,C),Z5)) ) ).

% Ex_prod_contract
tff(fact_5255_All__prod__contract,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o))] :
      ( ! [A6: B,X_12: C] : aa(C,$o,aa(B,fun(C,$o),Pa,A6),X_12)
    <=> ! [Z5: product_prod(B,C)] : aa(C,$o,aa(B,fun(C,$o),Pa,aa(product_prod(B,C),B,product_fst(B,C),Z5)),aa(product_prod(B,C),C,product_snd(B,C),Z5)) ) ).

% All_prod_contract
tff(fact_5256_sndE,axiom,
    ! [B: $tType,C: $tType,X: product_prod(B,C),A2: B,B2: C,Pa: fun(C,$o)] :
      ( ( X = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2) )
     => ( aa(C,$o,Pa,aa(product_prod(B,C),C,product_snd(B,C),X))
       => aa(C,$o,Pa,B2) ) ) ).

% sndE
tff(fact_5257_measure__snd,axiom,
    ! [C: $tType,B: $tType,F: fun(B,nat)] :
      ( fun_is_measure(B,F)
     => fun_is_measure(product_prod(C,B),aTP_Lamp_tj(fun(B,nat),fun(product_prod(C,B),nat),F)) ) ).

% measure_snd
tff(fact_5258_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o)),X: B,Y: C,A2: product_prod(B,C)] :
      ( aa(C,$o,aa(B,fun(C,$o),Pa,X),Y)
     => ( ( A2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) )
       => aa(C,$o,aa(B,fun(C,$o),Pa,aa(product_prod(B,C),B,product_fst(B,C),A2)),aa(product_prod(B,C),C,product_snd(B,C),A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5259_in__snd__imageE,axiom,
    ! [B: $tType,C: $tType,Y: B,S: set(product_prod(C,B))] :
      ( aa(set(B),$o,member(B,Y),aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),S))
     => ~ ! [X2: C] : ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X2),Y)),S) ) ).

% in_snd_imageE
tff(fact_5260_snd__diag__fst,axiom,
    ! [C: $tType,B: $tType] : aa(fun(product_prod(B,C),product_prod(B,B)),fun(product_prod(B,C),B),aa(fun(product_prod(B,B),B),fun(fun(product_prod(B,C),product_prod(B,B)),fun(product_prod(B,C),B)),comp(product_prod(B,B),B,product_prod(B,C)),product_snd(B,B)),aa(fun(product_prod(B,C),B),fun(product_prod(B,C),product_prod(B,B)),aa(fun(B,product_prod(B,B)),fun(fun(product_prod(B,C),B),fun(product_prod(B,C),product_prod(B,B))),comp(B,product_prod(B,B),product_prod(B,C)),aTP_Lamp_sk(B,product_prod(B,B))),product_fst(B,C))) = product_fst(B,C) ).

% snd_diag_fst
tff(fact_5261_fst__diag__snd,axiom,
    ! [C: $tType,B: $tType] : aa(fun(product_prod(B,C),product_prod(C,C)),fun(product_prod(B,C),C),aa(fun(product_prod(C,C),C),fun(fun(product_prod(B,C),product_prod(C,C)),fun(product_prod(B,C),C)),comp(product_prod(C,C),C,product_prod(B,C)),product_fst(C,C)),aa(fun(product_prod(B,C),C),fun(product_prod(B,C),product_prod(C,C)),aa(fun(C,product_prod(C,C)),fun(fun(product_prod(B,C),C),fun(product_prod(B,C),product_prod(C,C))),comp(C,product_prod(C,C),product_prod(B,C)),aTP_Lamp_tf(C,product_prod(C,C))),product_snd(B,C))) = product_snd(B,C) ).

% fst_diag_snd
tff(fact_5262_exE__realizer,axiom,
    ! [D: $tType,B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),P2: product_prod(C,B),Q: fun(D,$o),F: fun(C,fun(B,D))] :
      ( aa(C,$o,aa(B,fun(C,$o),Pa,aa(product_prod(C,B),B,product_snd(C,B),P2)),aa(product_prod(C,B),C,product_fst(C,B),P2))
     => ( ! [X2: C,Y2: B] :
            ( aa(C,$o,aa(B,fun(C,$o),Pa,Y2),X2)
           => aa(D,$o,Q,aa(B,D,aa(C,fun(B,D),F,X2),Y2)) )
       => aa(D,$o,Q,aa(product_prod(C,B),D,aa(fun(C,fun(B,D)),fun(product_prod(C,B),D),product_case_prod(C,B,D),F),P2)) ) ) ).

% exE_realizer
tff(fact_5263_split__comp__eq,axiom,
    ! [C: $tType,D: $tType,E: $tType,B: $tType,F: fun(E,fun(C,D)),G: fun(B,E)] : aa(fun(B,E),fun(product_prod(B,C),D),aTP_Lamp_tk(fun(E,fun(C,D)),fun(fun(B,E),fun(product_prod(B,C),D)),F),G) = aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),aa(fun(B,E),fun(B,fun(C,D)),aTP_Lamp_tl(fun(E,fun(C,D)),fun(fun(B,E),fun(B,fun(C,D))),F),G)) ).

% split_comp_eq
tff(fact_5264_case__prod__beta_H,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,fun(C,D)),X4: product_prod(B,C)] : aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),F),X4) = aa(C,D,aa(B,fun(C,D),F,aa(product_prod(B,C),B,product_fst(B,C),X4)),aa(product_prod(B,C),C,product_snd(B,C),X4)) ).

% case_prod_beta'
tff(fact_5265_case__prod__unfold,axiom,
    ! [D: $tType,C: $tType,B: $tType,X4: fun(B,fun(C,D)),Xa: product_prod(B,C)] : aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),X4),Xa) = aa(C,D,aa(B,fun(C,D),X4,aa(product_prod(B,C),B,product_fst(B,C),Xa)),aa(product_prod(B,C),C,product_snd(B,C),Xa)) ).

% case_prod_unfold
tff(fact_5266_divides__aux__def,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Qr: product_prod(B,B)] :
          ( unique5940410009612947441es_aux(B,Qr)
        <=> ( aa(product_prod(B,B),B,product_snd(B,B),Qr) = zero_zero(B) ) ) ) ).

% divides_aux_def
tff(fact_5267_snd__image__mp,axiom,
    ! [C: $tType,B: $tType,A4: set(product_prod(C,B)),B4: set(B),X: C,Y: B] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),A4)),B4)
     => ( aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)),A4)
       => aa(set(B),$o,member(B,Y),B4) ) ) ).

% snd_image_mp
tff(fact_5268_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R))) ).

% quotient_of_denom_pos'
tff(fact_5269_snd__diag__id,axiom,
    ! [B: $tType,Z: B] : aa(B,B,aa(fun(B,product_prod(B,B)),fun(B,B),aa(fun(product_prod(B,B),B),fun(fun(B,product_prod(B,B)),fun(B,B)),comp(product_prod(B,B),B,B),product_snd(B,B)),aTP_Lamp_sk(B,product_prod(B,B))),Z) = aa(B,B,id(B),Z) ).

% snd_diag_id
tff(fact_5270_case__prod__comp,axiom,
    ! [E: $tType,B: $tType,D: $tType,C: $tType,F: fun(E,fun(D,B)),G: fun(C,E),X: product_prod(C,D)] : aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),aa(fun(C,E),fun(C,fun(D,B)),aa(fun(E,fun(D,B)),fun(fun(C,E),fun(C,fun(D,B))),comp(E,fun(D,B),C),F),G)),X) = aa(D,B,aa(E,fun(D,B),F,aa(C,E,G,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(product_prod(C,D),D,product_snd(C,D),X)) ).

% case_prod_comp
tff(fact_5271_Eps__case__prod,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o))] : fChoice(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Pa)) = fChoice(product_prod(B,C),aTP_Lamp_tm(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),Pa)) ).

% Eps_case_prod
tff(fact_5272_The__case__prod,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o))] : the(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Pa)) = the(product_prod(B,C),aTP_Lamp_tm(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),Pa)) ).

% The_case_prod
tff(fact_5273_zip__eq__conv,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Zs: list(product_prod(B,C))] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( ( zip(B,C,Xs,Ys) = Zs )
      <=> ( ( aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Zs) = Xs )
          & ( aa(list(product_prod(B,C)),list(C),map(product_prod(B,C),C,product_snd(B,C)),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_5274_map__snd__zip__take,axiom,
    ! [C: $tType,B: $tType,Xs: list(C),Ys: list(B)] : aa(list(product_prod(C,B)),list(B),map(product_prod(C,B),B,product_snd(C,B)),zip(C,B,Xs,Ys)) = take(B,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(C),nat,size_size(list(C)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)),Ys) ).

% map_snd_zip_take
tff(fact_5275_ran__map__of,axiom,
    ! [B: $tType,C: $tType,Xs: list(product_prod(C,B))] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),ran(C,B,map_of(C,B,Xs))),aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Xs))) ).

% ran_map_of
tff(fact_5276_in__set__zip,axiom,
    ! [B: $tType,C: $tType,P2: product_prod(B,C),Xs: list(B),Ys: list(C)] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),P2),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,Xs,Ys)))
    <=> ? [N4: nat] :
          ( ( aa(nat,B,nth(B,Xs),N4) = aa(product_prod(B,C),B,product_fst(B,C),P2) )
          & ( aa(nat,C,nth(C,Ys),N4) = aa(product_prod(B,C),C,product_snd(B,C),P2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Xs))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(C),nat,size_size(list(C)),Ys)) ) ) ).

% in_set_zip
tff(fact_5277_Misc_Oran__distinct,axiom,
    ! [C: $tType,B: $tType,Al: list(product_prod(B,C))] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Al))
     => ( ran(B,C,map_of(B,C,Al)) = aa(set(product_prod(B,C)),set(C),image2(product_prod(B,C),C,product_snd(B,C)),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),Al)) ) ) ).

% Misc.ran_distinct
tff(fact_5278_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(nat,nat,aa(nat,fun(nat,nat),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,aa(nat,nat,aa(nat,fun(nat,nat),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,aa(nat,nat,aa(nat,fun(nat,nat),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_5279_minus__one__mod__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_mod_numeral
tff(fact_5280_one__mod__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).

% one_mod_minus_numeral
tff(fact_5281_fst__snd__flip,axiom,
    ! [C: $tType,B: $tType,Xy: product_prod(B,C)] : aa(product_prod(B,C),B,product_fst(B,C),Xy) = aa(product_prod(B,C),B,aa(fun(product_prod(B,C),product_prod(C,B)),fun(product_prod(B,C),B),aa(fun(product_prod(C,B),B),fun(fun(product_prod(B,C),product_prod(C,B)),fun(product_prod(B,C),B)),comp(product_prod(C,B),B,product_prod(B,C)),product_snd(C,B)),aa(fun(B,fun(C,product_prod(C,B))),fun(product_prod(B,C),product_prod(C,B)),product_case_prod(B,C,product_prod(C,B)),aTP_Lamp_te(B,fun(C,product_prod(C,B))))),Xy) ).

% fst_snd_flip
tff(fact_5282_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L) ).

% snd_divmod_integer
tff(fact_5283_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% snd_divmod_abs
tff(fact_5284_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_5285_effect__heapE,axiom,
    ! [B: $tType,F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,heap_Time_heap(B,F),H,H4,R,N)
     => ~ ( ( H4 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
         => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
           => ( R != aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),B,product_fst(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H)) ) ) ) ) ).

% effect_heapE
tff(fact_5286_effect__heapI,axiom,
    ! [B: $tType,N: nat,F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B] :
      ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
     => ( ( H4 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
       => ( ( R = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),B,product_fst(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H)) )
         => heap_Time_effect(B,heap_Time_heap(B,F),H,H4,R,N) ) ) ) ).

% effect_heapI
tff(fact_5287_effect__guardI,axiom,
    ! [B: $tType,Pa: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),H4: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),N: nat,R: B] :
      ( aa(heap_ext(product_unit),$o,Pa,H)
     => ( ( H4 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
       => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
         => ( ( R = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),B,product_fst(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H)) )
           => heap_Time_effect(B,heap_Time_guard(B,Pa,F),H,H4,R,N) ) ) ) ) ).

% effect_guardI
tff(fact_5288_effect__guardE,axiom,
    ! [B: $tType,Pa: fun(heap_ext(product_unit),$o),F: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,heap_Time_guard(B,Pa,F),H,H4,R,N)
     => ~ ( ( H4 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
         => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H))) )
           => ( ( R = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),B,product_fst(B,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),F,H)) )
             => ~ aa(heap_ext(product_unit),$o,Pa,H) ) ) ) ) ).

% effect_guardE
tff(fact_5289_effectE,axiom,
    ! [B: $tType,C2: heap_Time_Heap(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
      ( heap_Time_effect(B,C2,H,H4,R,N)
     => ~ ( ( R = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),B,product_fst(B,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),the2(product_prod(B,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,C2),H))) )
         => ( ( H4 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),the2(product_prod(B,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,C2),H)))) )
           => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),the2(product_prod(B,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,C2),H)))) )
             => ~ heap_Time_success(B,C2,H) ) ) ) ) ).

% effectE
tff(fact_5290_execute__bind__success,axiom,
    ! [C: $tType,B: $tType,F: heap_Time_Heap(B),H: heap_ext(product_unit),G: fun(B,heap_Time_Heap(C))] :
      ( heap_Time_success(B,F,H)
     => ( aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,heap_Time_bind(B,C,F,G)),H) = heap_Time_timeFrame(C,aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),the2(product_prod(B,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,F),H)))),aa(heap_ext(product_unit),option(product_prod(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,aa(B,heap_Time_Heap(C),G,aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),B,product_fst(B,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),the2(product_prod(B,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,F),H))))),aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(B,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),the2(product_prod(B,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,F),H)))))) ) ) ).

% execute_bind_success
tff(fact_5291_nths__shift__lemma__Suc,axiom,
    ! [B: $tType,Pa: fun(nat,$o),Xs: list(B),Is: list(nat)] : aa(list(product_prod(B,nat)),list(B),map(product_prod(B,nat),B,product_fst(B,nat)),filter2(product_prod(B,nat),aTP_Lamp_tn(fun(nat,$o),fun(product_prod(B,nat),$o),Pa),zip(B,nat,Xs,Is))) = aa(list(product_prod(B,nat)),list(B),map(product_prod(B,nat),B,product_fst(B,nat)),filter2(product_prod(B,nat),aTP_Lamp_to(fun(nat,$o),fun(product_prod(B,nat),$o),Pa),zip(B,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_5292_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R: int] :
      adjust_mod(L,R) = $ite(R = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R)) ).

% Divides.adjust_mod_def
tff(fact_5293_nths__shift__lemma,axiom,
    ! [B: $tType,A4: set(nat),Xs: list(B),I: nat] : aa(list(product_prod(B,nat)),list(B),map(product_prod(B,nat),B,product_fst(B,nat)),filter2(product_prod(B,nat),aTP_Lamp_tp(set(nat),fun(product_prod(B,nat),$o),A4),zip(B,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(B),nat,size_size(list(B)),Xs)))))) = aa(list(product_prod(B,nat)),list(B),map(product_prod(B,nat),B,product_fst(B,nat)),filter2(product_prod(B,nat),aa(nat,fun(product_prod(B,nat),$o),aTP_Lamp_tq(set(nat),fun(nat,fun(product_prod(B,nat),$o)),A4),I),zip(B,nat,Xs,upt(zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))))) ).

% nths_shift_lemma
tff(fact_5294_nths__def,axiom,
    ! [B: $tType,Xs: list(B),A4: set(nat)] : nths(B,Xs,A4) = aa(list(product_prod(B,nat)),list(B),map(product_prod(B,nat),B,product_fst(B,nat)),filter2(product_prod(B,nat),aTP_Lamp_tp(set(nat),fun(product_prod(B,nat),$o),A4),zip(B,nat,Xs,upt(zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))))) ).

% nths_def
tff(fact_5295_in__set__enumerate__eq,axiom,
    ! [B: $tType,P2: product_prod(nat,B),N: nat,Xs: list(B)] :
      ( aa(set(product_prod(nat,B)),$o,member(product_prod(nat,B),P2),aa(list(product_prod(nat,B)),set(product_prod(nat,B)),set2(product_prod(nat,B)),enumerate(B,N,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(product_prod(nat,B),nat,product_fst(nat,B),P2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,B),nat,product_fst(nat,B),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs)),N))
        & ( aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,B),nat,product_fst(nat,B),P2)),N)) = aa(product_prod(nat,B),B,product_snd(nat,B),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_5296_snd__fst__flip,axiom,
    ! [B: $tType,C: $tType,Xy: product_prod(C,B)] : aa(product_prod(C,B),B,product_snd(C,B),Xy) = aa(product_prod(C,B),B,aa(fun(product_prod(C,B),product_prod(B,C)),fun(product_prod(C,B),B),aa(fun(product_prod(B,C),B),fun(fun(product_prod(C,B),product_prod(B,C)),fun(product_prod(C,B),B)),comp(product_prod(B,C),B,product_prod(C,B)),product_fst(B,C)),aa(fun(C,fun(B,product_prod(B,C))),fun(product_prod(C,B),product_prod(B,C)),product_case_prod(C,B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))))),Xy) ).

% snd_fst_flip
tff(fact_5297_size__prod__simp,axiom,
    ! [B: $tType,C: $tType,F: fun(B,nat),G: fun(C,nat),P2: product_prod(B,C)] : basic_BNF_size_prod(B,C,F,G,P2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,F,aa(product_prod(B,C),B,product_fst(B,C),P2))),aa(C,nat,G,aa(product_prod(B,C),C,product_snd(B,C),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_5298_positive__def,axiom,
    positive = aa(fun(product_prod(int,int),$o),fun(rat,$o),map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o)),aTP_Lamp_tr(product_prod(int,int),$o)) ).

% positive_def
tff(fact_5299_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa2)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_5300_positive_Orep__eq,axiom,
    ! [X: rat] :
      ( aa(rat,$o,positive,X)
    <=> aa(int,$o,aa(int,fun(int,$o),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_5301_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa2) = Y )
     => ( 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))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa2),X),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)),aa(nat,product_prod(nat,nat),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)))) )
           => ~ 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_5302_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_ts(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_5303_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_tt(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_5304_one__rat__def,axiom,
    one_one(rat) = 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),one_one(int)),one_one(int))) ).

% one_rat_def
tff(fact_5305_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      aa(int,rat,aa(int,fun(int,rat),fract,Xa2),X) = aa(product_prod(int,int),rat,abs_Rat,
        $ite(X = 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),Xa2),X))) ).

% Fract.abs_eq
tff(fact_5306_zero__rat__def,axiom,
    zero_zero(rat) = 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),zero_zero(int)),one_one(int))) ).

% zero_rat_def
tff(fact_5307_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_tu(int,fun(int,product_prod(int,int)))) ).

% Fract_def
tff(fact_5308_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_tv(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_5309_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_tw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_5310_mergesort__by__rel__split__length,axiom,
    ! [B: $tType,Xs12: list(B),Xs23: list(B),Xs: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,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)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs12)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
      & ( aa(list(B),nat,size_size(list(B)),aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,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)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(B),nat,size_size(list(B)),Xs23)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_5311_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      normalize(P2) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
        $let(
          a2: int,
          a2:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a2: int,
            a2:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_5312_set__to__map__insert,axiom,
    ! [C: $tType,B: $tType,Kv: product_prod(B,C),S: set(product_prod(B,C))] :
      ( ~ aa(set(B),$o,member(B,aa(product_prod(B,C),B,product_fst(B,C),Kv)),aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),S))
     => ( set_to_map(B,C,aa(set(product_prod(B,C)),set(product_prod(B,C)),insert(product_prod(B,C),Kv),S)) = fun_upd(B,option(C),set_to_map(B,C,S),aa(product_prod(B,C),B,product_fst(B,C),Kv),aa(C,option(C),some(C),aa(product_prod(B,C),C,product_snd(B,C),Kv))) ) ) ).

% set_to_map_insert
tff(fact_5313_gcd__right__idem,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),B2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ).

% gcd_right_idem
tff(fact_5314_gcd__left__idem,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ).

% gcd_left_idem
tff(fact_5315_of__int__eq__id,axiom,
    ring_1_of_int(int) = id(int) ).

% of_int_eq_id
tff(fact_5316_gcd__eq__0__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = zero_zero(B) )
        <=> ( ( A2 = zero_zero(B) )
            & ( B2 = zero_zero(B) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_5317_gcd__add1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Ma: B,N: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Ma),N)),N) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),N) ) ).

% gcd_add1
tff(fact_5318_gcd__add2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Ma: B,N: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),aa(B,B,aa(B,fun(B,B),plus_plus(B),Ma),N)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),N) ) ).

% gcd_add2
tff(fact_5319_gcd_Obottom__left__bottom,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),one_one(B)),A2) = one_one(B) ) ).

% gcd.bottom_left_bottom
tff(fact_5320_gcd_Obottom__right__bottom,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),one_one(B)) = one_one(B) ) ).

% gcd.bottom_right_bottom
tff(fact_5321_gcd__neg2,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,uminus_uminus(B),B2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ).

% gcd_neg2
tff(fact_5322_gcd__neg1,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,uminus_uminus(B),A2)),B2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ).

% gcd_neg1
tff(fact_5323_gcd__greatest__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
            & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ) ).

% gcd_greatest_iff
tff(fact_5324_gcd__dvd2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),B2) ) ).

% gcd_dvd2
tff(fact_5325_gcd__dvd1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),A2) ) ).

% gcd_dvd1
tff(fact_5326_gcd__exp,axiom,
    ! [B: $tType] :
      ( semiri6843258321239162965malize(B)
     => ! [A2: B,N: nat,B2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),N) ) ).

% gcd_exp
tff(fact_5327_gcd__1__int,axiom,
    ! [Ma: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_5328_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% gcd_neg1_int
tff(fact_5329_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% gcd_neg2_int
tff(fact_5330_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,abs_abs(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% gcd_abs2_int
tff(fact_5331_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,abs_abs(int),X)),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% gcd_abs1_int
tff(fact_5332_abs__gcd__int,axiom,
    ! [X: int,Y: int] : aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% abs_gcd_int
tff(fact_5333_gcd__neg__numeral__1,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [N: num,A2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))),A2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(num,B,numeral_numeral(B),N)),A2) ) ).

% gcd_neg_numeral_1
tff(fact_5334_gcd__neg__numeral__2,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,N: num] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(num,B,numeral_numeral(B),N)) ) ).

% gcd_neg_numeral_2
tff(fact_5335_is__unit__gcd__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),one_one(B))
        <=> ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = one_one(B) ) ) ) ).

% is_unit_gcd_iff
tff(fact_5336_gcd__pos__int,axiom,
    ! [Ma: int,N: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),N))
    <=> ( ( Ma != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_5337_Gcd__insert,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A2: B,A4: set(B)] : gcd_Gcd(B,aa(set(B),set(B),insert(B,A2),A4)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),gcd_Gcd(B,A4)) ) ).

% Gcd_insert
tff(fact_5338_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(num,int,numeral_numeral(int),N)) ).

% gcd_neg_numeral_2_int
tff(fact_5339_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),X) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_5340_gcd__0__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),zero_zero(int)) = aa(int,int,abs_abs(int),X) ).

% gcd_0_int
tff(fact_5341_gcd__0__left__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),zero_zero(int)),X) = aa(int,int,abs_abs(int),X) ).

% gcd_0_left_int
tff(fact_5342_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Y),X)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,abs_abs(int),Y) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_5343_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,abs_abs(int),X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_5344_set__to__map__empty,axiom,
    ! [B: $tType,C: $tType,X4: B] : aa(B,option(C),set_to_map(B,C,bot_bot(set(product_prod(B,C)))),X4) = none(C) ).

% set_to_map_empty
tff(fact_5345_Gcd__2,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A2: B,B2: B] : gcd_Gcd(B,aa(set(B),set(B),insert(B,A2),aa(set(B),set(B),insert(B,B2),bot_bot(set(B))))) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ).

% Gcd_2
tff(fact_5346_mset__mergesort__by__rel__split,axiom,
    ! [B: $tType,Xs12: list(B),Xs23: list(B),Xs: list(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),mset(B,aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,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)),Xs12),Xs23),Xs)))),mset(B,aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,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)),Xs12),Xs23),Xs)))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),mset(B,Xs)),mset(B,Xs12))),mset(B,Xs23)) ).

% mset_mergesort_by_rel_split
tff(fact_5347_gcd__add__mult,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Ma: B,K: B,N: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),K),Ma)),N)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),N) ) ).

% gcd_add_mult
tff(fact_5348_gcd__idem__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),X) = aa(int,int,abs_abs(int),X) ).

% gcd_idem_int
tff(fact_5349_gcd__dvd__antisym,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),C2),D2))
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),C2),D2)),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2))
           => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),C2),D2) ) ) ) ) ).

% gcd_dvd_antisym
tff(fact_5350_gcd__greatest,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),B2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)) ) ) ) ).

% gcd_greatest
tff(fact_5351_gcd__dvdI2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),C2) ) ) ).

% gcd_dvdI2
tff(fact_5352_gcd__dvdI1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),C2) ) ) ).

% gcd_dvdI1
tff(fact_5353_dvd__gcdD2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2))
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ).

% dvd_gcdD2
tff(fact_5354_dvd__gcdD1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2))
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ).

% dvd_gcdD1
tff(fact_5355_gcd__mono,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),D2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),C2),D2)) ) ) ) ).

% gcd_mono
tff(fact_5356_gcd__red__int,axiom,
    ! [X: int,Y: int] : 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),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) ).

% gcd_red_int
tff(fact_5357_gcd_Oleft__commute,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,A2: B,C2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),C2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2)) ) ).

% gcd.left_commute
tff(fact_5358_gcd_Ocommute,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),A2) ) ).

% gcd.commute
tff(fact_5359_gcd_Oassoc,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2)) ) ).

% gcd.assoc
tff(fact_5360_gcd__diff1,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [Ma: B,N: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),Ma),N)),N) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),N) ) ).

% gcd_diff1
tff(fact_5361_gcd__diff2,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [N: B,Ma: B] : aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),N),Ma)),N) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),Ma),N) ) ).

% gcd_diff2
tff(fact_5362_gcd__dvd__prod,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,K: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_5363_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),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_5364_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V4: 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),V4),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% bezout_int
tff(fact_5365_gcd__mult__distrib__int,axiom,
    ! [K: int,Ma: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),N)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ma)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)) ).

% gcd_mult_distrib_int
tff(fact_5366_gcd__mult__unit1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2)),C2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_5367_gcd__mult__unit2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_5368_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [B: $tType,Xs12: list(B),Xs23: list(B),X12: B,X22: B,Xs: list(B)] : merges295452479951948502_split(B,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)),Xs12),Xs23),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),Xs))) = merges295452479951948502_split(B,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),X12),Xs12)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),Xs23)),Xs) ).

% mergesort_by_rel_split.simps(3)
tff(fact_5369_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [B: $tType,Xs12: list(B),Xs23: list(B)] : merges295452479951948502_split(B,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)),Xs12),Xs23),nil(B)) = 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)),Xs12),Xs23) ).

% mergesort_by_rel_split.simps(1)
tff(fact_5370_gcd__div__unit1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2)),C2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_5371_gcd__div__unit2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_5372_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),B2) ) ).

% gcd_le2_int
tff(fact_5373_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2) ) ).

% gcd_le1_int
tff(fact_5374_gcd__cases__int,axiom,
    ! [X: int,Y: int,Pa: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,Pa,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_5375_gcd__unique__int,axiom,
    ! [D2: int,A2: int,B2: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),A2)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),B2)
        & ! [E5: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E5),A2)
              & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E5),B2) )
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E5),D2) ) )
    <=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2) ) ) ).

% gcd_unique_int
tff(fact_5376_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_5377_gcd__code__int,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),L) = aa(int,int,abs_abs(int),
        $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))))) ).

% gcd_code_int
tff(fact_5378_set__to__map__empty__iff_I1_J,axiom,
    ! [C: $tType,B: $tType,S: set(product_prod(B,C))] :
      ( ! [X3: B] : aa(B,option(C),set_to_map(B,C,S),X3) = none(C)
    <=> ( S = bot_bot(set(product_prod(B,C))) ) ) ).

% set_to_map_empty_iff(1)
tff(fact_5379_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [B: $tType,Xs12: list(B),Xs23: list(B),X: B] : merges295452479951948502_split(B,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)),Xs12),Xs23),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) = 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),X),Xs12)),Xs23) ).

% mergesort_by_rel_split.simps(2)
tff(fact_5380_mergesort__by__rel__split_Oelims,axiom,
    ! [B: $tType,X: product_prod(list(B),list(B)),Xa2: list(B),Y: product_prod(list(B),list(B))] :
      ( ( merges295452479951948502_split(B,X,Xa2) = Y )
     => ( ! [Xs1: list(B),Xs22: list(B)] :
            ( ( X = 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)),Xs1),Xs22) )
           => ( ( Xa2 = nil(B) )
             => ( Y != 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)),Xs1),Xs22) ) ) )
       => ( ! [Xs1: list(B),Xs22: list(B)] :
              ( ( X = 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)),Xs1),Xs22) )
             => ! [X2: B] :
                  ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)) )
                 => ( Y != 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),X2),Xs1)),Xs22) ) ) )
         => ~ ! [Xs1: list(B),Xs22: list(B)] :
                ( ( X = 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)),Xs1),Xs22) )
               => ! [X1: B,X23: B,Xs2: list(B)] :
                    ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),Xs2)) )
                   => ( Y != merges295452479951948502_split(B,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),X1),Xs1)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X23),Xs22)),Xs2) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_5381_set__to__map__empty__iff_I2_J,axiom,
    ! [C: $tType,B: $tType,S: set(product_prod(B,C))] :
      ( ( aTP_Lamp_jv(B,option(C)) = set_to_map(B,C,S) )
    <=> ( S = bot_bot(set(product_prod(B,C))) ) ) ).

% set_to_map_empty_iff(2)
tff(fact_5382_set__to__map__ran,axiom,
    ! [B: $tType,C: $tType,S: set(product_prod(C,B))] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),ran(C,B,set_to_map(C,B,S))),aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),S)) ).

% set_to_map_ran
tff(fact_5383_of__rat__def,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ( field_char_0_of_rat(B) = aa(fun(product_prod(int,int),B),fun(rat,B),map_fun(rat,product_prod(int,int),B,B,rep_Rat,id(B)),aTP_Lamp_tx(product_prod(int,int),B)) ) ) ).

% of_rat_def
tff(fact_5384_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_ty(int,int)) ).

% Code_Numeral.dup_def
tff(fact_5385_lexord__take__index__conv,axiom,
    ! [B: $tType,X: list(B),Y: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),X),Y)),lexord(B,R))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),X)),aa(list(B),nat,size_size(list(B)),Y))
          & ( take(B,aa(list(B),nat,size_size(list(B)),X),Y) = X ) )
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),X)),aa(list(B),nat,size_size(list(B)),Y)))
            & ( take(B,I3,X) = take(B,I3,Y) )
            & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,nth(B,X),I3)),aa(nat,B,nth(B,Y),I3))),R) ) ) ) ).

% lexord_take_index_conv
tff(fact_5386_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_5387_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_5388_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_5389_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),zero_zero(nat)) = A2 ).

% gcd_nat.right_neutral
tff(fact_5390_gcd__0__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).

% gcd_0_nat
tff(fact_5391_gcd__0__left__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).

% gcd_0_left_nat
tff(fact_5392_gcd__1__nat,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_5393_gcd__nat_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb1
tff(fact_5394_gcd__nat_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb2
tff(fact_5395_gcd__nat_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2) ) ) ).

% gcd_nat.bounded_iff
tff(fact_5396_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),Y)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_5397_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Y),X)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = Y ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_5398_gcd__Suc__0,axiom,
    ! [Ma: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_5399_gcd__pos__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),N))
    <=> ( ( Ma != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_5400_gcd__int__int__eq,axiom,
    ! [Ma: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),Ma)),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),gcd_gcd(nat),Ma),N)) ).

% gcd_int_int_eq
tff(fact_5401_of__rat__0,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ( aa(rat,B,field_char_0_of_rat(B),zero_zero(rat)) = zero_zero(B) ) ) ).

% of_rat_0
tff(fact_5402_of__rat__eq__0__iff,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: rat] :
          ( ( aa(rat,B,field_char_0_of_rat(B),A2) = zero_zero(B) )
        <=> ( A2 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_5403_zero__eq__of__rat__iff,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: rat] :
          ( ( zero_zero(B) = aa(rat,B,field_char_0_of_rat(B),A2) )
        <=> ( zero_zero(rat) = A2 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_5404_of__rat__1,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ( aa(rat,B,field_char_0_of_rat(B),one_one(rat)) = one_one(B) ) ) ).

% of_rat_1
tff(fact_5405_of__rat__eq__1__iff,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: rat] :
          ( ( aa(rat,B,field_char_0_of_rat(B),A2) = one_one(B) )
        <=> ( A2 = one_one(rat) ) ) ) ).

% of_rat_eq_1_iff
tff(fact_5406_one__eq__of__rat__iff,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: rat] :
          ( ( one_one(B) = aa(rat,B,field_char_0_of_rat(B),A2) )
        <=> ( one_one(rat) = A2 ) ) ) ).

% one_eq_of_rat_iff
tff(fact_5407_of__rat__of__nat__eq,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [N: nat] : aa(rat,B,field_char_0_of_rat(B),aa(nat,rat,semiring_1_of_nat(rat),N)) = aa(nat,B,semiring_1_of_nat(B),N) ) ).

% of_rat_of_nat_eq
tff(fact_5408_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).

% gcd_nat_abs_left_eq
tff(fact_5409_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_5410_zero__less__of__rat__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(rat,B,field_char_0_of_rat(B),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R) ) ) ).

% zero_less_of_rat_iff
tff(fact_5411_of__rat__less__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(rat,B,field_char_0_of_rat(B),R)),zero_zero(B))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R),zero_zero(rat)) ) ) ).

% of_rat_less_0_iff
tff(fact_5412_of__rat__le__0__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(rat,B,field_char_0_of_rat(B),R)),zero_zero(B))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R),zero_zero(rat)) ) ) ).

% of_rat_le_0_iff
tff(fact_5413_zero__le__of__rat__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(rat,B,field_char_0_of_rat(B),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),R) ) ) ).

% zero_le_of_rat_iff
tff(fact_5414_of__rat__neg__one,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ( aa(rat,B,field_char_0_of_rat(B),aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(B,B,uminus_uminus(B),one_one(B)) ) ) ).

% of_rat_neg_one
tff(fact_5415_one__less__of__rat__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(rat,B,field_char_0_of_rat(B),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R) ) ) ).

% one_less_of_rat_iff
tff(fact_5416_of__rat__less__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(rat,B,field_char_0_of_rat(B),R)),one_one(B))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_5417_of__rat__le__1__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(rat,B,field_char_0_of_rat(B),R)),one_one(B))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R),one_one(rat)) ) ) ).

% of_rat_le_1_iff
tff(fact_5418_one__le__of__rat__iff,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(rat,B,field_char_0_of_rat(B),R))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R) ) ) ).

% one_le_of_rat_iff
tff(fact_5419_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Ma: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),N)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ma)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% gcd_mult_distrib_nat
tff(fact_5420_gcd__nat_Omono,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),D2)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),C2),D2)) ) ) ).

% gcd_nat.mono
tff(fact_5421_gcd__nat_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_nat.orderE
tff(fact_5422_gcd__nat_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2) ) ).

% gcd_nat.orderI
tff(fact_5423_gcd__nat_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb3
tff(fact_5424_gcd__nat_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
        & ( B2 != A2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb4
tff(fact_5425_gcd__nat_OboundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2) ) ) ).

% gcd_nat.boundedE
tff(fact_5426_gcd__nat_OboundedI,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)) ) ) ).

% gcd_nat.boundedI
tff(fact_5427_gcd__nat_Oorder__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
    <=> ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_nat.order_iff
tff(fact_5428_gcd__nat_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ).

% gcd_nat.cobounded1
tff(fact_5429_gcd__nat_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ).

% gcd_nat.cobounded2
tff(fact_5430_gcd__nat_Oabsorb__iff1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = A2 ) ) ).

% gcd_nat.absorb_iff1
tff(fact_5431_gcd__nat_Oabsorb__iff2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),A2)
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = B2 ) ) ).

% gcd_nat.absorb_iff2
tff(fact_5432_gcd__nat_OcoboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2) ) ).

% gcd_nat.coboundedI1
tff(fact_5433_gcd__nat_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2) ) ).

% gcd_nat.coboundedI2
tff(fact_5434_gcd__nat_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))
        & ( A2 != aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2) ) )
     => ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
            & ( A2 != B2 ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
              & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_5435_gcd__nat_Ostrict__order__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),B2)
        & ( A2 != B2 ) )
    <=> ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_5436_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),C2)
        & ( A2 != C2 ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2)
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_5437_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),C2)
        & ( B2 != C2 ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),C2)
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_5438_gcd__unique__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),B2)
        & ! [E5: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),E5),A2)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),E5),B2) )
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),E5),D2) ) )
    <=> ( D2 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ).

% gcd_unique_nat
tff(fact_5439_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)) ).

% gcd_red_nat
tff(fact_5440_of__rat__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( field_char_0(B)
     => ! [F: fun(C,rat),A4: set(C)] : aa(rat,B,field_char_0_of_rat(B),aa(set(C),rat,aa(fun(C,rat),fun(set(C),rat),groups7311177749621191930dd_sum(C,rat),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aTP_Lamp_tz(fun(C,rat),fun(C,B),F)),A4) ) ).

% of_rat_sum
tff(fact_5441_of__rat__prod,axiom,
    ! [B: $tType,C: $tType] :
      ( field_char_0(B)
     => ! [F: fun(C,rat),A4: set(C)] : aa(rat,B,field_char_0_of_rat(B),aa(set(C),rat,aa(fun(C,rat),fun(set(C),rat),groups7121269368397514597t_prod(C,rat),F),A4)) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aTP_Lamp_tz(fun(C,rat),fun(C,B),F)),A4) ) ).

% of_rat_prod
tff(fact_5442_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ) ).

% gcd_le1_nat
tff(fact_5443_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_5444_gcd__diff2__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),N) ) ) ).

% gcd_diff2_nat
tff(fact_5445_gcd__diff1__nat,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),Ma)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ma),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),N) ) ) ).

% gcd_diff1_nat
tff(fact_5446_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_5447_gcd__nat_Osimps,axiom,
    ! [X: nat,Y: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = $ite(Y = zero_zero(nat),X,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y))) ).

% gcd_nat.simps
tff(fact_5448_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( Y = $ite(Xa2 = zero_zero(nat),X,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2))) ) ) ).

% gcd_nat.elims
tff(fact_5449_gcd__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),gcd_gcd(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% gcd_integer.rep_eq
tff(fact_5450_gcd__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(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),gcd_gcd(int),Xa2),X)) ).

% gcd_integer.abs_eq
tff(fact_5451_Code__Numeral_Odup__code_I1_J,axiom,
    aa(code_integer,code_integer,code_dup,zero_zero(code_integer)) = zero_zero(code_integer) ).

% Code_Numeral.dup_code(1)
tff(fact_5452_of__rat__less__eq,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [R: rat,S2: rat] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(rat,B,field_char_0_of_rat(B),R)),aa(rat,B,field_char_0_of_rat(B),S2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R),S2) ) ) ).

% of_rat_less_eq
tff(fact_5453_Gcd__in,axiom,
    ! [A4: set(nat)] :
      ( ! [A3: nat,B3: nat] :
          ( aa(set(nat),$o,member(nat,A3),A4)
         => ( aa(set(nat),$o,member(nat,B3),A4)
           => aa(set(nat),$o,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3)),A4) ) )
     => ( ( A4 != bot_bot(set(nat)) )
       => aa(set(nat),$o,member(nat,gcd_Gcd(nat,A4)),A4) ) ) ).

% Gcd_in
tff(fact_5454_of__rat__add,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: rat,B2: rat] : aa(rat,B,field_char_0_of_rat(B),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),A2),B2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(rat,B,field_char_0_of_rat(B),A2)),aa(rat,B,field_char_0_of_rat(B),B2)) ) ).

% of_rat_add
tff(fact_5455_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X2: nat,Y2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_5456_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X2: nat,Y2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_5457_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),K),L) = aa(code_integer,code_integer,abs_abs(code_integer),
        $ite(L = zero_zero(code_integer),K,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),L),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))))) ).

% gcd_code_integer
tff(fact_5458_lexord__sufE,axiom,
    ! [B: $tType,Xs: list(B),Zs: list(B),Ys: list(B),Qs: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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(list(B),fun(list(B),list(B)),append(B),Xs),Zs)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Qs))),lexord(B,R))
     => ( ( Xs != Ys )
       => ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(B),nat,size_size(list(B)),Qs) )
           => aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),lexord(B,R)) ) ) ) ) ).

% lexord_sufE
tff(fact_5459_lexord__lex,axiom,
    ! [B: $tType,X: list(B),Y: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),X),Y)),lex(B,R))
    <=> ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),X),Y)),lexord(B,R))
        & ( aa(list(B),nat,size_size(list(B)),X) = aa(list(B),nat,size_size(list(B)),Y) ) ) ) ).

% lexord_lex
tff(fact_5460_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_5461_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_5462_gcd__int__def,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ).

% gcd_int_def
tff(fact_5463_of__rat__rat,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [B2: int,A2: int] :
          ( ( B2 != zero_zero(int) )
         => ( aa(rat,B,field_char_0_of_rat(B),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(int,B,ring_1_of_int(B),A2)),aa(int,B,ring_1_of_int(B),B2)) ) ) ) ).

% of_rat_rat
tff(fact_5464_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_ba(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_5465_lexord__sufI,axiom,
    ! [B: $tType,U: list(B),W: list(B),R: set(product_prod(B,B)),V2: list(B),Z: list(B)] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),U),W)),lexord(B,R))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),W)),aa(list(B),nat,size_size(list(B)),U))
       => aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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(list(B),fun(list(B),list(B)),append(B),U),V2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),W),Z))),lexord(B,R)) ) ) ).

% lexord_sufI
tff(fact_5466_gcd__integer__def,axiom,
    gcd_gcd(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)),gcd_gcd(int)) ).

% gcd_integer_def
tff(fact_5467_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_5468_List_Olexordp__def,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),Xs: list(B),Ys: list(B)] :
      ( lexordp(B,R,Xs,Ys)
    <=> aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Xs),Ys)),lexord(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),R)))) ) ).

% List.lexordp_def
tff(fact_5469_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero(nat),X,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_5470_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa2),Xa2)
     => ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),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_5471_gcd__idem__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),X) = X ).

% gcd_idem_nat
tff(fact_5472_gcd__nat_Oright__idem,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ).

% gcd_nat.right_idem
tff(fact_5473_gcd__nat_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ).

% gcd_nat.left_idem
tff(fact_5474_gcd__nat_Oidem,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),A2) = A2 ).

% gcd_nat.idem
tff(fact_5475_ratrel__iff,axiom,
    ! [X: product_prod(int,int),Y: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),Y)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Y) != 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),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y)),aa(product_prod(int,int),int,product_snd(int,int),X)) ) ) ) ).

% ratrel_iff
tff(fact_5476_one__rat_Orsp,axiom,
    aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

% one_rat.rsp
tff(fact_5477_zero__rat_Orsp,axiom,
    aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,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),zero_zero(int)),one_one(int))) ).

% zero_rat.rsp
tff(fact_5478_ratrel__def,axiom,
    ! [X4: product_prod(int,int),Xa: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X4),Xa)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X4) != 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),X4)),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),X4)) ) ) ) ).

% ratrel_def
tff(fact_5479_positive_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
     => ( aa(rat,$o,positive,aa(product_prod(int,int),rat,abs_Rat,X))
      <=> aa(int,$o,aa(int,fun(int,$o),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_5480_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,
            $ite(aa(product_prod(int,int),int,product_fst(int,int),X) = 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),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_5481_set__relcomp,axiom,
    ! [C: $tType,D: $tType,B: $tType,Xys: list(product_prod(B,D)),Yzs: list(product_prod(D,C))] : relcomp(B,D,C,aa(list(product_prod(B,D)),set(product_prod(B,D)),set2(product_prod(B,D)),Xys),aa(list(product_prod(D,C)),set(product_prod(D,C)),set2(product_prod(D,C)),Yzs)) = aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),concat(product_prod(B,C),aa(list(product_prod(B,D)),list(list(product_prod(B,C))),map(product_prod(B,D),list(product_prod(B,C)),aTP_Lamp_ub(list(product_prod(D,C)),fun(product_prod(B,D),list(product_prod(B,C))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_5482_Code__Numeral_Osub__code_I9_J,axiom,
    ! [Ma: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,Ma)),aa(num,num,bit1,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Ma),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(9)
tff(fact_5483_Code__Numeral_Osub__code_I8_J,axiom,
    ! [Ma: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,Ma)),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,Ma),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(8)
tff(fact_5484_Code__Numeral_Osub__code_I1_J,axiom,
    aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),one2) = zero_zero(code_integer) ).

% Code_Numeral.sub_code(1)
tff(fact_5485_Code__Numeral_Osub__code_I6_J,axiom,
    ! [Ma: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,Ma)),aa(num,num,bit0,N)) = aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Ma),N)) ).

% Code_Numeral.sub_code(6)
tff(fact_5486_Code__Numeral_Osub__code_I7_J,axiom,
    ! [Ma: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,Ma)),aa(num,num,bit1,N)) = aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Ma),N)) ).

% Code_Numeral.sub_code(7)
tff(fact_5487_sub_Orep__eq,axiom,
    ! [X: num,Xa2: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,X),Xa2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),X)),aa(num,int,numeral_numeral(int),Xa2)) ).

% sub.rep_eq
tff(fact_5488_sub_Oabs__eq,axiom,
    ! [Xa2: num,X: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Xa2),X) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xa2)),aa(num,int,numeral_numeral(int),X))) ).

% sub.abs_eq
tff(fact_5489_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_uc(num,fun(num,int))) ).

% Code_Numeral.sub_def
tff(fact_5490_map__to__set__map__of,axiom,
    ! [C: $tType,B: $tType,L: list(product_prod(B,C))] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),L))
     => ( map_to_set(B,C,map_of(B,C,L)) = aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),L) ) ) ).

% map_to_set_map_of
tff(fact_5491_map__of__map__to__set,axiom,
    ! [C: $tType,B: $tType,L: list(product_prod(B,C)),Ma: fun(B,option(C))] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),L))
     => ( ( map_of(B,C,L) = Ma )
      <=> ( aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),L) = map_to_set(B,C,Ma) ) ) ) ).

% map_of_map_to_set
tff(fact_5492_map__of__distinct__upd,axiom,
    ! [B: $tType,C: $tType,X: B,Xs: list(product_prod(B,C)),Y: C] :
      ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs)))
     => ( map_add(B,C,fun_upd(B,option(C),aTP_Lamp_jv(B,option(C)),X,aa(C,option(C),some(C),Y)),map_of(B,C,Xs)) = fun_upd(B,option(C),map_of(B,C,Xs),X,aa(C,option(C),some(C),Y)) ) ) ).

% map_of_distinct_upd
tff(fact_5493_empty__eq__map__add__iff,axiom,
    ! [B: $tType,C: $tType,F: fun(B,option(C)),G: fun(B,option(C))] :
      ( ( aTP_Lamp_jv(B,option(C)) = map_add(B,C,F,G) )
    <=> ( ! [X3: B] : aa(B,option(C),F,X3) = none(C)
        & ! [X3: B] : aa(B,option(C),G,X3) = none(C) ) ) ).

% empty_eq_map_add_iff
tff(fact_5494_map__add__empty,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : map_add(B,C,Ma,aTP_Lamp_jv(B,option(C))) = Ma ).

% map_add_empty
tff(fact_5495_empty__map__add,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : map_add(B,C,aTP_Lamp_jv(B,option(C)),Ma) = Ma ).

% empty_map_add
tff(fact_5496_map__to__set__empty,axiom,
    ! [C: $tType,B: $tType] : map_to_set(B,C,aTP_Lamp_jv(B,option(C))) = bot_bot(set(product_prod(B,C))) ).

% map_to_set_empty
tff(fact_5497_map__add__first__le,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [Ma: fun(B,option(C)),M3: fun(B,option(C)),N: fun(B,option(C))] :
          ( aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),Ma),M3)
         => aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),map_add(B,C,Ma,N)),map_add(B,C,M3,N)) ) ) ).

% map_add_first_le
tff(fact_5498_map__add__left__None,axiom,
    ! [B: $tType,C: $tType,F: fun(C,option(B)),K: C,G: fun(C,option(B))] :
      ( ( aa(C,option(B),F,K) = none(B) )
     => ( aa(C,option(B),map_add(C,B,F,G),K) = aa(C,option(B),G,K) ) ) ).

% map_add_left_None
tff(fact_5499_map__add__find__left,axiom,
    ! [B: $tType,C: $tType,G: fun(C,option(B)),K: C,F: fun(C,option(B))] :
      ( ( aa(C,option(B),G,K) = none(B) )
     => ( aa(C,option(B),map_add(C,B,F,G),K) = aa(C,option(B),F,K) ) ) ).

% map_add_find_left
tff(fact_5500_map__to__set__inverse,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : set_to_map(B,C,map_to_set(B,C,Ma)) = Ma ).

% map_to_set_inverse
tff(fact_5501_map__add__def,axiom,
    ! [C: $tType,B: $tType,M1: fun(B,option(C)),M22: fun(B,option(C)),X4: B] : aa(B,option(C),map_add(B,C,M1,M22),X4) = case_option(option(C),C,aa(B,option(C),M1,X4),some(C),aa(B,option(C),M22,X4)) ).

% map_add_def
tff(fact_5502_map__to__set__empty__iff_I2_J,axiom,
    ! [B: $tType,C: $tType,Ma: fun(B,option(C))] :
      ( ( bot_bot(set(product_prod(B,C))) = map_to_set(B,C,Ma) )
    <=> ! [X3: B] : aa(B,option(C),Ma,X3) = none(C) ) ).

% map_to_set_empty_iff(2)
tff(fact_5503_map__to__set__empty__iff_I1_J,axiom,
    ! [B: $tType,C: $tType,Ma: fun(B,option(C))] :
      ( ( map_to_set(B,C,Ma) = bot_bot(set(product_prod(B,C))) )
    <=> ! [X3: B] : aa(B,option(C),Ma,X3) = none(C) ) ).

% map_to_set_empty_iff(1)
tff(fact_5504_map__to__set__def,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : map_to_set(B,C,Ma) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_ud(fun(B,option(C)),fun(B,fun(C,$o)),Ma))) ).

% map_to_set_def
tff(fact_5505_map__add__map__of__foldr,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),Ps: list(product_prod(B,C))] : map_add(B,C,Ma,map_of(B,C,Ps)) = aa(fun(B,option(C)),fun(B,option(C)),foldr(product_prod(B,C),fun(B,option(C)),aa(fun(B,fun(C,fun(fun(B,option(C)),fun(B,option(C))))),fun(product_prod(B,C),fun(fun(B,option(C)),fun(B,option(C)))),product_case_prod(B,C,fun(fun(B,option(C)),fun(B,option(C)))),aTP_Lamp_ue(B,fun(C,fun(fun(B,option(C)),fun(B,option(C)))))),Ps),Ma) ).

% map_add_map_of_foldr
tff(fact_5506_map__of__concat,axiom,
    ! [C: $tType,B: $tType,Xss: list(list(product_prod(B,C)))] : map_of(B,C,concat(product_prod(B,C),Xss)) = aa(fun(B,option(C)),fun(B,option(C)),foldr(list(product_prod(B,C)),fun(B,option(C)),aTP_Lamp_uf(list(product_prod(B,C)),fun(fun(B,option(C)),fun(B,option(C)))),Xss),aTP_Lamp_jv(B,option(C))) ).

% map_of_concat
tff(fact_5507_map__to__set__ran,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B))] : ran(C,B,Ma) = aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),map_to_set(C,B,Ma)) ).

% map_to_set_ran
tff(fact_5508_cr__rat__def,axiom,
    ! [X4: product_prod(int,int),Xa: rat] :
      ( cr_rat(X4,Xa)
    <=> ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X4),X4)
        & ( aa(product_prod(int,int),rat,abs_Rat,X4) = Xa ) ) ) ).

% cr_rat_def
tff(fact_5509_set__zip,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] : aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,Xs,Ys)) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(list(C),fun(product_prod(B,C),$o),aTP_Lamp_ug(list(B),fun(list(C),fun(product_prod(B,C),$o)),Xs),Ys)) ).

% set_zip
tff(fact_5510_mergesort__by__rel_Opinduct,axiom,
    ! [B: $tType,A0: fun(B,fun(B,$o)),A12: list(B),Pa: fun(fun(B,fun(B,$o)),fun(list(B),$o))] :
      ( accp(product_prod(fun(B,fun(B,$o)),list(B)),mergesort_by_rel_rel(B),aa(list(B),product_prod(fun(B,fun(B,$o)),list(B)),aa(fun(B,fun(B,$o)),fun(list(B),product_prod(fun(B,fun(B,$o)),list(B))),product_Pair(fun(B,fun(B,$o)),list(B)),A0),A12))
     => ( ! [R4: fun(B,fun(B,$o)),Xs2: list(B)] :
            ( accp(product_prod(fun(B,fun(B,$o)),list(B)),mergesort_by_rel_rel(B),aa(list(B),product_prod(fun(B,fun(B,$o)),list(B)),aa(fun(B,fun(B,$o)),fun(list(B),product_prod(fun(B,fun(B,$o)),list(B))),product_Pair(fun(B,fun(B,$o)),list(B)),R4),Xs2))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
               => aa(list(B),$o,aa(fun(B,fun(B,$o)),fun(list(B),$o),Pa,R4),aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xs2))) )
             => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
                 => aa(list(B),$o,aa(fun(B,fun(B,$o)),fun(list(B),$o),Pa,R4),aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B)),Xs2))) )
               => aa(list(B),$o,aa(fun(B,fun(B,$o)),fun(list(B),$o),Pa,R4),Xs2) ) ) )
       => aa(list(B),$o,aa(fun(B,fun(B,$o)),fun(list(B),$o),Pa,A0),A12) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_5511_finite__Collect__bounded__ex,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),Q: fun(C,fun(B,$o))] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
     => ( aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_uh(fun(B,$o),fun(fun(C,fun(B,$o)),fun(C,$o)),Pa),Q)))
      <=> ! [Y5: B] :
            ( aa(B,$o,Pa,Y5)
           => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5512_eq__or__mem__image__simp,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A2: C,B4: set(C)] : aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aa(C,fun(set(C),fun(B,$o)),aTP_Lamp_uj(fun(C,B),fun(C,fun(set(C),fun(B,$o))),F),A2),B4)) = aa(set(B),set(B),insert(B,aa(C,B,F,A2)),aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_uk(fun(C,B),fun(set(C),fun(B,$o)),F),B4))) ).

% eq_or_mem_image_simp
tff(fact_5513_map__to__set__upd,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),K: B,V2: C] : map_to_set(B,C,fun_upd(B,option(C),Ma,K,aa(C,option(C),some(C),V2))) = aa(set(product_prod(B,C)),set(product_prod(B,C)),insert(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),K),V2)),aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),minus_minus(set(product_prod(B,C))),map_to_set(B,C,Ma)),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aTP_Lamp_ul(B,fun(product_prod(B,C),$o),K)))) ).

% map_to_set_upd
tff(fact_5514_fs__contract,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(B,fun(C,D)),S: set(D)] : aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(set(D),fun(product_prod(B,C),$o),aTP_Lamp_um(fun(B,fun(C,D)),fun(set(D),fun(product_prod(B,C),$o)),F),S))) = aa(fun(B,$o),set(B),collect(B),aa(set(D),fun(B,$o),aTP_Lamp_un(fun(B,fun(C,D)),fun(set(D),fun(B,$o)),F),S)) ).

% fs_contract
tff(fact_5515_set__Cons__def,axiom,
    ! [B: $tType,A4: set(B),XS: set(list(B))] : set_Cons(B,A4,XS) = aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(set(list(B)),fun(list(B),$o),aTP_Lamp_uo(set(B),fun(set(list(B)),fun(list(B),$o)),A4),XS)) ).

% set_Cons_def
tff(fact_5516_finite__image__set,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),F: fun(B,C)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
     => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(B,C),fun(C,$o),aTP_Lamp_up(fun(B,$o),fun(fun(B,C),fun(C,$o)),Pa),F))) ) ).

% finite_image_set
tff(fact_5517_finite__image__set2,axiom,
    ! [D: $tType,C: $tType,B: $tType,Pa: fun(B,$o),Q: fun(C,$o),F: fun(B,fun(C,D))] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
     => ( aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),Q))
       => aa(set(D),$o,finite_finite2(D),aa(fun(D,$o),set(D),collect(D),aa(fun(B,fun(C,D)),fun(D,$o),aa(fun(C,$o),fun(fun(B,fun(C,D)),fun(D,$o)),aTP_Lamp_uq(fun(B,$o),fun(fun(C,$o),fun(fun(B,fun(C,D)),fun(D,$o))),Pa),Q),F))) ) ) ).

% finite_image_set2
tff(fact_5518_setcompr__eq__image,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Pa: fun(C,$o)] : aa(fun(B,$o),set(B),collect(B),aa(fun(C,$o),fun(B,$o),aTP_Lamp_ur(fun(C,B),fun(fun(C,$o),fun(B,$o)),F),Pa)) = aa(set(C),set(B),image2(C,B,F),aa(fun(C,$o),set(C),collect(C),Pa)) ).

% setcompr_eq_image
tff(fact_5519_Setcompr__eq__image,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A4: set(C)] : aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_uk(fun(C,B),fun(set(C),fun(B,$o)),F),A4)) = aa(set(C),set(B),image2(C,B,F),A4) ).

% Setcompr_eq_image
tff(fact_5520_exE__some,axiom,
    ! [B: $tType,Pa: fun(B,$o),C2: B] :
      ( ? [X_1: B] : aa(B,$o,Pa,X_1)
     => ( ( C2 = fChoice(B,Pa) )
       => aa(B,$o,Pa,C2) ) ) ).

% exE_some
tff(fact_5521_mergesort__by__rel_Ocases,axiom,
    ! [B: $tType,X: product_prod(fun(B,fun(B,$o)),list(B))] :
      ~ ! [R4: fun(B,fun(B,$o)),Xs2: list(B)] : X != aa(list(B),product_prod(fun(B,fun(B,$o)),list(B)),aa(fun(B,fun(B,$o)),fun(list(B),product_prod(fun(B,fun(B,$o)),list(B))),product_Pair(fun(B,fun(B,$o)),list(B)),R4),Xs2) ).

% mergesort_by_rel.cases
tff(fact_5522_ran__def,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B))] : ran(C,B,Ma) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_us(fun(C,option(B)),fun(B,$o),Ma)) ).

% ran_def
tff(fact_5523_some__theI,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o))] :
      ( ? [A8: B,X_1: C] : aa(C,$o,aa(B,fun(C,$o),Pa,A8),X_1)
     => ( ! [B1: C,B22: C] :
            ( ? [A3: B] : aa(C,$o,aa(B,fun(C,$o),Pa,A3),B1)
           => ( ? [A3: B] : aa(C,$o,aa(B,fun(C,$o),Pa,A3),B22)
             => ( B1 = B22 ) ) )
       => aa(C,$o,aa(B,fun(C,$o),Pa,fChoice(B,aTP_Lamp_ut(fun(B,fun(C,$o)),fun(B,$o),Pa))),the(C,aTP_Lamp_uu(fun(B,fun(C,$o)),fun(C,$o),Pa))) ) ) ).

% some_theI
tff(fact_5524_listrel1__def,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : listrel1(B,R) = aa(fun(product_prod(list(B),list(B)),$o),set(product_prod(list(B),list(B))),collect(product_prod(list(B),list(B))),aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_uv(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),R))) ).

% listrel1_def
tff(fact_5525_lexord__def,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : lexord(B,R) = aa(fun(product_prod(list(B),list(B)),$o),set(product_prod(list(B),list(B))),collect(product_prod(list(B),list(B))),aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_uw(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),R))) ).

% lexord_def
tff(fact_5526_lex__conv,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : lex(B,R) = aa(fun(product_prod(list(B),list(B)),$o),set(product_prod(list(B),list(B))),collect(product_prod(list(B),list(B))),aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_ux(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),R))) ).

% lex_conv
tff(fact_5527_relcomp__unfold,axiom,
    ! [B: $tType,C: $tType,D: $tType,R: set(product_prod(B,D)),S2: set(product_prod(D,C))] : relcomp(B,D,C,R,S2) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(set(product_prod(D,C)),fun(B,fun(C,$o)),aTP_Lamp_uy(set(product_prod(B,D)),fun(set(product_prod(D,C)),fun(B,fun(C,$o))),R),S2))) ).

% relcomp_unfold
tff(fact_5528_set__map__filter,axiom,
    ! [B: $tType,C: $tType,G: fun(C,option(B)),Xs: list(C)] : aa(list(B),set(B),set2(B),map_filter(C,B,G,Xs)) = aa(fun(B,$o),set(B),collect(B),aa(list(C),fun(B,$o),aTP_Lamp_uz(fun(C,option(B)),fun(list(C),fun(B,$o)),G),Xs)) ).

% set_map_filter
tff(fact_5529_set__conv__nth,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),set(B),set2(B),Xs) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_va(list(B),fun(B,$o),Xs)) ).

% set_conv_nth
tff(fact_5530_set__nths,axiom,
    ! [B: $tType,Xs: list(B),I5: set(nat)] : aa(list(B),set(B),set2(B),nths(B,Xs,I5)) = aa(fun(B,$o),set(B),collect(B),aa(set(nat),fun(B,$o),aTP_Lamp_vb(list(B),fun(set(nat),fun(B,$o)),Xs),I5)) ).

% set_nths
tff(fact_5531_set__drop__conv,axiom,
    ! [B: $tType,N: nat,L: list(B)] : aa(list(B),set(B),set2(B),drop(B,N,L)) = aa(fun(B,$o),set(B),collect(B),aa(list(B),fun(B,$o),aTP_Lamp_vc(nat,fun(list(B),fun(B,$o)),N),L)) ).

% set_drop_conv
tff(fact_5532_arg__min__list_Opelims,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [X: fun(B,C),Xa2: list(B),Y: B] :
          ( ( arg_min_list(B,C,X,Xa2) = Y )
         => ( accp(product_prod(fun(B,C),list(B)),arg_min_list_rel(B,C),aa(list(B),product_prod(fun(B,C),list(B)),aa(fun(B,C),fun(list(B),product_prod(fun(B,C),list(B))),product_Pair(fun(B,C),list(B)),X),Xa2))
           => ( ! [X2: B] :
                  ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)) )
                 => ( ( Y = X2 )
                   => ~ accp(product_prod(fun(B,C),list(B)),arg_min_list_rel(B,C),aa(list(B),product_prod(fun(B,C),list(B)),aa(fun(B,C),fun(list(B),product_prod(fun(B,C),list(B))),product_Pair(fun(B,C),list(B)),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),nil(B)))) ) )
             => ( ! [X2: B,Y2: B,Zs2: list(B)] :
                    ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Zs2)) )
                   => ( ( Y = $let(
                            m2: B,
                            m2:= arg_min_list(B,C,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Zs2)),
                            $ite(aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,X,X2)),aa(B,C,X,m2)),X2,m2) ) )
                     => ~ accp(product_prod(fun(B,C),list(B)),arg_min_list_rel(B,C),aa(list(B),product_prod(fun(B,C),list(B)),aa(fun(B,C),fun(list(B),product_prod(fun(B,C),list(B))),product_Pair(fun(B,C),list(B)),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Zs2)))) ) )
               => ~ ( ( Xa2 = nil(B) )
                   => ( ( Y = undefined(B) )
                     => ~ accp(product_prod(fun(B,C),list(B)),arg_min_list_rel(B,C),aa(list(B),product_prod(fun(B,C),list(B)),aa(fun(B,C),fun(list(B),product_prod(fun(B,C),list(B))),product_Pair(fun(B,C),list(B)),X),nil(B))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_5533_lexn__conv,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),N: nat] : aa(nat,set(product_prod(list(B),list(B))),lexn(B,R),N) = aa(fun(product_prod(list(B),list(B)),$o),set(product_prod(list(B),list(B))),collect(product_prod(list(B),list(B))),aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aa(nat,fun(list(B),fun(list(B),$o)),aTP_Lamp_vd(set(product_prod(B,B)),fun(nat,fun(list(B),fun(list(B),$o))),R),N))) ).

% lexn_conv
tff(fact_5534_pairself__image__eq,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Pa: fun(C,fun(C,$o))] : aa(set(product_prod(C,C)),set(product_prod(B,B)),image2(product_prod(C,C),product_prod(B,B),pairself(C,B,F)),aa(fun(product_prod(C,C),$o),set(product_prod(C,C)),collect(product_prod(C,C)),aa(fun(C,fun(C,$o)),fun(product_prod(C,C),$o),product_case_prod(C,C,$o),Pa))) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(C,fun(C,$o)),fun(product_prod(B,B),$o),aTP_Lamp_ve(fun(C,B),fun(fun(C,fun(C,$o)),fun(product_prod(B,B),$o)),F),Pa)) ).

% pairself_image_eq
tff(fact_5535_lexn_Osimps_I1_J,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : aa(nat,set(product_prod(list(B),list(B))),lexn(B,R),zero_zero(nat)) = bot_bot(set(product_prod(list(B),list(B)))) ).

% lexn.simps(1)
tff(fact_5536_pairself_Osimps,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A2: C,B2: C] : aa(product_prod(C,C),product_prod(B,B),pairself(C,B,F),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),A2),B2)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(C,B,F,A2)),aa(C,B,F,B2)) ).

% pairself.simps
tff(fact_5537_pairself_Oelims,axiom,
    ! [B: $tType,C: $tType,X: fun(C,B),Xa2: product_prod(C,C),Y: product_prod(B,B)] :
      ( ( aa(product_prod(C,C),product_prod(B,B),pairself(C,B,X),Xa2) = Y )
     => ~ ! [A3: C,B3: C] :
            ( ( Xa2 = aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),A3),B3) )
           => ( Y != aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(C,B,X,A3)),aa(C,B,X,B3)) ) ) ) ).

% pairself.elims
tff(fact_5538_lexn__length,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B),R: set(product_prod(B,B)),N: nat] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),aa(nat,set(product_prod(list(B),list(B))),lexn(B,R),N))
     => ( ( aa(list(B),nat,size_size(list(B)),Xs) = N )
        & ( aa(list(B),nat,size_size(list(B)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_5539_mergesort__by__rel_Opelims,axiom,
    ! [B: $tType,X: fun(B,fun(B,$o)),Xa2: list(B),Y: list(B)] :
      ( ( aa(list(B),list(B),mergesort_by_rel(B,X),Xa2) = Y )
     => ( accp(product_prod(fun(B,fun(B,$o)),list(B)),mergesort_by_rel_rel(B),aa(list(B),product_prod(fun(B,fun(B,$o)),list(B)),aa(fun(B,fun(B,$o)),fun(list(B),product_prod(fun(B,fun(B,$o)),list(B))),product_Pair(fun(B,fun(B,$o)),list(B)),X),Xa2))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xa2,merges9089515139780605204_merge(B,X,aa(list(B),list(B),mergesort_by_rel(B,X),aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B)),Xa2))),aa(list(B),list(B),mergesort_by_rel(B,X),aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B)),Xa2))))) )
           => ~ accp(product_prod(fun(B,fun(B,$o)),list(B)),mergesort_by_rel_rel(B),aa(list(B),product_prod(fun(B,fun(B,$o)),list(B)),aa(fun(B,fun(B,$o)),fun(list(B),product_prod(fun(B,fun(B,$o)),list(B))),product_Pair(fun(B,fun(B,$o)),list(B)),X),Xa2)) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_5540_mergesort__by__rel_Opsimps,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] :
      ( accp(product_prod(fun(B,fun(B,$o)),list(B)),mergesort_by_rel_rel(B),aa(list(B),product_prod(fun(B,fun(B,$o)),list(B)),aa(fun(B,fun(B,$o)),fun(list(B),product_prod(fun(B,fun(B,$o)),list(B))),product_Pair(fun(B,fun(B,$o)),list(B)),R2),Xs))
     => ( aa(list(B),list(B),mergesort_by_rel(B,R2),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xs,merges9089515139780605204_merge(B,R2,aa(list(B),list(B),mergesort_by_rel(B,R2),aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xs))),aa(list(B),list(B),mergesort_by_rel(B,R2),aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xs))))) ) ) ).

% mergesort_by_rel.psimps
tff(fact_5541_Eps__Opt__def,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      eps_Opt(B,Pa) = $ite(
        ? [X7: B] : aa(B,$o,Pa,X7),
        aa(B,option(B),some(B),fChoice(B,Pa)),
        none(B) ) ).

% Eps_Opt_def
tff(fact_5542_mergesort__by__rel__simps_I1_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o))] : aa(list(B),list(B),mergesort_by_rel(B,R2),nil(B)) = nil(B) ).

% mergesort_by_rel_simps(1)
tff(fact_5543_set__mergesort__by__rel,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),mergesort_by_rel(B,R2),Xs)) = aa(list(B),set(B),set2(B),Xs) ).

% set_mergesort_by_rel
tff(fact_5544_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Ys: list(B)] : merges9089515139780605204_merge(B,R2,nil(B),Ys) = Ys ).

% mergesort_by_rel_merge_simps(3)
tff(fact_5545_sorted__wrt__mergesort__by__rel__merge,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B),Ys: list(B)] :
      ( ! [X2: B,Y2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),R2,X2),Y2)
          | aa(B,$o,aa(B,fun(B,$o),R2,Y2),X2) )
     => ( ! [X2: B,Y2: B,Z2: B] :
            ( aa(B,$o,aa(B,fun(B,$o),R2,X2),Y2)
           => ( aa(B,$o,aa(B,fun(B,$o),R2,Y2),Z2)
             => aa(B,$o,aa(B,fun(B,$o),R2,X2),Z2) ) )
       => ( sorted_wrt(B,R2,merges9089515139780605204_merge(B,R2,Xs,Ys))
        <=> ( sorted_wrt(B,R2,Xs)
            & sorted_wrt(B,R2,Ys) ) ) ) ) ).

% sorted_wrt_mergesort_by_rel_merge
tff(fact_5546_mergesort__by__rel__permutes,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] : mset(B,aa(list(B),list(B),mergesort_by_rel(B,R2),Xs)) = mset(B,Xs) ).

% mergesort_by_rel_permutes
tff(fact_5547_some__opt__sym__eq__trivial,axiom,
    ! [B: $tType,X: B] : eps_Opt(B,aa(B,fun(B,$o),fequal(B),X)) = aa(B,option(B),some(B),X) ).

% some_opt_sym_eq_trivial
tff(fact_5548_some__opt__eq__trivial,axiom,
    ! [B: $tType,X: B] : eps_Opt(B,aTP_Lamp_af(B,fun(B,$o),X)) = aa(B,option(B),some(B),X) ).

% some_opt_eq_trivial
tff(fact_5549_some__opt__false__trivial,axiom,
    ! [B: $tType] : eps_Opt(B,aTP_Lamp_hh(B,$o)) = none(B) ).

% some_opt_false_trivial
tff(fact_5550_mergesort__by__rel__simps_I2_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),X: B] : aa(list(B),list(B),mergesort_by_rel(B,R2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)) ).

% mergesort_by_rel_simps(2)
tff(fact_5551_mset__mergesort__by__rel__merge,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B),Ys: list(B)] : mset(B,merges9089515139780605204_merge(B,R2,Xs,Ys)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),mset(B,Xs)),mset(B,Ys)) ).

% mset_mergesort_by_rel_merge
tff(fact_5552_Eps__Opt__eq__None,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ( eps_Opt(B,Pa) = none(B) )
    <=> ~ ? [X_12: B] : aa(B,$o,Pa,X_12) ) ).

% Eps_Opt_eq_None
tff(fact_5553_mergesort__by__rel__simps_I3_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),X12: B,X22: B,Xs: list(B)] : aa(list(B),list(B),mergesort_by_rel(B,R2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X12),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),Xs))) = aa(product_prod(list(B),list(B)),list(B),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)),aTP_Lamp_vf(fun(B,fun(B,$o)),fun(list(B),fun(list(B),list(B))),R2)),merges295452479951948502_split(B,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),X12),nil(B))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X22),nil(B))),Xs)) ).

% mergesort_by_rel_simps(3)
tff(fact_5554_sorted__wrt__mergesort__by__rel,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] :
      ( ! [X2: B,Y2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),R2,X2),Y2)
          | aa(B,$o,aa(B,fun(B,$o),R2,Y2),X2) )
     => ( ! [X2: B,Y2: B,Z2: B] :
            ( aa(B,$o,aa(B,fun(B,$o),R2,X2),Y2)
           => ( aa(B,$o,aa(B,fun(B,$o),R2,Y2),Z2)
             => aa(B,$o,aa(B,fun(B,$o),R2,X2),Z2) ) )
       => sorted_wrt(B,R2,aa(list(B),list(B),mergesort_by_rel(B,R2),Xs)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_5555_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] : merges9089515139780605204_merge(B,R2,Xs,nil(B)) = Xs ).

% mergesort_by_rel_merge_simps(2)
tff(fact_5556_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),X: B,Xs: list(B),Y: B,Ys: list(B)] :
      merges9089515139780605204_merge(B,R2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),R2,X),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),merges9089515139780605204_merge(B,R2,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),merges9089515139780605204_merge(B,R2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),Ys))) ).

% mergesort_by_rel_merge_simps(1)
tff(fact_5557_Eps__Opt__eq__Some,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B] :
      ( ! [X8: B] :
          ( aa(B,$o,Pa,X)
         => ( aa(B,$o,Pa,X8)
           => ( X8 = X ) ) )
     => ( ( eps_Opt(B,Pa) = aa(B,option(B),some(B),X) )
      <=> aa(B,$o,Pa,X) ) ) ).

% Eps_Opt_eq_Some
tff(fact_5558_Eps__Opt__eq__Some__implies,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B] :
      ( ( eps_Opt(B,Pa) = aa(B,option(B),some(B),X) )
     => aa(B,$o,Pa,X) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_5559_sorted__mergesort__by__rel,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),mergesort_by_rel(B,ord_less_eq(B)),Xs)) ) ).

% sorted_mergesort_by_rel
tff(fact_5560_mergesort__by__rel__merge_Oelims,axiom,
    ! [B: $tType,X: fun(B,fun(B,$o)),Xa2: list(B),Xb: list(B),Y: list(B)] :
      ( ( merges9089515139780605204_merge(B,X,Xa2,Xb) = Y )
     => ( ! [X2: B,Xs2: list(B)] :
            ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
           => ! [Y2: B,Ys3: list(B)] :
                ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
               => ( Y != $ite(aa(B,$o,aa(B,fun(B,$o),X,X2),Y2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),merges9089515139780605204_merge(B,X,Xs2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),merges9089515139780605204_merge(B,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2),Ys3))) ) ) )
       => ( ( ( Xb = nil(B) )
           => ( Y != Xa2 ) )
         => ~ ( ( Xa2 = nil(B) )
             => ! [V4: B,Va: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
                 => ( Y != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
tff(fact_5561_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),V2: B,Va2: list(B)] : merges9089515139780605204_merge(B,R2,nil(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2) ).

% mergesort_by_rel_merge.simps(3)
tff(fact_5562_sort__mergesort__by__rel,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( linorder_sort_key(B,B,aTP_Lamp_hr(B,B)) = mergesort_by_rel(B,ord_less_eq(B)) ) ) ).

% sort_mergesort_by_rel
tff(fact_5563_mergesort__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ( mergesort(B) = mergesort_by_rel(B,ord_less_eq(B)) ) ) ).

% mergesort_def
tff(fact_5564_set__to__map__def,axiom,
    ! [B: $tType,C: $tType,S: set(product_prod(C,B)),K: C] : aa(C,option(B),set_to_map(C,B,S),K) = eps_Opt(B,aa(C,fun(B,$o),aTP_Lamp_mh(set(product_prod(C,B)),fun(C,fun(B,$o)),S),K)) ).

% set_to_map_def
tff(fact_5565_mergesort__by__rel_Oelims,axiom,
    ! [B: $tType,X: fun(B,fun(B,$o)),Xa2: list(B),Y: list(B)] :
      ( ( aa(list(B),list(B),mergesort_by_rel(B,X),Xa2) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xa2,merges9089515139780605204_merge(B,X,aa(list(B),list(B),mergesort_by_rel(B,X),aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xa2))),aa(list(B),list(B),mergesort_by_rel(B,X),aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xa2))))) ) ) ).

% mergesort_by_rel.elims
tff(fact_5566_mergesort__by__rel_Osimps,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B)] :
      aa(list(B),list(B),mergesort_by_rel(B,R2),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xs,merges9089515139780605204_merge(B,R2,aa(list(B),list(B),mergesort_by_rel(B,R2),aa(product_prod(list(B),list(B)),list(B),product_fst(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xs))),aa(list(B),list(B),mergesort_by_rel(B,R2),aa(product_prod(list(B),list(B)),list(B),product_snd(list(B),list(B)),merges295452479951948502_split(B,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)),nil(B)),nil(B)),Xs))))) ).

% mergesort_by_rel.simps
tff(fact_5567_image2__def,axiom,
    ! [B: $tType,C: $tType,D: $tType,A4: set(D),F: fun(D,B),G: fun(D,C)] : bNF_Greatest_image2(D,B,C,A4,F,G) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(D,C),fun(product_prod(B,C),$o),aa(fun(D,B),fun(fun(D,C),fun(product_prod(B,C),$o)),aTP_Lamp_vg(set(D),fun(fun(D,B),fun(fun(D,C),fun(product_prod(B,C),$o))),A4),F),G)) ).

% image2_def
tff(fact_5568_image__mset__map__of,axiom,
    ! [C: $tType,B: $tType,Xs: list(product_prod(B,C))] :
      ( distinct(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs))
     => ( aa(multiset(B),multiset(C),image_mset(B,C,aTP_Lamp_vh(list(product_prod(B,C)),fun(B,C),Xs)),mset(B,aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Xs))) = mset(C,aa(list(product_prod(B,C)),list(C),map(product_prod(B,C),C,product_snd(B,C)),Xs)) ) ) ).

% image_mset_map_of
tff(fact_5569_finite__range__prod,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(C,product_prod(B,D))] :
      ( aa(set(B),$o,finite_finite2(B),aa(set(C),set(B),image2(C,B,aa(fun(C,product_prod(B,D)),fun(C,B),aa(fun(product_prod(B,D),B),fun(fun(C,product_prod(B,D)),fun(C,B)),comp(product_prod(B,D),B,C),product_fst(B,D)),F)),top_top(set(C))))
     => ( aa(set(D),$o,finite_finite2(D),aa(set(C),set(D),image2(C,D,aa(fun(C,product_prod(B,D)),fun(C,D),aa(fun(product_prod(B,D),D),fun(fun(C,product_prod(B,D)),fun(C,D)),comp(product_prod(B,D),D,C),product_snd(B,D)),F)),top_top(set(C))))
       => aa(set(product_prod(B,D)),$o,finite_finite2(product_prod(B,D)),aa(set(C),set(product_prod(B,D)),image2(C,product_prod(B,D),F),top_top(set(C)))) ) ) ).

% finite_range_prod
tff(fact_5570_top__apply,axiom,
    ! [C: $tType,B: $tType] :
      ( top(B)
     => ! [X: C] : aa(C,B,top_top(fun(C,B)),X) = top_top(B) ) ).

% top_apply
tff(fact_5571_atMost__UNIV__triv,axiom,
    ! [B: $tType] : aa(set(B),set(set(B)),set_ord_atMost(set(B)),top_top(set(B))) = top_top(set(set(B))) ).

% atMost_UNIV_triv
tff(fact_5572_card__UNIV__unit,axiom,
    aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_5573_image__mset_Oidentity,axiom,
    ! [B: $tType] : image_mset(B,B,aTP_Lamp_il(B,B)) = id(multiset(B)) ).

% image_mset.identity
tff(fact_5574_card__eq__UNIV2,axiom,
    ! [B: $tType] :
      ( finite_finite(B)
     => ! [S: set(B)] :
          ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) = aa(set(B),nat,finite_card(B),S) )
        <=> ( S = top_top(set(B)) ) ) ) ).

% card_eq_UNIV2
tff(fact_5575_card__eq__UNIV,axiom,
    ! [B: $tType] :
      ( finite_finite(B)
     => ! [S: set(B)] :
          ( ( aa(set(B),nat,finite_card(B),S) = aa(set(B),nat,finite_card(B),top_top(set(B))) )
        <=> ( S = top_top(set(B)) ) ) ) ).

% card_eq_UNIV
tff(fact_5576_min__top,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_min(B),top_top(B)),X) = X ) ).

% min_top
tff(fact_5577_min__top2,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_min(B),X),top_top(B)) = X ) ).

% min_top2
tff(fact_5578_max__top,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),top_top(B)),X) = top_top(B) ) ).

% max_top
tff(fact_5579_max__top2,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [X: B] : aa(B,B,aa(B,fun(B,B),ord_max(B),X),top_top(B)) = top_top(B) ) ).

% max_top2
tff(fact_5580_image__mset__union,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),M: multiset(C),N5: multiset(C)] : aa(multiset(C),multiset(B),image_mset(C,B,F),aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),plus_plus(multiset(C)),M),N5)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(C),multiset(B),image_mset(C,B,F),M)),aa(multiset(C),multiset(B),image_mset(C,B,F),N5)) ).

% image_mset_union
tff(fact_5581_restrict__map__UNIV,axiom,
    ! [C: $tType,B: $tType,F: fun(B,option(C))] : restrict_map(B,C,F,top_top(set(B))) = F ).

% restrict_map_UNIV
tff(fact_5582_size__image__mset,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),M: multiset(C)] : aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(C),multiset(B),image_mset(C,B,F),M)) = aa(multiset(C),nat,size_size(multiset(C)),M) ).

% size_image_mset
tff(fact_5583_Collect__const,axiom,
    ! [B: $tType,Pa: $o] :
      aa(fun(B,$o),set(B),collect(B),aTP_Lamp_vi($o,fun(B,$o),(Pa))) = $ite((Pa),top_top(set(B)),bot_bot(set(B))) ).

% Collect_const
tff(fact_5584_finite__Collect__not,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)))
      <=> aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_Collect_not
tff(fact_5585_Collect__const__case__prod,axiom,
    ! [C: $tType,B: $tType,Pa: $o] :
      aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_vj($o,fun(B,fun(C,$o)),(Pa)))) = $ite((Pa),top_top(set(product_prod(B,C))),bot_bot(set(product_prod(B,C)))) ).

% Collect_const_case_prod
tff(fact_5586_surj__plus,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),top_top(set(B))) = top_top(set(B)) ) ).

% surj_plus
tff(fact_5587_card__ge__UNIV,axiom,
    ! [B: $tType] :
      ( finite_finite(B)
     => ! [S: set(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),top_top(set(B)))),aa(set(B),nat,finite_card(B),S))
        <=> ( S = top_top(set(B)) ) ) ) ).

% card_ge_UNIV
tff(fact_5588_surj__fn,axiom,
    ! [B: $tType,F: fun(B,B),N: nat] :
      ( ( aa(set(B),set(B),image2(B,B,F),top_top(set(B))) = top_top(set(B)) )
     => ( aa(set(B),set(B),image2(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F)),top_top(set(B))) = top_top(set(B)) ) ) ).

% surj_fn
tff(fact_5589_Gcd__UNIV,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ( gcd_Gcd(B,top_top(set(B))) = one_one(B) ) ) ).

% Gcd_UNIV
tff(fact_5590_surj__diff__right,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B] : aa(set(B),set(B),image2(B,B,aTP_Lamp_jj(B,fun(B,B),A2)),top_top(set(B))) = top_top(set(B)) ) ).

% surj_diff_right
tff(fact_5591_range__constant,axiom,
    ! [C: $tType,B: $tType,X: B] : aa(set(C),set(B),image2(C,B,aa(B,fun(C,B),aTP_Lamp_it(B,fun(C,B)),X)),top_top(set(C))) = aa(set(B),set(B),insert(B,X),bot_bot(set(B))) ).

% range_constant
tff(fact_5592_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [B: $tType] :
      ( bounded_lattice(B)
     => ! [X: B,Y: B] :
          ( ( set_or1337092689740270186AtMost(B,X,Y) = top_top(set(B)) )
        <=> ( ( X = bot_bot(B) )
            & ( Y = top_top(B) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_5593_finite__fun__UNIVD1,axiom,
    ! [C: $tType,B: $tType] :
      ( aa(set(fun(B,C)),$o,finite_finite2(fun(B,C)),top_top(set(fun(B,C))))
     => ( ( aa(set(C),nat,finite_card(C),top_top(set(C))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_fun_UNIVD1
tff(fact_5594_mset__map__split__orig__le,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),Pa: multiset(C),M13: multiset(B),M24: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(C),multiset(B),image_mset(C,B,F),Pa)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M13),M24))
     => ~ ! [P12: multiset(C),P22: multiset(C)] :
            ( ( Pa = aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),plus_plus(multiset(C)),P12),P22) )
           => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(C),multiset(B),image_mset(C,B,F),P12)),M13)
             => ~ aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(C),multiset(B),image_mset(C,B,F),P22)),M24) ) ) ) ).

% mset_map_split_orig_le
tff(fact_5595_top_Oextremum__uniqueI,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),top_top(B)),A2)
         => ( A2 = top_top(B) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5596_top_Oextremum__unique,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),top_top(B)),A2)
        <=> ( A2 = top_top(B) ) ) ) ).

% top.extremum_unique
tff(fact_5597_top__greatest,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),top_top(B)) ) ).

% top_greatest
tff(fact_5598_top_Oextremum__strict,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [A2: B] : ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),top_top(B)),A2) ) ).

% top.extremum_strict
tff(fact_5599_top_Onot__eq__extremum,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [A2: B] :
          ( ( A2 != top_top(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),top_top(B)) ) ) ).

% top.not_eq_extremum
tff(fact_5600_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite2(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_5601_not__UNIV__eq__Iic,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [H4: B] : top_top(set(B)) != aa(B,set(B),set_ord_atMost(B),H4) ) ).

% not_UNIV_eq_Iic
tff(fact_5602_atMost__eq__UNIV__iff,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [X: B] :
          ( ( aa(B,set(B),set_ord_atMost(B),X) = top_top(set(B)) )
        <=> ( X = top_top(B) ) ) ) ).

% atMost_eq_UNIV_iff
tff(fact_5603_mset__map__id,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),G: fun(B,C),X5: multiset(B)] :
      ( ! [X2: B] : aa(C,B,F,aa(B,C,G,X2)) = X2
     => ( aa(multiset(C),multiset(B),image_mset(C,B,F),aa(multiset(B),multiset(C),image_mset(B,C,G),X5)) = X5 ) ) ).

% mset_map_id
tff(fact_5604_not__UNIV__eq__Icc,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [L3: B,H4: B] : top_top(set(B)) != set_or1337092689740270186AtMost(B,L3,H4) ) ).

% not_UNIV_eq_Icc
tff(fact_5605_integer__of__int__cases,axiom,
    ! [X: code_integer] :
      ~ ! [Y2: int] :
          ( ( X = aa(int,code_integer,code_integer_of_int,Y2) )
         => ~ aa(set(int),$o,member(int,Y2),top_top(set(int))) ) ).

% integer_of_int_cases
tff(fact_5606_integer__of__int__induct,axiom,
    ! [Pa: fun(code_integer,$o),X: code_integer] :
      ( ! [Y2: int] :
          ( aa(set(int),$o,member(int,Y2),top_top(set(int)))
         => aa(code_integer,$o,Pa,aa(int,code_integer,code_integer_of_int,Y2)) )
     => aa(code_integer,$o,Pa,X) ) ).

% integer_of_int_induct
tff(fact_5607_integer__of__int__inject,axiom,
    ! [X: int,Y: int] :
      ( aa(set(int),$o,member(int,X),top_top(set(int)))
     => ( aa(set(int),$o,member(int,Y),top_top(set(int)))
       => ( ( aa(int,code_integer,code_integer_of_int,X) = aa(int,code_integer,code_integer_of_int,Y) )
        <=> ( X = Y ) ) ) ) ).

% integer_of_int_inject
tff(fact_5608_int__of__integer,axiom,
    ! [X: code_integer] : aa(set(int),$o,member(int,aa(code_integer,int,code_int_of_integer,X)),top_top(set(int))) ).

% int_of_integer
tff(fact_5609_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( aa(set(int),$o,member(int,Y),top_top(set(int)))
     => ~ ! [X2: code_integer] : Y != aa(code_integer,int,code_int_of_integer,X2) ) ).

% int_of_integer_cases
tff(fact_5610_int__of__integer__induct,axiom,
    ! [Y: int,Pa: fun(int,$o)] :
      ( aa(set(int),$o,member(int,Y),top_top(set(int)))
     => ( ! [X2: code_integer] : aa(int,$o,Pa,aa(code_integer,int,code_int_of_integer,X2))
       => aa(int,$o,Pa,Y) ) ) ).

% int_of_integer_induct
tff(fact_5611_mset__map__split__orig,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),Pa: multiset(C),M13: multiset(B),M24: multiset(B)] :
      ( ( aa(multiset(C),multiset(B),image_mset(C,B,F),Pa) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M13),M24) )
     => ~ ! [P12: multiset(C),P22: multiset(C)] :
            ( ( Pa = aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),plus_plus(multiset(C)),P12),P22) )
           => ( ( aa(multiset(C),multiset(B),image_mset(C,B,F),P12) = M13 )
             => ( aa(multiset(C),multiset(B),image_mset(C,B,F),P22) != M24 ) ) ) ) ).

% mset_map_split_orig
tff(fact_5612_UNIV__def,axiom,
    ! [B: $tType] : top_top(set(B)) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_jq(B,$o)) ).

% UNIV_def
tff(fact_5613_multiset_Omap__ident,axiom,
    ! [B: $tType,Ta: multiset(B)] : aa(multiset(B),multiset(B),image_mset(B,B,aTP_Lamp_il(B,B)),Ta) = Ta ).

% multiset.map_ident
tff(fact_5614_rangeE,axiom,
    ! [B: $tType,C: $tType,B2: B,F: fun(C,B)] :
      ( aa(set(B),$o,member(B,B2),aa(set(C),set(B),image2(C,B,F),top_top(set(C))))
     => ~ ! [X2: C] : B2 != aa(C,B,F,X2) ) ).

% rangeE
tff(fact_5615_range__composition,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(D,B),G: fun(C,D)] : aa(set(C),set(B),image2(C,B,aa(fun(C,D),fun(C,B),aTP_Lamp_vk(fun(D,B),fun(fun(C,D),fun(C,B)),F),G)),top_top(set(C))) = aa(set(D),set(B),image2(D,B,F),aa(set(C),set(D),image2(C,D,G),top_top(set(C)))) ).

% range_composition
tff(fact_5616_not__UNIV__le__Icc,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [L: B,H: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),top_top(set(B))),set_or1337092689740270186AtMost(B,L,H)) ) ).

% not_UNIV_le_Icc
tff(fact_5617_not__UNIV__le__Iic,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [H: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),top_top(set(B))),aa(B,set(B),set_ord_atMost(B),H)) ) ).

% not_UNIV_le_Iic
tff(fact_5618_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( aa(set(int),$o,member(int,Y),top_top(set(int)))
     => ( aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,Y)) = Y ) ) ).

% integer_of_int_inverse
tff(fact_5619_finite__range__imageI,axiom,
    ! [B: $tType,D: $tType,C: $tType,G: fun(C,B),F: fun(B,D)] :
      ( aa(set(B),$o,finite_finite2(B),aa(set(C),set(B),image2(C,B,G),top_top(set(C))))
     => aa(set(D),$o,finite_finite2(D),aa(set(C),set(D),image2(C,D,aa(fun(B,D),fun(C,D),aTP_Lamp_vl(fun(C,B),fun(fun(B,D),fun(C,D)),G),F)),top_top(set(C)))) ) ).

% finite_range_imageI
tff(fact_5620_full__SetCompr__eq,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B)] : aa(fun(B,$o),set(B),collect(B),aTP_Lamp_vm(fun(C,B),fun(B,$o),F)) = aa(set(C),set(B),image2(C,B,F),top_top(set(C))) ).

% full_SetCompr_eq
tff(fact_5621_top_Oordering__top__axioms,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ordering_top(B,ord_less_eq(B),ord_less(B),top_top(B)) ) ).

% top.ordering_top_axioms
tff(fact_5622_image2__eqI,axiom,
    ! [B: $tType,D: $tType,C: $tType,B2: B,F: fun(C,B),X: C,C2: D,G: fun(C,D),A4: set(C)] :
      ( ( B2 = aa(C,B,F,X) )
     => ( ( C2 = aa(C,D,G,X) )
       => ( aa(set(C),$o,member(C,X),A4)
         => aa(set(product_prod(B,D)),$o,member(product_prod(B,D),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),C2)),bNF_Greatest_image2(C,B,D,A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_5623_Ints__def,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ( ring_1_Ints(B) = aa(set(int),set(B),image2(int,B,ring_1_of_int(B)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_5624_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => folding_insort_key(B,B,ord_less_eq(B),ord_less(B),top_top(set(B)),aTP_Lamp_hr(B,B)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_5625_int__in__range__abs,axiom,
    ! [N: nat] : aa(set(int),$o,member(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int)))) ).

% int_in_range_abs
tff(fact_5626_inf__top_Osemilattice__neutr__order__axioms,axiom,
    ! [B: $tType] :
      ( bounde4346867609351753570nf_top(B)
     => semila1105856199041335345_order(B,inf_inf(B),top_top(B),ord_less_eq(B),ord_less(B)) ) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_5627_finite__UNIV__card__ge__0,axiom,
    ! [B: $tType] :
      ( aa(set(B),$o,finite_finite2(B),top_top(set(B)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),top_top(set(B)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_5628_card__range__greater__zero,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B)] :
      ( aa(set(B),$o,finite_finite2(B),aa(set(C),set(B),image2(C,B,F),top_top(set(C))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),aa(set(C),set(B),image2(C,B,F),top_top(set(C))))) ) ).

% card_range_greater_zero
tff(fact_5629_range__mod,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_vn(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_5630_range__prod,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(D,product_prod(B,C))] : aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),aa(set(D),set(product_prod(B,C)),image2(D,product_prod(B,C),F),top_top(set(D)))),product_Sigma(B,C,aa(set(D),set(B),image2(D,B,aa(fun(D,product_prod(B,C)),fun(D,B),aa(fun(product_prod(B,C),B),fun(fun(D,product_prod(B,C)),fun(D,B)),comp(product_prod(B,C),B,D),product_fst(B,C)),F)),top_top(set(D))),aTP_Lamp_vo(fun(D,product_prod(B,C)),fun(B,set(C)),F))) ).

% range_prod
tff(fact_5631_dual__max,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( max(B,aTP_Lamp_kh(B,fun(B,$o))) = ord_min(B) ) ) ).

% dual_max
tff(fact_5632_card__UN__disjoint,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),I5)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),I5)
           => aa(set(C),$o,finite_finite2(C),aa(B,set(C),A4,X2)) )
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),I5)
             => ! [Xa3: B] :
                  ( aa(set(B),$o,member(B,Xa3),I5)
                 => ( ( X2 != Xa3 )
                   => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,X2)),aa(B,set(C),A4,Xa3)) = bot_bot(set(C)) ) ) ) )
         => ( aa(set(C),nat,finite_card(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aTP_Lamp_vp(fun(B,set(C)),fun(B,nat),A4)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_5633_Sup__atMost,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Y: B] : aa(set(B),B,complete_Sup_Sup(B),aa(B,set(B),set_ord_atMost(B),Y)) = Y ) ).

% Sup_atMost
tff(fact_5634_Sup__lessThan,axiom,
    ! [B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & dense_linorder(B) )
     => ! [Y: B] : aa(set(B),B,complete_Sup_Sup(B),aa(B,set(B),set_ord_lessThan(B),Y)) = Y ) ).

% Sup_lessThan
tff(fact_5635_Collect__case__prod,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),Q: fun(C,$o)] : aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(fun(C,$o),fun(B,fun(C,$o)),aTP_Lamp_vq(fun(B,$o),fun(fun(C,$o),fun(B,fun(C,$o))),Pa),Q))) = product_Sigma(B,C,aa(fun(B,$o),set(B),collect(B),Pa),aTP_Lamp_vr(fun(C,$o),fun(B,set(C)),Q)) ).

% Collect_case_prod
tff(fact_5636_Sup__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(set(B),B,complete_Sup_Sup(B),set_or1337092689740270186AtMost(B,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_5637_cSup__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(set(B),B,complete_Sup_Sup(B),set_or1337092689740270186AtMost(B,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_5638_Sup__atLeastLessThan,axiom,
    ! [B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & dense_linorder(B) )
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(set(B),B,complete_Sup_Sup(B),set_or7035219750837199246ssThan(B,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_5639_Sup__greaterThanAtMost,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(set(B),B,complete_Sup_Sup(B),set_or3652927894154168847AtMost(B,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5640_Sup__greaterThanLessThan,axiom,
    ! [B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & dense_linorder(B) )
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(set(B),B,complete_Sup_Sup(B),set_or5935395276787703475ssThan(B,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_5641_cSUP__const,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),C2: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,aTP_Lamp_vs(C,fun(B,C),C2)),A4)) = C2 ) ) ) ).

% cSUP_const
tff(fact_5642_finite__UN__I,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ! [A3: B] :
            ( aa(set(B),$o,member(B,A3),A4)
           => aa(set(C),$o,finite_finite2(C),aa(B,set(C),B4,A3)) )
       => aa(set(C),$o,finite_finite2(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ) ).

% finite_UN_I
tff(fact_5643_Sigma__empty2,axiom,
    ! [C: $tType,B: $tType,A4: set(B)] : product_Sigma(B,C,A4,aTP_Lamp_vt(B,set(C))) = bot_bot(set(product_prod(B,C))) ).

% Sigma_empty2
tff(fact_5644_Times__empty,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( ( product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)) = bot_bot(set(product_prod(B,C))) )
    <=> ( ( A4 = bot_bot(set(B)) )
        | ( B4 = bot_bot(set(C)) ) ) ) ).

% Times_empty
tff(fact_5645_Compl__Times__UNIV2,axiom,
    ! [C: $tType,B: $tType,A4: set(B)] : aa(set(product_prod(B,C)),set(product_prod(B,C)),uminus_uminus(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vv(B,set(C)))) = product_Sigma(B,C,aa(set(B),set(B),uminus_uminus(set(B)),A4),aTP_Lamp_vv(B,set(C))) ).

% Compl_Times_UNIV2
tff(fact_5646_Compl__Times__UNIV1,axiom,
    ! [B: $tType,C: $tType,A4: set(C)] : aa(set(product_prod(B,C)),set(product_prod(B,C)),uminus_uminus(set(product_prod(B,C))),product_Sigma(B,C,top_top(set(B)),aTP_Lamp_vu(set(C),fun(B,set(C)),A4))) = product_Sigma(B,C,top_top(set(B)),aTP_Lamp_vw(set(C),fun(B,set(C)),A4)) ).

% Compl_Times_UNIV1
tff(fact_5647_finite__SigmaI,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ! [A3: B] :
            ( aa(set(B),$o,member(B,A3),A4)
           => aa(set(C),$o,finite_finite2(C),aa(B,set(C),B4,A3)) )
       => aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,B4)) ) ) ).

% finite_SigmaI
tff(fact_5648_UNIV__Times__UNIV,axiom,
    ! [C: $tType,B: $tType] : product_Sigma(B,C,top_top(set(B)),aTP_Lamp_vv(B,set(C))) = top_top(set(product_prod(B,C))) ).

% UNIV_Times_UNIV
tff(fact_5649_fst__image__times,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(C)] :
      aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) = $ite(B4 = bot_bot(set(C)),bot_bot(set(B)),A4) ).

% fst_image_times
tff(fact_5650_snd__image__times,axiom,
    ! [C: $tType,B: $tType,A4: set(C),B4: set(B)] :
      aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),product_Sigma(C,B,A4,aTP_Lamp_vx(set(B),fun(C,set(B)),B4))) = $ite(A4 = bot_bot(set(C)),bot_bot(set(B)),B4) ).

% snd_image_times
tff(fact_5651_card__SigmaI,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => aa(set(C),$o,finite_finite2(C),aa(B,set(C),B4,X2)) )
       => ( aa(set(product_prod(B,C)),nat,finite_card(product_prod(B,C)),product_Sigma(B,C,A4,B4)) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aTP_Lamp_vp(fun(B,set(C)),fun(B,nat),B4)),A4) ) ) ) ).

% card_SigmaI
tff(fact_5652_Sigma__UNIV__cancel,axiom,
    ! [C: $tType,B: $tType,A4: set(B),X5: set(C)] : aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),minus_minus(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),X5))),product_Sigma(B,C,A4,aTP_Lamp_vv(B,set(C)))) = bot_bot(set(product_prod(B,C))) ).

% Sigma_UNIV_cancel
tff(fact_5653_set__product,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C)] : aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),product(B,C,Xs,Ys)) = product_Sigma(B,C,aa(list(B),set(B),set2(B),Xs),aTP_Lamp_vy(list(C),fun(B,set(C)),Ys)) ).

% set_product
tff(fact_5654_pairself__image__cart,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A4: set(C),B4: set(C)] : aa(set(product_prod(C,C)),set(product_prod(B,B)),image2(product_prod(C,C),product_prod(B,B),pairself(C,B,F)),product_Sigma(C,C,A4,aTP_Lamp_vz(set(C),fun(C,set(C)),B4))) = product_Sigma(B,B,aa(set(C),set(B),image2(C,B,F),A4),aa(set(C),fun(B,set(B)),aTP_Lamp_wa(fun(C,B),fun(set(C),fun(B,set(B))),F),B4)) ).

% pairself_image_cart
tff(fact_5655_set__concat,axiom,
    ! [B: $tType,Xs: list(list(B))] : aa(list(B),set(B),set2(B),concat(B,Xs)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(list(B)),set(set(B)),image2(list(B),set(B),set2(B)),aa(list(list(B)),set(list(B)),set2(list(B)),Xs))) ).

% set_concat
tff(fact_5656_Sigma__def,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: fun(B,set(C))] : product_Sigma(B,C,A4,B4) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(B),set(set(product_prod(B,C))),image2(B,set(product_prod(B,C)),aTP_Lamp_wc(fun(B,set(C)),fun(B,set(product_prod(B,C))),B4)),A4)) ).

% Sigma_def
tff(fact_5657_Some__SUP,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( aa(C,option(C),some(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))) = aa(set(option(C)),option(C),complete_Sup_Sup(option(C)),aa(set(B),set(option(C)),image2(B,option(C),aTP_Lamp_wd(fun(B,C),fun(B,option(C)),F)),A4)) ) ) ) ).

% Some_SUP
tff(fact_5658_le__cSup__finite,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X5: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),X5)
         => ( aa(set(B),$o,member(B,X),X5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,complete_Sup_Sup(B),X5)) ) ) ) ).

% le_cSup_finite
tff(fact_5659_inf__Sup,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A2: B,B4: set(B)] : aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),aa(set(B),B,complete_Sup_Sup(B),B4)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),inf_inf(B),A2)),B4)) ) ).

% inf_Sup
tff(fact_5660_snd__image__Sigma,axiom,
    ! [B: $tType,C: $tType,A4: set(C),B4: fun(C,set(B))] : aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),product_Sigma(C,B,A4,B4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) ).

% snd_image_Sigma
tff(fact_5661_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_5662_Sigma__Union,axiom,
    ! [C: $tType,B: $tType,X5: set(set(B)),B4: fun(B,set(C))] : product_Sigma(B,C,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),X5),B4) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(set(B)),set(set(product_prod(B,C))),image2(set(B),set(product_prod(B,C)),aTP_Lamp_we(fun(B,set(C)),fun(set(B),set(product_prod(B,C))),B4)),X5)) ).

% Sigma_Union
tff(fact_5663_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_5664_Union__natural,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C)] : aa(fun(set(set(B)),set(set(C))),fun(set(set(B)),set(C)),aa(fun(set(set(C)),set(C)),fun(fun(set(set(B)),set(set(C))),fun(set(set(B)),set(C))),comp(set(set(C)),set(C),set(set(B))),complete_Sup_Sup(set(C))),image2(set(B),set(C),image2(B,C,F))) = aa(fun(set(set(B)),set(B)),fun(set(set(B)),set(C)),aa(fun(set(B),set(C)),fun(fun(set(set(B)),set(B)),fun(set(set(B)),set(C))),comp(set(B),set(C),set(set(B))),image2(B,C,F)),complete_Sup_Sup(set(B))) ).

% Union_natural
tff(fact_5665_relcomp__UNION__distrib,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,S2: set(product_prod(B,D)),R: fun(E,set(product_prod(D,C))),I5: set(E)] : relcomp(B,D,C,S2,aa(set(set(product_prod(D,C))),set(product_prod(D,C)),complete_Sup_Sup(set(product_prod(D,C))),aa(set(E),set(set(product_prod(D,C))),image2(E,set(product_prod(D,C)),R),I5))) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(E),set(set(product_prod(B,C))),image2(E,set(product_prod(B,C)),aa(fun(E,set(product_prod(D,C))),fun(E,set(product_prod(B,C))),aTP_Lamp_wf(set(product_prod(B,D)),fun(fun(E,set(product_prod(D,C))),fun(E,set(product_prod(B,C)))),S2),R)),I5)) ).

% relcomp_UNION_distrib
tff(fact_5666_relcomp__UNION__distrib2,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,R: fun(E,set(product_prod(B,D))),I5: set(E),S2: set(product_prod(D,C))] : relcomp(B,D,C,aa(set(set(product_prod(B,D))),set(product_prod(B,D)),complete_Sup_Sup(set(product_prod(B,D))),aa(set(E),set(set(product_prod(B,D))),image2(E,set(product_prod(B,D)),R),I5)),S2) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(E),set(set(product_prod(B,C))),image2(E,set(product_prod(B,C)),aa(set(product_prod(D,C)),fun(E,set(product_prod(B,C))),aTP_Lamp_wg(fun(E,set(product_prod(B,D))),fun(set(product_prod(D,C)),fun(E,set(product_prod(B,C)))),R),S2)),I5)) ).

% relcomp_UNION_distrib2
tff(fact_5667_top__empty__eq2,axiom,
    ! [C: $tType,B: $tType,X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),top_top(fun(B,fun(C,$o))),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),top_top(set(product_prod(B,C)))) ) ).

% top_empty_eq2
tff(fact_5668_UN__Times__distrib,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,E6: fun(D,set(B)),F3: fun(E,set(C)),A4: set(D),B4: set(E)] : aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(product_prod(D,E)),set(set(product_prod(B,C))),image2(product_prod(D,E),set(product_prod(B,C)),aa(fun(D,fun(E,set(product_prod(B,C)))),fun(product_prod(D,E),set(product_prod(B,C))),product_case_prod(D,E,set(product_prod(B,C))),aa(fun(E,set(C)),fun(D,fun(E,set(product_prod(B,C)))),aTP_Lamp_wi(fun(D,set(B)),fun(fun(E,set(C)),fun(D,fun(E,set(product_prod(B,C))))),E6),F3))),product_Sigma(D,E,A4,aTP_Lamp_wj(set(E),fun(D,set(E)),B4)))) = product_Sigma(B,C,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),E6),A4)),aa(set(E),fun(B,set(C)),aTP_Lamp_wk(fun(E,set(C)),fun(set(E),fun(B,set(C))),F3),B4)) ).

% UN_Times_distrib
tff(fact_5669_Times__eq__cancel2,axiom,
    ! [B: $tType,C: $tType,X: B,C6: set(B),A4: set(C),B4: set(C)] :
      ( aa(set(B),$o,member(B,X),C6)
     => ( ( product_Sigma(C,B,A4,aTP_Lamp_vx(set(B),fun(C,set(B)),C6)) = product_Sigma(C,B,B4,aTP_Lamp_vx(set(B),fun(C,set(B)),C6)) )
      <=> ( A4 = B4 ) ) ) ).

% Times_eq_cancel2
tff(fact_5670_Sigma__cong,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(B),C6: fun(B,set(C)),D3: fun(B,set(C))] :
      ( ( A4 = B4 )
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),B4)
           => ( aa(B,set(C),C6,X2) = aa(B,set(C),D3,X2) ) )
       => ( product_Sigma(B,C,A4,C6) = product_Sigma(B,C,B4,D3) ) ) ) ).

% Sigma_cong
tff(fact_5671_ord_Omax__def,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),A2: B,B2: B] :
      aa(B,B,aa(B,fun(B,B),max(B,Less_eq),A2),B2) = $ite(aa(B,$o,aa(B,fun(B,$o),Less_eq,A2),B2),B2,A2) ).

% ord.max_def
tff(fact_5672_ord_Omax_Ocong,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o))] : max(B,Less_eq) = max(B,Less_eq) ).

% ord.max.cong
tff(fact_5673_cSup__eq__maximum,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Z: B,X5: set(B)] :
          ( aa(set(B),$o,member(B,Z),X5)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Z) )
           => ( aa(set(B),B,complete_Sup_Sup(B),X5) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_5674_cSup__eq,axiom,
    ! [B: $tType] :
      ( ( condit1219197933456340205attice(B)
        & no_bot(B) )
     => ! [X5: set(B),A2: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),X5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),A2) )
         => ( ! [Y2: B] :
                ( ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),X5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y2) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Y2) )
           => ( aa(set(B),B,complete_Sup_Sup(B),X5) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_5675_in__Union__o__assoc,axiom,
    ! [C: $tType,B: $tType,D: $tType,X: B,Gset: fun(C,set(set(B))),Gmap: fun(D,C),A4: D] :
      ( aa(set(B),$o,member(B,X),aa(D,set(B),aa(fun(D,C),fun(D,set(B)),aa(fun(C,set(B)),fun(fun(D,C),fun(D,set(B))),comp(C,set(B),D),aa(fun(C,set(set(B))),fun(C,set(B)),aa(fun(set(set(B)),set(B)),fun(fun(C,set(set(B))),fun(C,set(B))),comp(set(set(B)),set(B),C),complete_Sup_Sup(set(B))),Gset)),Gmap),A4))
     => aa(set(B),$o,member(B,X),aa(D,set(B),aa(fun(D,set(set(B))),fun(D,set(B)),aa(fun(set(set(B)),set(B)),fun(fun(D,set(set(B))),fun(D,set(B))),comp(set(set(B)),set(B),D),complete_Sup_Sup(set(B))),aa(fun(D,C),fun(D,set(set(B))),aa(fun(C,set(set(B))),fun(fun(D,C),fun(D,set(set(B)))),comp(C,set(set(B)),D),Gset),Gmap)),A4)) ) ).

% in_Union_o_assoc
tff(fact_5676_Collect__case__prod__Sigma,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),Q: fun(B,fun(C,$o))] : aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_wl(fun(B,$o),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),Pa),Q))) = product_Sigma(B,C,aa(fun(B,$o),set(B),collect(B),Pa),aTP_Lamp_wm(fun(B,fun(C,$o)),fun(B,set(C)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_5677_Sup__inf__eq__bot__iff,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [B4: set(B),A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Sup_Sup(B),B4)),A2) = bot_bot(B) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),B4)
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X3),A2) = bot_bot(B) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_5678_times__eq__iff,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C),C6: set(B),D3: set(C)] :
      ( ( product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)) = product_Sigma(B,C,C6,aTP_Lamp_vu(set(C),fun(B,set(C)),D3)) )
    <=> ( ( ( A4 = C6 )
          & ( B4 = D3 ) )
        | ( ( ( A4 = bot_bot(set(B)) )
            | ( B4 = bot_bot(set(C)) ) )
          & ( ( C6 = bot_bot(set(B)) )
            | ( D3 = bot_bot(set(C)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_5679_cSup__eq__non__empty,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X5: set(B),A2: B] :
          ( ( X5 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),A2) )
           => ( ! [Y2: B] :
                  ( ! [X4: B] :
                      ( aa(set(B),$o,member(B,X4),X5)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y2) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Y2) )
             => ( aa(set(B),B,complete_Sup_Sup(B),X5) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5680_cSup__least,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X5: set(B),Z: B] :
          ( ( X5 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Z) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),X5)),Z) ) ) ) ).

% cSup_least
tff(fact_5681_finite__UNION__then__finite,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C),A2: C] :
      ( aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))
     => ( aa(set(C),$o,member(C,A2),A4)
       => aa(set(B),$o,finite_finite2(B),aa(C,set(B),B4,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_5682_card__Union__le__sum__card,axiom,
    ! [B: $tType,U4: set(set(B))] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),U4))),aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),U4)) ).

% card_Union_le_sum_card
tff(fact_5683_Product__Type_Oproduct__def,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] : product_product(B,C,A4,B4) = product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)) ).

% Product_Type.product_def
tff(fact_5684_member__product,axiom,
    ! [B: $tType,C: $tType,X: product_prod(B,C),A4: set(B),B4: set(C)] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X),product_product(B,C,A4,B4))
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) ) ).

% member_product
tff(fact_5685_Times__subset__cancel2,axiom,
    ! [B: $tType,C: $tType,X: B,C6: set(B),A4: set(C),B4: set(C)] :
      ( aa(set(B),$o,member(B,X),C6)
     => ( aa(set(product_prod(C,B)),$o,aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),$o),ord_less_eq(set(product_prod(C,B))),product_Sigma(C,B,A4,aTP_Lamp_vx(set(B),fun(C,set(B)),C6))),product_Sigma(C,B,B4,aTP_Lamp_vx(set(B),fun(C,set(B)),C6)))
      <=> aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),A4),B4) ) ) ).

% Times_subset_cancel2
tff(fact_5686_mem__Times__iff,axiom,
    ! [B: $tType,C: $tType,X: product_prod(B,C),A4: set(B),B4: set(C)] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
    <=> ( aa(set(B),$o,member(B,aa(product_prod(B,C),B,product_fst(B,C),X)),A4)
        & aa(set(C),$o,member(C,aa(product_prod(B,C),C,product_snd(B,C),X)),B4) ) ) ).

% mem_Times_iff
tff(fact_5687_in__prod__fst__sndI,axiom,
    ! [B: $tType,C: $tType,X: product_prod(B,C),A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,member(B,aa(product_prod(B,C),B,product_fst(B,C),X)),A4)
     => ( aa(set(C),$o,member(C,aa(product_prod(B,C),C,product_snd(B,C),X)),B4)
       => aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) ) ) ).

% in_prod_fst_sndI
tff(fact_5688_card__cartesian__product,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] : aa(set(product_prod(B,C)),nat,finite_card(product_prod(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(C),nat,finite_card(C),B4)) ).

% card_cartesian_product
tff(fact_5689_Sigma__empty__iff,axiom,
    ! [B: $tType,C: $tType,I5: set(B),X5: fun(B,set(C))] :
      ( ( product_Sigma(B,C,I5,X5) = bot_bot(set(product_prod(B,C))) )
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),I5)
         => ( aa(B,set(C),X5,X3) = bot_bot(set(C)) ) ) ) ).

% Sigma_empty_iff
tff(fact_5690_SUP__inf__distrib2,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [F: fun(C,B),A4: set(C),G: fun(D,B),B4: set(D)] : aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,G),B4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(set(D),fun(C,B),aa(fun(D,B),fun(set(D),fun(C,B)),aTP_Lamp_wo(fun(C,B),fun(fun(D,B),fun(set(D),fun(C,B))),F),G),B4)),A4)) ) ).

% SUP_inf_distrib2
tff(fact_5691_inf__SUP,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A2: B,F: fun(C,B),B4: set(C)] : aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),B4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_wp(B,fun(fun(C,B),fun(C,B)),A2),F)),B4)) ) ).

% inf_SUP
tff(fact_5692_Sup__inf,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [B4: set(B),A2: B] : aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Sup_Sup(B),B4)),A2) = aa(set(B),B,complete_Sup_Sup(B),aa(set(B),set(B),image2(B,B,aTP_Lamp_wq(B,fun(B,B),A2)),B4)) ) ).

% Sup_inf
tff(fact_5693_SUP__inf,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [F: fun(C,B),B4: set(C),A2: B] : aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),B4))),A2) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(B,fun(C,B),aTP_Lamp_wr(fun(C,B),fun(B,fun(C,B)),F),A2)),B4)) ) ).

% SUP_inf
tff(fact_5694_Times__Int__Times,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C),C6: set(B),D3: set(C)] : aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))),product_Sigma(B,C,C6,aTP_Lamp_vu(set(C),fun(B,set(C)),D3))) = product_Sigma(B,C,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),C6),aa(set(C),fun(B,set(C)),aTP_Lamp_ws(set(C),fun(set(C),fun(B,set(C))),B4),D3)) ).

% Times_Int_Times
tff(fact_5695_Sigma__Int__distrib2,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C)),B4: fun(B,set(C))] : product_Sigma(B,C,I5,aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_wt(fun(B,set(C)),fun(fun(B,set(C)),fun(B,set(C))),A4),B4)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),product_Sigma(B,C,I5,A4)),product_Sigma(B,C,I5,B4)) ).

% Sigma_Int_distrib2
tff(fact_5696_Times__Int__distrib1,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(B),C6: set(C)] : product_Sigma(B,C,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4),aTP_Lamp_vu(set(C),fun(B,set(C)),C6)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))),product_Sigma(B,C,B4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))) ).

% Times_Int_distrib1
tff(fact_5697_finite__cartesian__product,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) ) ) ).

% finite_cartesian_product
tff(fact_5698_relcomp__subset__Sigma,axiom,
    ! [C: $tType,B: $tType,D: $tType,R: set(product_prod(B,C)),A4: set(B),B4: set(C),S2: set(product_prod(C,D)),C6: set(D)] :
      ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),R),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
     => ( aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),S2),product_Sigma(C,D,B4,aTP_Lamp_wu(set(D),fun(C,set(D)),C6)))
       => aa(set(product_prod(B,D)),$o,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),$o),ord_less_eq(set(product_prod(B,D))),relcomp(B,C,D,R,S2)),product_Sigma(B,D,A4,aTP_Lamp_wv(set(D),fun(B,set(D)),C6))) ) ) ).

% relcomp_subset_Sigma
tff(fact_5699_Times__Diff__distrib1,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(B),C6: set(C)] : product_Sigma(B,C,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4),aTP_Lamp_vu(set(C),fun(B,set(C)),C6)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),minus_minus(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))),product_Sigma(B,C,B4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))) ).

% Times_Diff_distrib1
tff(fact_5700_Sigma__Diff__distrib2,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C)),B4: fun(B,set(C))] : product_Sigma(B,C,I5,aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_ww(fun(B,set(C)),fun(fun(B,set(C)),fun(B,set(C))),A4),B4)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),minus_minus(set(product_prod(B,C))),product_Sigma(B,C,I5,A4)),product_Sigma(B,C,I5,B4)) ).

% Sigma_Diff_distrib2
tff(fact_5701_UN__image__subset,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(C,set(B)),G: fun(D,set(C)),X: D,X5: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),F),aa(D,set(C),G,X)))),X5)
    <=> aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(D,set(C),G,X)),aa(fun(C,$o),set(C),collect(C),aa(set(B),fun(C,$o),aTP_Lamp_wx(fun(C,set(B)),fun(set(B),fun(C,$o)),F),X5))) ) ).

% UN_image_subset
tff(fact_5702_UN__UN__finite__eq,axiom,
    ! [B: $tType,A4: fun(nat,set(B))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),aTP_Lamp_wy(fun(nat,set(B)),fun(nat,set(B)),A4)),top_top(set(nat)))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_5703_cSUP__least,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),F: fun(B,C),M: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),M) )
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),M) ) ) ) ).

% cSUP_least
tff(fact_5704_cSup__abs__le,axiom,
    ! [B: $tType] :
      ( ( condit6923001295902523014norder(B)
        & linordered_idom(B) )
     => ! [S: set(B),A2: B] :
          ( ( S != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),X2)),A2) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(B),B,complete_Sup_Sup(B),S))),A2) ) ) ) ).

% cSup_abs_le
tff(fact_5705_sum_OUnion__comp,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [B4: set(set(B)),G: fun(B,C)] :
          ( ! [X2: set(B)] :
              ( aa(set(set(B)),$o,member(set(B),X2),B4)
             => aa(set(B),$o,finite_finite2(B),X2) )
         => ( ! [A13: set(B)] :
                ( aa(set(set(B)),$o,member(set(B),A13),B4)
               => ! [A24: set(B)] :
                    ( aa(set(set(B)),$o,member(set(B),A24),B4)
                   => ( ( A13 != A24 )
                     => ! [X2: B] :
                          ( aa(set(B),$o,member(B,X2),A13)
                         => ( aa(set(B),$o,member(B,X2),A24)
                           => ( aa(B,C,G,X2) = zero_zero(C) ) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)) = aa(set(set(B)),C,aa(fun(B,C),fun(set(set(B)),C),aa(fun(fun(B,C),fun(set(B),C)),fun(fun(B,C),fun(set(set(B)),C)),aa(fun(fun(set(B),C),fun(set(set(B)),C)),fun(fun(fun(B,C),fun(set(B),C)),fun(fun(B,C),fun(set(set(B)),C))),comp(fun(set(B),C),fun(set(set(B)),C),fun(B,C)),groups7311177749621191930dd_sum(set(B),C)),groups7311177749621191930dd_sum(B,C)),G),B4) ) ) ) ) ).

% sum.Union_comp
tff(fact_5706_prod_OUnion__comp,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [B4: set(set(B)),G: fun(B,C)] :
          ( ! [X2: set(B)] :
              ( aa(set(set(B)),$o,member(set(B),X2),B4)
             => aa(set(B),$o,finite_finite2(B),X2) )
         => ( ! [A13: set(B)] :
                ( aa(set(set(B)),$o,member(set(B),A13),B4)
               => ! [A24: set(B)] :
                    ( aa(set(set(B)),$o,member(set(B),A24),B4)
                   => ( ( A13 != A24 )
                     => ! [X2: B] :
                          ( aa(set(B),$o,member(B,X2),A13)
                         => ( aa(set(B),$o,member(B,X2),A24)
                           => ( aa(B,C,G,X2) = one_one(C) ) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)) = aa(set(set(B)),C,aa(fun(B,C),fun(set(set(B)),C),aa(fun(fun(B,C),fun(set(B),C)),fun(fun(B,C),fun(set(set(B)),C)),aa(fun(fun(set(B),C),fun(set(set(B)),C)),fun(fun(fun(B,C),fun(set(B),C)),fun(fun(B,C),fun(set(set(B)),C))),comp(fun(set(B),C),fun(set(set(B)),C),fun(B,C)),groups7121269368397514597t_prod(set(B),C)),groups7121269368397514597t_prod(B,C)),G),B4) ) ) ) ) ).

% prod.Union_comp
tff(fact_5707_card__Union__le__sum__card__weak,axiom,
    ! [B: $tType,U4: set(set(B))] :
      ( ! [X2: set(B)] :
          ( aa(set(set(B)),$o,member(set(B),X2),U4)
         => aa(set(B),$o,finite_finite2(B),X2) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),U4))),aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),U4)) ) ).

% card_Union_le_sum_card_weak
tff(fact_5708_lex__def,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : lex(B,R) = aa(set(set(product_prod(list(B),list(B)))),set(product_prod(list(B),list(B))),complete_Sup_Sup(set(product_prod(list(B),list(B)))),aa(set(nat),set(set(product_prod(list(B),list(B)))),image2(nat,set(product_prod(list(B),list(B))),lexn(B,R)),top_top(set(nat)))) ).

% lex_def
tff(fact_5709_card__cartesian__product__singleton,axiom,
    ! [B: $tType,C: $tType,X: B,A4: set(C)] : aa(set(product_prod(B,C)),nat,finite_card(product_prod(B,C)),product_Sigma(B,C,aa(set(B),set(B),insert(B,X),bot_bot(set(B))),aTP_Lamp_vu(set(C),fun(B,set(C)),A4))) = aa(set(C),nat,finite_card(C),A4) ).

% card_cartesian_product_singleton
tff(fact_5710_times__subset__iff,axiom,
    ! [B: $tType,C: $tType,A4: set(B),C6: set(C),B4: set(B),D3: set(C)] :
      ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))),product_Sigma(B,C,B4,aTP_Lamp_vu(set(C),fun(B,set(C)),D3)))
    <=> ( ( A4 = bot_bot(set(B)) )
        | ( C6 = bot_bot(set(C)) )
        | ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
          & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),C6),D3) ) ) ) ).

% times_subset_iff
tff(fact_5711_finite__SigmaI2,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,set(C)),fun(B,$o),aTP_Lamp_wz(set(B),fun(fun(B,set(C)),fun(B,$o)),A4),B4)))
     => ( ! [A3: B] :
            ( aa(set(B),$o,member(B,A3),A4)
           => aa(set(C),$o,finite_finite2(C),aa(B,set(C),B4,A3)) )
       => aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,B4)) ) ) ).

% finite_SigmaI2
tff(fact_5712_finite__cartesian__productD1,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
     => ( ( B4 != bot_bot(set(C)) )
       => aa(set(B),$o,finite_finite2(B),A4) ) ) ).

% finite_cartesian_productD1
tff(fact_5713_finite__cartesian__productD2,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
     => ( ( A4 != bot_bot(set(B)) )
       => aa(set(C),$o,finite_finite2(C),B4) ) ) ).

% finite_cartesian_productD2
tff(fact_5714_finite__cartesian__product__iff,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
    <=> ( ( A4 = bot_bot(set(B)) )
        | ( B4 = bot_bot(set(C)) )
        | ( aa(set(B),$o,finite_finite2(B),A4)
          & aa(set(C),$o,finite_finite2(C),B4) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_5715_fst__image__Sigma,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: fun(B,set(C))] : aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),product_Sigma(B,C,A4,B4)) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,set(C)),fun(B,$o),aTP_Lamp_wz(set(B),fun(fun(B,set(C)),fun(B,$o)),A4),B4)) ).

% fst_image_Sigma
tff(fact_5716_finite__subset__Union,axiom,
    ! [B: $tType,A4: set(B),B11: set(set(B))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B11))
       => ~ ! [F7: set(set(B))] :
              ( aa(set(set(B)),$o,finite_finite2(set(B)),F7)
             => ( aa(set(set(B)),$o,aa(set(set(B)),fun(set(set(B)),$o),ord_less_eq(set(set(B))),F7),B11)
               => ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),F7)) ) ) ) ) ).

% finite_subset_Union
tff(fact_5717_set__zip__cart,axiom,
    ! [B: $tType,C: $tType,X: product_prod(B,C),L: list(B),L3: list(C)] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,L,L3)))
     => aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X),product_Sigma(B,C,aa(list(B),set(B),set2(B),L),aTP_Lamp_vy(list(C),fun(B,set(C)),L3))) ) ).

% set_zip_cart
tff(fact_5718_UN__finite__subset,axiom,
    ! [B: $tType,A4: fun(nat,set(B)),C6: set(B)] :
      ( ! [N2: nat] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C6)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),top_top(set(nat))))),C6) ) ).

% UN_finite_subset
tff(fact_5719_UN__finite2__eq,axiom,
    ! [B: $tType,A4: fun(nat,set(B)),B4: fun(nat,set(B)),K: nat] :
      ( ! [N2: nat] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),top_top(set(nat)))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),B4),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5720_swap__product,axiom,
    ! [B: $tType,C: $tType,A4: set(C),B4: set(B)] : aa(set(product_prod(C,B)),set(product_prod(B,C)),image2(product_prod(C,B),product_prod(B,C),aa(fun(C,fun(B,product_prod(B,C))),fun(product_prod(C,B),product_prod(B,C)),product_case_prod(C,B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))))),product_Sigma(C,B,A4,aTP_Lamp_vx(set(B),fun(C,set(B)),B4))) = product_Sigma(B,C,B4,aTP_Lamp_vu(set(C),fun(B,set(C)),A4)) ).

% swap_product
tff(fact_5721_subset__mset_Omax__add__distrib__left,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B),Z: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),max(multiset(B),subseteq_mset(B)),X),Y)),Z) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),max(multiset(B),subseteq_mset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Z)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Y),Z)) ).

% subset_mset.max_add_distrib_left
tff(fact_5722_subset__mset_Omax__add__distrib__right,axiom,
    ! [B: $tType,X: multiset(B),Y: multiset(B),Z: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),max(multiset(B),subseteq_mset(B)),Y),Z)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),max(multiset(B),subseteq_mset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Y)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Z)) ).

% subset_mset.max_add_distrib_right
tff(fact_5723_UN__le__add__shift,axiom,
    ! [B: $tType,M: fun(nat,set(B)),K: nat,N: nat] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),aa(nat,fun(nat,set(B)),aTP_Lamp_xa(fun(nat,set(B)),fun(nat,fun(nat,set(B))),M),K)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),M),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift
tff(fact_5724_UN__le__add__shift__strict,axiom,
    ! [B: $tType,M: fun(nat,set(B)),K: nat,N: nat] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),aa(nat,fun(nat,set(B)),aTP_Lamp_xa(fun(nat,set(B)),fun(nat,fun(nat,set(B))),M),K)),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),M),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift_strict
tff(fact_5725_cSup__asclose,axiom,
    ! [B: $tType] :
      ( ( condit6923001295902523014norder(B)
        & linordered_idom(B) )
     => ! [S: set(B),L: B,E2: B] :
          ( ( S != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X2),L))),E2) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(B),B,complete_Sup_Sup(B),S)),L))),E2) ) ) ) ).

% cSup_asclose
tff(fact_5726_sum_Ocartesian__product,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,fun(D,B)),B4: set(D),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(set(D),fun(C,B),aTP_Lamp_bl(fun(C,fun(D,B)),fun(set(D),fun(C,B)),G),B4)),A4) = aa(set(product_prod(C,D)),B,aa(fun(product_prod(C,D),B),fun(set(product_prod(C,D)),B),groups7311177749621191930dd_sum(product_prod(C,D),B),aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),G)),product_Sigma(C,D,A4,aTP_Lamp_wu(set(D),fun(C,set(D)),B4))) ) ).

% sum.cartesian_product
tff(fact_5727_prod_Ocartesian__product,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,fun(D,B)),B4: set(D),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(set(D),fun(C,B),aTP_Lamp_dr(fun(C,fun(D,B)),fun(set(D),fun(C,B)),G),B4)),A4) = aa(set(product_prod(C,D)),B,aa(fun(product_prod(C,D),B),fun(set(product_prod(C,D)),B),groups7121269368397514597t_prod(product_prod(C,D),B),aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),G)),product_Sigma(C,D,A4,aTP_Lamp_wu(set(D),fun(C,set(D)),B4))) ) ).

% prod.cartesian_product
tff(fact_5728_subset__fst__snd,axiom,
    ! [C: $tType,B: $tType,A4: set(product_prod(B,C))] : aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),A4),product_Sigma(B,C,aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),A4),aTP_Lamp_xb(set(product_prod(B,C)),fun(B,set(C)),A4))) ).

% subset_fst_snd
tff(fact_5729_image__paired__Times,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,F: fun(D,B),G: fun(E,C),A4: set(D),B4: set(E)] : aa(set(product_prod(D,E)),set(product_prod(B,C)),image2(product_prod(D,E),product_prod(B,C),aa(fun(D,fun(E,product_prod(B,C))),fun(product_prod(D,E),product_prod(B,C)),product_case_prod(D,E,product_prod(B,C)),aa(fun(E,C),fun(D,fun(E,product_prod(B,C))),aTP_Lamp_sn(fun(D,B),fun(fun(E,C),fun(D,fun(E,product_prod(B,C)))),F),G))),product_Sigma(D,E,A4,aTP_Lamp_wj(set(E),fun(D,set(E)),B4))) = product_Sigma(B,C,aa(set(D),set(B),image2(D,B,F),A4),aa(set(E),fun(B,set(C)),aTP_Lamp_xc(fun(E,C),fun(set(E),fun(B,set(C))),G),B4)) ).

% image_paired_Times
tff(fact_5730_UN__finite2__subset,axiom,
    ! [B: $tType,A4: fun(nat,set(B)),B4: fun(nat,set(B)),K: nat] :
      ( ! [N2: nat] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)))))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),A4),top_top(set(nat))))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),B4),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_5731_sum_OSigma,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [A4: set(B),B4: fun(B,set(C)),G: fun(B,fun(C,D))] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(set(C),$o,finite_finite2(C),aa(B,set(C),B4,X2)) )
           => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(B,fun(C,D)),fun(B,D),aTP_Lamp_xd(fun(B,set(C)),fun(fun(B,fun(C,D)),fun(B,D)),B4),G)),A4) = aa(set(product_prod(B,C)),D,aa(fun(product_prod(B,C),D),fun(set(product_prod(B,C)),D),groups7311177749621191930dd_sum(product_prod(B,C),D),aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G)),product_Sigma(B,C,A4,B4)) ) ) ) ) ).

% sum.Sigma
tff(fact_5732_prod_OSigma,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [A4: set(B),B4: fun(B,set(C)),G: fun(B,fun(C,D))] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(set(C),$o,finite_finite2(C),aa(B,set(C),B4,X2)) )
           => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(fun(B,fun(C,D)),fun(B,D),aTP_Lamp_xe(fun(B,set(C)),fun(fun(B,fun(C,D)),fun(B,D)),B4),G)),A4) = aa(set(product_prod(B,C)),D,aa(fun(product_prod(B,C),D),fun(set(product_prod(B,C)),D),groups7121269368397514597t_prod(product_prod(B,C),D),aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G)),product_Sigma(B,C,A4,B4)) ) ) ) ) ).

% prod.Sigma
tff(fact_5733_sum_OUNION__disjoint,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [I5: set(B),A4: fun(B,set(C)),G: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),I5)
               => aa(set(C),$o,finite_finite2(C),aa(B,set(C),A4,X2)) )
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),I5)
                 => ! [Xa3: B] :
                      ( aa(set(B),$o,member(B,Xa3),I5)
                     => ( ( X2 != Xa3 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,X2)),aa(B,set(C),A4,Xa3)) = bot_bot(set(C)) ) ) ) )
             => ( aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_xf(fun(B,set(C)),fun(fun(C,D),fun(B,D)),A4),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_5734_prod_OUNION__disjoint,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [I5: set(B),A4: fun(B,set(C)),G: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),I5)
               => aa(set(C),$o,finite_finite2(C),aa(B,set(C),A4,X2)) )
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),I5)
                 => ! [Xa3: B] :
                      ( aa(set(B),$o,member(B,Xa3),I5)
                     => ( ( X2 != Xa3 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,X2)),aa(B,set(C),A4,Xa3)) = bot_bot(set(C)) ) ) ) )
             => ( aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_xg(fun(B,set(C)),fun(fun(C,D),fun(B,D)),A4),G)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_5735_card__UN__le,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(C),nat,finite_card(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)))),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aTP_Lamp_vp(fun(B,set(C)),fun(B,nat),A4)),I5)) ) ).

% card_UN_le
tff(fact_5736_lists__length__Suc__eq,axiom,
    ! [B: $tType,A4: set(B),N: nat] : aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_xh(set(B),fun(nat,fun(list(B),$o)),A4),N)) = aa(set(product_prod(list(B),B)),set(list(B)),image2(product_prod(list(B),B),list(B),aa(fun(list(B),fun(B,list(B))),fun(product_prod(list(B),B),list(B)),product_case_prod(list(B),B,list(B)),aTP_Lamp_qj(list(B),fun(B,list(B))))),product_Sigma(list(B),B,aa(fun(list(B),$o),set(list(B)),collect(list(B)),aa(nat,fun(list(B),$o),aTP_Lamp_ha(set(B),fun(nat,fun(list(B),$o)),A4),N)),aTP_Lamp_xi(set(B),fun(list(B),set(B)),A4))) ).

% lists_length_Suc_eq
tff(fact_5737_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [B: $tType,F: fun(nat,set(B)),S: set(B)] :
      ( ! [I2: nat] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(nat,set(B),F,I2)),S)
     => ( aa(set(B),$o,finite_finite2(B),S)
       => ( ? [N10: nat] :
              ( ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N10)
                 => ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N10)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N2)
                       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(nat,set(B),F,M2)),aa(nat,set(B),F,N2)) ) ) )
              & ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),N2)
                 => ( aa(nat,set(B),F,N10) = aa(nat,set(B),F,N2) ) ) )
         => ( aa(nat,set(B),F,aa(set(B),nat,finite_card(B),S)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),F),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5738_UN__singleton,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(B),set(set(B)),image2(B,set(B),aTP_Lamp_xj(B,set(B))),A4)) = A4 ).

% UN_singleton
tff(fact_5739_UN__simps_I1_J,axiom,
    ! [B: $tType,C: $tType,A2: B,B4: fun(C,set(B)),C6: set(C)] :
      aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_xk(B,fun(fun(C,set(B)),fun(C,set(B))),A2),B4)),C6)) = $ite(C6 = bot_bot(set(C)),bot_bot(set(B)),aa(set(B),set(B),insert(B,A2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6)))) ).

% UN_simps(1)
tff(fact_5740_SUP__eq__top__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [F: fun(C,B),A4: set(C)] :
          ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4)) = top_top(B) )
        <=> ! [X3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),top_top(B))
             => ? [Xa4: C] :
                  ( aa(set(C),$o,member(C,Xa4),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),aa(C,B,F,Xa4)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_5741_subset__mset_OcSUP__const,axiom,
    ! [B: $tType,C: $tType,A4: set(B),C2: multiset(C)] :
      ( ( A4 != bot_bot(set(B)) )
     => ( aa(set(multiset(C)),multiset(C),complete_Sup_Sup(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),aTP_Lamp_xl(multiset(C),fun(B,multiset(C)),C2)),A4)) = C2 ) ) ).

% subset_mset.cSUP_const
tff(fact_5742_Sup__apply,axiom,
    ! [B: $tType,C: $tType] :
      ( complete_Sup(B)
     => ! [A4: set(fun(C,B)),X: C] : aa(C,B,aa(set(fun(C,B)),fun(C,B),complete_Sup_Sup(fun(C,B)),A4),X) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(C,B)),set(B),image2(fun(C,B),B,aTP_Lamp_xm(C,fun(fun(C,B),B),X)),A4)) ) ).

% Sup_apply
tff(fact_5743_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_5744_top2I,axiom,
    ! [B: $tType,C: $tType,X: B,Y: C] : aa(C,$o,aa(B,fun(C,$o),top_top(fun(B,fun(C,$o))),X),Y) ).

% top2I
tff(fact_5745_SUP__apply,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( complete_Sup(B)
     => ! [F: fun(D,fun(C,B)),A4: set(D),X: C] : aa(C,B,aa(set(fun(C,B)),fun(C,B),complete_Sup_Sup(fun(C,B)),aa(set(D),set(fun(C,B)),image2(D,fun(C,B),F),A4)),X) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),aTP_Lamp_xn(fun(D,fun(C,B)),fun(C,fun(D,B)),F),X)),A4)) ) ).

% SUP_apply
tff(fact_5746_SUP__identity__eq,axiom,
    ! [B: $tType] :
      ( complete_Sup(B)
     => ! [A4: set(B)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(B),set(B),image2(B,B,aTP_Lamp_xo(B,B)),A4)) = aa(set(B),B,complete_Sup_Sup(B),A4) ) ).

% SUP_identity_eq
tff(fact_5747_UN__iff,axiom,
    ! [B: $tType,C: $tType,B2: B,B4: fun(C,set(B)),A4: set(C)] :
      ( aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),A4)
          & aa(set(B),$o,member(B,B2),aa(C,set(B),B4,X3)) ) ) ).

% UN_iff
tff(fact_5748_UN__I,axiom,
    ! [C: $tType,B: $tType,A2: B,A4: set(B),B2: C,B4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,A2),A4)
     => ( aa(set(C),$o,member(C,B2),aa(B,set(C),B4,A2))
       => aa(set(C),$o,member(C,B2),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ) ).

% UN_I
tff(fact_5749_SUP__bot__conv_I2_J,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: fun(C,B),A4: set(C)] :
          ( ( bot_bot(B) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,B4),A4)) )
        <=> ! [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
             => ( aa(C,B,B4,X3) = bot_bot(B) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_5750_SUP__bot__conv_I1_J,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: fun(C,B),A4: set(C)] :
          ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,B4),A4)) = bot_bot(B) )
        <=> ! [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
             => ( aa(C,B,B4,X3) = bot_bot(B) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_5751_SUP__bot,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(C)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_xp(C,B)),A4)) = bot_bot(B) ) ).

% SUP_bot
tff(fact_5752_SUP__const,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),F: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,aTP_Lamp_xq(C,fun(B,C),F)),A4)) = F ) ) ) ).

% SUP_const
tff(fact_5753_UN__constant,axiom,
    ! [C: $tType,B: $tType,C2: set(B),A4: set(C)] :
      aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_vx(set(B),fun(C,set(B)),C2)),A4)) = $ite(A4 = bot_bot(set(C)),bot_bot(set(B)),C2) ).

% UN_constant
tff(fact_5754_SUP__Sup__eq,axiom,
    ! [B: $tType,S: set(set(B)),X4: B] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(set(B)),set(fun(B,$o)),image2(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o))),S)),X4)
    <=> aa(set(B),$o,member(B,X4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S)) ) ).

% SUP_Sup_eq
tff(fact_5755_Sup__set__def,axiom,
    ! [B: $tType,A4: set(set(B))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A4) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_xr(set(set(B)),fun(B,$o),A4)) ).

% Sup_set_def
tff(fact_5756_SUP__UN__eq2,axiom,
    ! [B: $tType,C: $tType,D: $tType,R: fun(D,set(product_prod(B,C))),S: set(D),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(D),set(fun(B,fun(C,$o))),image2(D,fun(B,fun(C,$o)),aTP_Lamp_xs(fun(D,set(product_prod(B,C))),fun(D,fun(B,fun(C,$o))),R)),S)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(D),set(set(product_prod(B,C))),image2(D,set(product_prod(B,C)),R),S))) ) ).

% SUP_UN_eq2
tff(fact_5757_SUP__Sup__eq2,axiom,
    ! [C: $tType,B: $tType,S: set(set(product_prod(B,C))),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(set(product_prod(B,C))),set(fun(B,fun(C,$o))),image2(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o)))),S)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),S)) ) ).

% SUP_Sup_eq2
tff(fact_5758_Sup__SUP__eq2,axiom,
    ! [C: $tType,B: $tType,S: set(fun(B,fun(C,$o))),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),S),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(fun(product_prod(B,C),$o)),set(set(product_prod(B,C))),image2(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C))),aa(set(fun(B,fun(C,$o))),set(fun(product_prod(B,C),$o)),image2(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o)),S)))) ) ).

% Sup_SUP_eq2
tff(fact_5759_SUP__UN__eq,axiom,
    ! [B: $tType,C: $tType,R: fun(C,set(B)),S: set(C),X4: B] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(C),set(fun(B,$o)),image2(C,fun(B,$o),aTP_Lamp_xt(fun(C,set(B)),fun(C,fun(B,$o)),R)),S)),X4)
    <=> aa(set(B),$o,member(B,X4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),R),S))) ) ).

% SUP_UN_eq
tff(fact_5760_Inf_OINF__identity__eq,axiom,
    ! [B: $tType,Inf: fun(set(B),B),A4: set(B)] : aa(set(B),B,Inf,aa(set(B),set(B),image2(B,B,aTP_Lamp_il(B,B)),A4)) = aa(set(B),B,Inf,A4) ).

% Inf.INF_identity_eq
tff(fact_5761_Sup_OSUP__identity__eq,axiom,
    ! [B: $tType,Sup: fun(set(B),B),A4: set(B)] : aa(set(B),B,Sup,aa(set(B),set(B),image2(B,B,aTP_Lamp_il(B,B)),A4)) = aa(set(B),B,Sup,A4) ).

% Sup.SUP_identity_eq
tff(fact_5762_Sup__fun__def,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [A4: set(fun(B,C)),X4: B] : aa(B,C,aa(set(fun(B,C)),fun(B,C),complete_Sup_Sup(fun(B,C)),A4),X4) = aa(set(C),C,complete_Sup_Sup(C),aa(set(fun(B,C)),set(C),image2(fun(B,C),C,aTP_Lamp_xu(B,fun(fun(B,C),C),X4)),A4)) ) ).

% Sup_fun_def
tff(fact_5763_Sup__eqI,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),X: B] :
          ( ! [Y2: B] :
              ( aa(set(B),$o,member(B,Y2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X) )
         => ( ! [Y2: B] :
                ( ! [Z3: B] :
                    ( aa(set(B),$o,member(B,Z3),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z3),Y2) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2) )
           => ( aa(set(B),B,complete_Sup_Sup(B),A4) = X ) ) ) ) ).

% Sup_eqI
tff(fact_5764_Sup__mono,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( ! [A3: B] :
              ( aa(set(B),$o,member(B,A3),A4)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X4) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),aa(set(B),B,complete_Sup_Sup(B),B4)) ) ) ).

% Sup_mono
tff(fact_5765_Sup__least,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),Z: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Z) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),Z) ) ) ).

% Sup_least
tff(fact_5766_Sup__upper,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B,A4: set(B)] :
          ( aa(set(B),$o,member(B,X),A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ).

% Sup_upper
tff(fact_5767_Sup__le__iff,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),B2)
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2) ) ) ) ).

% Sup_le_iff
tff(fact_5768_Sup__upper2,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [U: B,A4: set(B),V2: B] :
          ( aa(set(B),$o,member(B,U),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),V2),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),V2),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ) ).

% Sup_upper2
tff(fact_5769_SUP__commute,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,fun(D,B)),B4: set(D),A4: set(C)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(set(D),fun(C,B),aTP_Lamp_xv(fun(C,fun(D,B)),fun(set(D),fun(C,B)),F),B4)),A4)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,aa(set(C),fun(D,B),aTP_Lamp_xx(fun(C,fun(D,B)),fun(set(C),fun(D,B)),F),A4)),B4)) ) ).

% SUP_commute
tff(fact_5770_image__Union,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),S: set(set(C))] : aa(set(C),set(B),image2(C,B,F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),S)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(C)),set(set(B)),image2(set(C),set(B),image2(C,B,F)),S)) ).

% image_Union
tff(fact_5771_Union__SetCompr__eq,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),Pa: fun(C,$o)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aa(fun(C,$o),fun(set(B),$o),aTP_Lamp_xy(fun(C,set(B)),fun(fun(C,$o),fun(set(B),$o)),F),Pa))) = aa(fun(B,$o),set(B),collect(B),aa(fun(C,$o),fun(B,$o),aTP_Lamp_xz(fun(C,set(B)),fun(fun(C,$o),fun(B,$o)),F),Pa)) ).

% Union_SetCompr_eq
tff(fact_5772_Int__Union,axiom,
    ! [B: $tType,A4: set(B),B4: set(set(B))] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4)),B4)) ).

% Int_Union
tff(fact_5773_Int__Union2,axiom,
    ! [B: $tType,B4: set(set(B)),A4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)),A4) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aTP_Lamp_ya(set(B),fun(set(B),set(B)),A4)),B4)) ).

% Int_Union2
tff(fact_5774_UN__UN__flatten,axiom,
    ! [C: $tType,B: $tType,D: $tType,C6: fun(C,set(B)),B4: fun(D,set(C)),A4: set(D)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),C6),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4)))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_yb(fun(C,set(B)),fun(fun(D,set(C)),fun(D,set(B))),C6),B4)),A4)) ).

% UN_UN_flatten
tff(fact_5775_UN__E,axiom,
    ! [B: $tType,C: $tType,B2: B,B4: fun(C,set(B)),A4: set(C)] :
      ( aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))
     => ~ ! [X2: C] :
            ( aa(set(C),$o,member(C,X2),A4)
           => ~ aa(set(B),$o,member(B,B2),aa(C,set(B),B4,X2)) ) ) ).

% UN_E
tff(fact_5776_UN__extend__simps_I8_J,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(set(C))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(C)),set(set(B)),image2(set(C),set(B),aTP_Lamp_yc(fun(C,set(B)),fun(set(C),set(B)),B4)),A4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),A4))) ).

% UN_extend_simps(8)
tff(fact_5777_UN__extend__simps_I9_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,C6: fun(D,set(B)),B4: fun(C,set(D)),A4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(D)),fun(C,set(B)),aTP_Lamp_yd(fun(D,set(B)),fun(fun(C,set(D)),fun(C,set(B))),C6),B4)),A4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),C6),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B4),A4)))) ).

% UN_extend_simps(9)
tff(fact_5778_le__Sup__iff,axiom,
    ! [B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [X: B,A4: set(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,complete_Sup_Sup(B),A4))
        <=> ! [Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),X)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),X3) ) ) ) ) ).

% le_Sup_iff
tff(fact_5779_SUP__eq,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comple6319245703460814977attice(D)
     => ! [A4: set(B),B4: set(C),F: fun(B,D),G: fun(C,D)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => ? [X4: C] :
                  ( aa(set(C),$o,member(C,X4),B4)
                  & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(B,D,F,I2)),aa(C,D,G,X4)) ) )
         => ( ! [J2: C] :
                ( aa(set(C),$o,member(C,J2),B4)
               => ? [X4: B] :
                    ( aa(set(B),$o,member(B,X4),A4)
                    & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(C,D,G,J2)),aa(B,D,F,X4)) ) )
           => ( aa(set(D),D,complete_Sup_Sup(D),aa(set(B),set(D),image2(B,D,F),A4)) = aa(set(D),D,complete_Sup_Sup(D),aa(set(C),set(D),image2(C,D,G),B4)) ) ) ) ) ).

% SUP_eq
tff(fact_5780_less__eq__Sup,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),U: B] :
          ( ! [V4: B] :
              ( aa(set(B),$o,member(B,V4),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),V4) )
         => ( ( A4 != bot_bot(set(B)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ) ).

% less_eq_Sup
tff(fact_5781_Sup__subset__mono,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),aa(set(B),B,complete_Sup_Sup(B),B4)) ) ) ).

% Sup_subset_mono
tff(fact_5782_SUP__eqI,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),F: fun(B,C),X: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),X) )
         => ( ! [Y2: C] :
                ( ! [I4: B] :
                    ( aa(set(B),$o,member(B,I4),A4)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I4)),Y2) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),X),Y2) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_5783_SUP__mono,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comple6319245703460814977attice(D)
     => ! [A4: set(B),B4: set(C),F: fun(B,D),G: fun(C,D)] :
          ( ! [N2: B] :
              ( aa(set(B),$o,member(B,N2),A4)
             => ? [X4: C] :
                  ( aa(set(C),$o,member(C,X4),B4)
                  & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(B,D,F,N2)),aa(C,D,G,X4)) ) )
         => aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(set(D),D,complete_Sup_Sup(D),aa(set(B),set(D),image2(B,D,F),A4))),aa(set(D),D,complete_Sup_Sup(D),aa(set(C),set(D),image2(C,D,G),B4))) ) ) ).

% SUP_mono
tff(fact_5784_SUP__least,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),F: fun(B,C),U: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),U) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),U) ) ) ).

% SUP_least
tff(fact_5785_SUP__mono_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [F: fun(B,C),G: fun(B,C),A4: set(B)] :
          ( ! [X2: B] : aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2))
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),A4))) ) ) ).

% SUP_mono'
tff(fact_5786_SUP__upper,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I: B,A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,member(B,I),A4)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I)),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ).

% SUP_upper
tff(fact_5787_SUP__le__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A4: set(C),U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))),U)
        <=> ! [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X3)),U) ) ) ) ).

% SUP_le_iff
tff(fact_5788_SUP__upper2,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I: B,A4: set(B),U: C,F: fun(B,C)] :
          ( aa(set(B),$o,member(B,I),A4)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),U),aa(B,C,F,I))
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),U),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ) ).

% SUP_upper2
tff(fact_5789_SUP__lessD,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A4: set(C),Y: B,I: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))),Y)
         => ( aa(set(C),$o,member(C,I),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_5790_less__SUP__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [A2: B,F: fun(C,B),A4: set(C)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4)))
        <=> ? [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(C,B,F,X3)) ) ) ) ).

% less_SUP_iff
tff(fact_5791_UN__extend__simps_I10_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,B4: fun(D,set(B)),F: fun(C,D),A4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,D),fun(C,set(B)),aTP_Lamp_ye(fun(D,set(B)),fun(fun(C,D),fun(C,set(B))),B4),F)),A4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),B4),aa(set(C),set(D),image2(C,D,F),A4))) ).

% UN_extend_simps(10)
tff(fact_5792_image__UN,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,B),B4: fun(D,set(C)),A4: set(D)] : aa(set(C),set(B),image2(C,B,F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_yf(fun(C,B),fun(fun(D,set(C)),fun(D,set(B))),F),B4)),A4)) ).

% image_UN
tff(fact_5793_UNION__empty__conv_I2_J,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,set(B)),A4: set(C)] :
      ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) = bot_bot(set(B)) )
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),A4)
         => ( aa(C,set(B),B4,X3) = bot_bot(set(B)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_5794_UNION__empty__conv_I1_J,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,set(B)),A4: set(C)] :
      ( ( bot_bot(set(B)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) )
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),A4)
         => ( aa(C,set(B),B4,X3) = bot_bot(set(B)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_5795_UN__empty,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,set(B))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),bot_bot(set(C)))) = bot_bot(set(B)) ).

% UN_empty
tff(fact_5796_UN__empty2,axiom,
    ! [C: $tType,B: $tType,A4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_yg(C,set(B))),A4)) = bot_bot(set(B)) ).

% UN_empty2
tff(fact_5797_UN__subset__iff,axiom,
    ! [C: $tType,B: $tType,A4: fun(C,set(B)),I5: set(C),B4: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))),B4)
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),I5)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),A4,X3)),B4) ) ) ).

% UN_subset_iff
tff(fact_5798_UN__upper,axiom,
    ! [C: $tType,B: $tType,A2: B,A4: set(B),B4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,A2),A4)
     => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),B4,A2)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ).

% UN_upper
tff(fact_5799_UN__least,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(B,set(C)),C6: set(C)] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),B4,X2)),C6) )
     => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))),C6) ) ).

% UN_least
tff(fact_5800_UN__mono,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(B),F: fun(B,set(C)),G: fun(B,set(C))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),F,X2)),aa(B,set(C),G,X2)) )
       => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),A4))),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),G),B4))) ) ) ).

% UN_mono
tff(fact_5801_UN__insert__distrib,axiom,
    ! [C: $tType,B: $tType,U: B,A4: set(B),A2: C,B4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,U),A4)
     => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_yh(C,fun(fun(B,set(C)),fun(B,set(C))),A2),B4)),A4)) = aa(set(C),set(C),insert(C,A2),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ) ).

% UN_insert_distrib
tff(fact_5802_Int__UN__distrib2,axiom,
    ! [D: $tType,B: $tType,C: $tType,A4: fun(C,set(B)),I5: set(C),B4: fun(D,set(B)),J4: set(D)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),B4),J4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(D),fun(C,set(B)),aa(fun(D,set(B)),fun(set(D),fun(C,set(B))),aTP_Lamp_yj(fun(C,set(B)),fun(fun(D,set(B)),fun(set(D),fun(C,set(B)))),A4),B4),J4)),I5)) ).

% Int_UN_distrib2
tff(fact_5803_Int__UN__distrib,axiom,
    ! [B: $tType,C: $tType,B4: set(B),A4: fun(C,set(B)),I5: set(C)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_yk(set(B),fun(fun(C,set(B)),fun(C,set(B))),B4),A4)),I5)) ).

% Int_UN_distrib
tff(fact_5804_UN__extend__simps_I4_J,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),C6: set(C),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_yl(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6)) ).

% UN_extend_simps(4)
tff(fact_5805_UN__extend__simps_I5_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_yk(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) ).

% UN_extend_simps(5)
tff(fact_5806_UN__extend__simps_I6_J,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),C6: set(C),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_ym(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6)) ).

% UN_extend_simps(6)
tff(fact_5807_le__SUP__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [X: B,F: fun(C,B),A4: set(C)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4)))
        <=> ! [Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),X)
             => ? [X3: C] :
                  ( aa(set(C),$o,member(C,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),aa(C,B,F,X3)) ) ) ) ) ).

% le_SUP_iff
tff(fact_5808_SUP__eq__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I5: set(B),C2: C,F: fun(B,C)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),I5)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),C2),aa(B,C,F,I2)) )
           => ( ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),I5)) = C2 )
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),I5)
                 => ( aa(B,C,F,X3) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_5809_Sup__inter__less__eq,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),B4: set(B)] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Sup_Sup(B),A4)),aa(set(B),B,complete_Sup_Sup(B),B4))) ) ).

% Sup_inter_less_eq
tff(fact_5810_SUP__subset__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),B4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2)) )
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B4))) ) ) ) ).

% SUP_subset_mono
tff(fact_5811_SUP__constant,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [C2: B,A4: set(C)] :
          aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_yn(B,fun(C,B),C2)),A4)) = $ite(A4 = bot_bot(set(C)),bot_bot(B),C2) ) ).

% SUP_constant
tff(fact_5812_SUP__empty,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),bot_bot(set(C)))) = bot_bot(B) ) ).

% SUP_empty
tff(fact_5813_UN__extend__simps_I1_J,axiom,
    ! [B: $tType,C: $tType,A2: B,B4: fun(C,set(B)),C6: set(C)] :
      aa(set(B),set(B),insert(B,A2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = $ite(C6 = bot_bot(set(C)),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_xk(B,fun(fun(C,set(B)),fun(C,set(B))),A2),B4)),C6))) ).

% UN_extend_simps(1)
tff(fact_5814_SUP__UNION,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),G: fun(D,set(C)),A4: set(D)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(D),set(set(C)),image2(D,set(C),G),A4)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,aa(fun(D,set(C)),fun(D,B),aTP_Lamp_yo(fun(C,B),fun(fun(D,set(C)),fun(D,B)),F),G)),A4)) ) ).

% SUP_UNION
tff(fact_5815_Collect__ex__eq,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o))] : aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ut(fun(B,fun(C,$o)),fun(B,$o),Pa)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_yq(fun(B,fun(C,$o)),fun(C,set(B)),Pa)),top_top(set(C)))) ).

% Collect_ex_eq
tff(fact_5816_UNION__singleton__eq__range,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_yr(fun(C,B),fun(C,set(B)),F)),A4)) = aa(set(C),set(B),image2(C,B,F),A4) ).

% UNION_singleton_eq_range
tff(fact_5817_card__UNION,axiom,
    ! [B: $tType,A4: set(set(B))] :
      ( aa(set(set(B)),$o,finite_finite2(set(B)),A4)
     => ( ! [X2: set(B)] :
            ( aa(set(set(B)),$o,member(set(B),X2),A4)
           => aa(set(B),$o,finite_finite2(B),X2) )
       => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A4)) = aa(int,nat,nat2,aa(set(set(set(B))),int,aa(fun(set(set(B)),int),fun(set(set(set(B))),int),groups7311177749621191930dd_sum(set(set(B)),int),aTP_Lamp_ys(set(set(B)),int)),aa(fun(set(set(B)),$o),set(set(set(B))),collect(set(set(B))),aTP_Lamp_yt(set(set(B)),fun(set(set(B)),$o),A4)))) ) ) ) ).

% card_UNION
tff(fact_5818_UN__UN__split__split__eq,axiom,
    ! [E: $tType,F6: $tType,B: $tType,D: $tType,C: $tType,A4: fun(C,fun(D,fun(E,fun(F6,set(B))))),Y4: set(product_prod(E,F6)),X5: set(product_prod(C,D))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(product_prod(C,D)),set(set(B)),image2(product_prod(C,D),set(B),aa(fun(C,fun(D,set(B))),fun(product_prod(C,D),set(B)),product_case_prod(C,D,set(B)),aa(set(product_prod(E,F6)),fun(C,fun(D,set(B))),aTP_Lamp_yu(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(set(product_prod(E,F6)),fun(C,fun(D,set(B)))),A4),Y4))),X5)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(product_prod(C,D)),set(set(B)),image2(product_prod(C,D),set(B),aa(set(product_prod(E,F6)),fun(product_prod(C,D),set(B)),aTP_Lamp_yx(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(set(product_prod(E,F6)),fun(product_prod(C,D),set(B))),A4),Y4)),X5)) ).

% UN_UN_split_split_eq
tff(fact_5819_infinite__cartesian__product,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ( ~ aa(set(C),$o,finite_finite2(C),B4)
       => ~ aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) ) ) ).

% infinite_cartesian_product
tff(fact_5820_INF__identity__eq,axiom,
    ! [B: $tType] :
      ( complete_Inf(B)
     => ! [A4: set(B)] : aa(set(B),B,complete_Inf_Inf(B),aa(set(B),set(B),image2(B,B,aTP_Lamp_yy(B,B)),A4)) = aa(set(B),B,complete_Inf_Inf(B),A4) ) ).

% INF_identity_eq
tff(fact_5821_INT__I,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B2: C,B4: fun(B,set(C))] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(set(C),$o,member(C,B2),aa(B,set(C),B4,X2)) )
     => aa(set(C),$o,member(C,B2),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ).

% INT_I
tff(fact_5822_INT__iff,axiom,
    ! [B: $tType,C: $tType,B2: B,B4: fun(C,set(B)),A4: set(C)] :
      ( aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),A4)
         => aa(set(B),$o,member(B,B2),aa(C,set(B),B4,X3)) ) ) ).

% INT_iff
tff(fact_5823_Inf__apply,axiom,
    ! [B: $tType,C: $tType] :
      ( complete_Inf(B)
     => ! [A4: set(fun(C,B)),X: C] : aa(C,B,aa(set(fun(C,B)),fun(C,B),complete_Inf_Inf(fun(C,B)),A4),X) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(C,B)),set(B),image2(fun(C,B),B,aTP_Lamp_yz(C,fun(fun(C,B),B),X)),A4)) ) ).

% Inf_apply
tff(fact_5824_SUP2__I,axiom,
    ! [C: $tType,B: $tType,D: $tType,A2: B,A4: set(B),B4: fun(B,fun(C,fun(D,$o))),B2: C,C2: D] :
      ( aa(set(B),$o,member(B,A2),A4)
     => ( aa(D,$o,aa(C,fun(D,$o),aa(B,fun(C,fun(D,$o)),B4,A2),B2),C2)
       => aa(D,$o,aa(C,fun(D,$o),aa(set(fun(C,fun(D,$o))),fun(C,fun(D,$o)),complete_Sup_Sup(fun(C,fun(D,$o))),aa(set(B),set(fun(C,fun(D,$o))),image2(B,fun(C,fun(D,$o)),B4),A4)),B2),C2) ) ) ).

% SUP2_I
tff(fact_5825_Inf__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(set(B),B,complete_Inf_Inf(B),set_or1337092689740270186AtMost(B,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_5826_cInf__atLeastAtMost,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(set(B),B,complete_Inf_Inf(B),set_or1337092689740270186AtMost(B,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_5827_Inf__atLeastLessThan,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(set(B),B,complete_Inf_Inf(B),set_or7035219750837199246ssThan(B,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_5828_SUP1__I,axiom,
    ! [B: $tType,C: $tType,A2: B,A4: set(B),B4: fun(B,fun(C,$o)),B2: C] :
      ( aa(set(B),$o,member(B,A2),A4)
     => ( aa(C,$o,aa(B,fun(C,$o),B4,A2),B2)
       => aa(C,$o,aa(set(fun(C,$o)),fun(C,$o),complete_Sup_Sup(fun(C,$o)),aa(set(B),set(fun(C,$o)),image2(B,fun(C,$o),B4),A4)),B2) ) ) ).

% SUP1_I
tff(fact_5829_Inf__atMost,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B] : aa(set(B),B,complete_Inf_Inf(B),aa(B,set(B),set_ord_atMost(B),X)) = bot_bot(B) ) ).

% Inf_atMost
tff(fact_5830_Inf__greaterThanAtMost,axiom,
    ! [B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & dense_linorder(B) )
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(set(B),B,complete_Inf_Inf(B),set_or3652927894154168847AtMost(B,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5831_Inf__greaterThanLessThan,axiom,
    ! [B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & dense_linorder(B) )
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y)
         => ( aa(set(B),B,complete_Inf_Inf(B),set_or5935395276787703475ssThan(B,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_5832_cINF__const,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),C2: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,aTP_Lamp_vs(C,fun(B,C),C2)),A4)) = C2 ) ) ) ).

% cINF_const
tff(fact_5833_INF__const,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),F: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,aTP_Lamp_xq(C,fun(B,C),F)),A4)) = F ) ) ) ).

% INF_const
tff(fact_5834_INF__top,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(C)] : aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_za(C,B)),A4)) = top_top(B) ) ).

% INF_top
tff(fact_5835_INF__top__conv_I1_J,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: fun(C,B),A4: set(C)] :
          ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,B4),A4)) = top_top(B) )
        <=> ! [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
             => ( aa(C,B,B4,X3) = top_top(B) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_5836_INF__top__conv_I2_J,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: fun(C,B),A4: set(C)] :
          ( ( top_top(B) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,B4),A4)) )
        <=> ! [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
             => ( aa(C,B,B4,X3) = top_top(B) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_5837_finite__INT,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C))] :
      ( ? [X4: B] :
          ( aa(set(B),$o,member(B,X4),I5)
          & aa(set(C),$o,finite_finite2(C),aa(B,set(C),A4,X4)) )
     => aa(set(C),$o,finite_finite2(C),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) ) ).

% finite_INT
tff(fact_5838_INF__apply,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( complete_Inf(B)
     => ! [F: fun(D,fun(C,B)),A4: set(D),X: C] : aa(C,B,aa(set(fun(C,B)),fun(C,B),complete_Inf_Inf(fun(C,B)),aa(set(D),set(fun(C,B)),image2(D,fun(C,B),F),A4)),X) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),aTP_Lamp_zb(fun(D,fun(C,B)),fun(C,fun(D,B)),F),X)),A4)) ) ).

% INF_apply
tff(fact_5839_INF__eq__bot__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [F: fun(C,B),A4: set(C)] :
          ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4)) = bot_bot(B) )
        <=> ! [X3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),bot_bot(B)),X3)
             => ? [Xa4: C] :
                  ( aa(set(C),$o,member(C,Xa4),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,Xa4)),X3) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_5840_INT__constant,axiom,
    ! [C: $tType,B: $tType,C2: set(B),A4: set(C)] :
      aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_vx(set(B),fun(C,set(B)),C2)),A4)) = $ite(A4 = bot_bot(set(C)),top_top(set(B)),C2) ).

% INT_constant
tff(fact_5841_INT__insert,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A2: C,A4: set(C)] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),aa(set(C),set(C),insert(C,A2),A4))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(C,set(B),B4,A2)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4))) ).

% INT_insert
tff(fact_5842_Compl__INT,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C)] : aa(set(B),set(B),uminus_uminus(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_zc(fun(C,set(B)),fun(C,set(B)),B4)),A4)) ).

% Compl_INT
tff(fact_5843_Compl__UN,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C)] : aa(set(B),set(B),uminus_uminus(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_zc(fun(C,set(B)),fun(C,set(B)),B4)),A4)) ).

% Compl_UN
tff(fact_5844_Inf__atMostLessThan,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),top_top(B)),X)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(B,set(B),set_ord_lessThan(B),X)) = bot_bot(B) ) ) ) ).

% Inf_atMostLessThan
tff(fact_5845_INT__simps_I1_J,axiom,
    ! [C: $tType,B: $tType,A4: fun(C,set(B)),B4: set(B),C6: set(C)] :
      aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_yl(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6)) = $ite(C6 = bot_bot(set(C)),top_top(set(B)),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(C),set(set(B)),image2(C,set(B),A4),C6))),B4)) ).

% INT_simps(1)
tff(fact_5846_INT__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] :
      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(B)),fun(C,set(B)),aTP_Lamp_yk(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) = $ite(C6 = bot_bot(set(C)),top_top(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6)))) ).

% INT_simps(2)
tff(fact_5847_INT__simps_I3_J,axiom,
    ! [C: $tType,B: $tType,A4: fun(C,set(B)),B4: set(B),C6: set(C)] :
      aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_ym(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6)) = $ite(C6 = bot_bot(set(C)),top_top(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4)) ).

% INT_simps(3)
tff(fact_5848_INT__simps_I4_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] :
      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(B)),fun(C,set(B)),aTP_Lamp_zd(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) = $ite(C6 = bot_bot(set(C)),top_top(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6)))) ).

% INT_simps(4)
tff(fact_5849_cInf__le__finite,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X5: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),X5)
         => ( aa(set(B),$o,member(B,X),X5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),X5)),X) ) ) ) ).

% cInf_le_finite
tff(fact_5850_INT__bool__eq,axiom,
    ! [B: $tType,A4: fun($o,set(B))] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set($o),set(set(B)),image2($o,set(B),A4),top_top(set($o)))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa($o,set(B),A4,$true)),aa($o,set(B),A4,$false)) ).

% INT_bool_eq
tff(fact_5851_INF__UNIV__bool__expand,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: fun($o,B)] : aa(set(B),B,complete_Inf_Inf(B),aa(set($o),set(B),image2($o,B,A4),top_top(set($o)))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa($o,B,A4,$true)),aa($o,B,A4,$false)) ) ).

% INF_UNIV_bool_expand
tff(fact_5852_INF__commute,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,fun(D,B)),B4: set(D),A4: set(C)] : aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(set(D),fun(C,B),aTP_Lamp_ze(fun(C,fun(D,B)),fun(set(D),fun(C,B)),F),B4)),A4)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,aa(set(C),fun(D,B),aTP_Lamp_zf(fun(C,fun(D,B)),fun(set(C),fun(D,B)),F),A4)),B4)) ) ).

% INF_commute
tff(fact_5853_INT__D,axiom,
    ! [B: $tType,C: $tType,B2: B,B4: fun(C,set(B)),A4: set(C),A2: C] :
      ( aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))
     => ( aa(set(C),$o,member(C,A2),A4)
       => aa(set(B),$o,member(B,B2),aa(C,set(B),B4,A2)) ) ) ).

% INT_D
tff(fact_5854_INT__E,axiom,
    ! [B: $tType,C: $tType,B2: B,B4: fun(C,set(B)),A4: set(C),A2: C] :
      ( aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))
     => ( ~ aa(set(B),$o,member(B,B2),aa(C,set(B),B4,A2))
       => ~ aa(set(C),$o,member(C,A2),A4) ) ) ).

% INT_E
tff(fact_5855_SUP1__E,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,fun(B,$o)),A4: set(C),B2: B] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(C),set(fun(B,$o)),image2(C,fun(B,$o),B4),A4)),B2)
     => ~ ! [X2: C] :
            ( aa(set(C),$o,member(C,X2),A4)
           => ~ aa(B,$o,aa(C,fun(B,$o),B4,X2),B2) ) ) ).

% SUP1_E
tff(fact_5856_SUP2__E,axiom,
    ! [B: $tType,D: $tType,C: $tType,B4: fun(D,fun(B,fun(C,$o))),A4: set(D),B2: B,C2: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(D),set(fun(B,fun(C,$o))),image2(D,fun(B,fun(C,$o)),B4),A4)),B2),C2)
     => ~ ! [X2: D] :
            ( aa(set(D),$o,member(D,X2),A4)
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(D,fun(B,fun(C,$o)),B4,X2),B2),C2) ) ) ).

% SUP2_E
tff(fact_5857_Inf__fun__def,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [A4: set(fun(B,C)),X4: B] : aa(B,C,aa(set(fun(B,C)),fun(B,C),complete_Inf_Inf(fun(B,C)),A4),X4) = aa(set(C),C,complete_Inf_Inf(C),aa(set(fun(B,C)),set(C),image2(fun(B,C),C,aTP_Lamp_zg(B,fun(fun(B,C),C),X4)),A4)) ) ).

% Inf_fun_def
tff(fact_5858_Some__INF,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A4: set(C)] : aa(B,option(B),some(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))) = aa(set(option(B)),option(B),complete_Inf_Inf(option(B)),aa(set(C),set(option(B)),image2(C,option(B),aTP_Lamp_zh(fun(C,B),fun(C,option(B)),F)),A4)) ) ).

% Some_INF
tff(fact_5859_INF__eq,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comple6319245703460814977attice(D)
     => ! [A4: set(B),B4: set(C),G: fun(C,D),F: fun(B,D)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => ? [X4: C] :
                  ( aa(set(C),$o,member(C,X4),B4)
                  & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(C,D,G,X4)),aa(B,D,F,I2)) ) )
         => ( ! [J2: C] :
                ( aa(set(C),$o,member(C,J2),B4)
               => ? [X4: B] :
                    ( aa(set(B),$o,member(B,X4),A4)
                    & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(B,D,F,X4)),aa(C,D,G,J2)) ) )
           => ( aa(set(D),D,complete_Inf_Inf(D),aa(set(B),set(D),image2(B,D,F),A4)) = aa(set(D),D,complete_Inf_Inf(D),aa(set(C),set(D),image2(C,D,G),B4)) ) ) ) ) ).

% INF_eq
tff(fact_5860_cInf__greatest,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X5: set(B),Z: B] :
          ( ( X5 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X2) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),aa(set(B),B,complete_Inf_Inf(B),X5)) ) ) ) ).

% cInf_greatest
tff(fact_5861_cInf__eq__non__empty,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X5: set(B),A2: B] :
          ( ( X5 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2) )
           => ( ! [Y2: B] :
                  ( ! [X4: B] :
                      ( aa(set(B),$o,member(B,X4),X5)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X4) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),A2) )
             => ( aa(set(B),B,complete_Inf_Inf(B),X5) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5862_Inf__less__eq,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),U: B] :
          ( ! [V4: B] :
              ( aa(set(B),$o,member(B,V4),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),V4),U) )
         => ( ( A4 != bot_bot(set(B)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),U) ) ) ) ).

% Inf_less_eq
tff(fact_5863_Inf__le__iff,axiom,
    ! [B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [A4: set(B),X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),X)
        <=> ! [Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y5)
             => ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y5) ) ) ) ) ).

% Inf_le_iff
tff(fact_5864_Inf__eqI,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),X: B] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),I2) )
         => ( ! [Y2: B] :
                ( ! [I4: B] :
                    ( aa(set(B),$o,member(B,I4),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),I4) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X) )
           => ( aa(set(B),B,complete_Inf_Inf(B),A4) = X ) ) ) ) ).

% Inf_eqI
tff(fact_5865_Inf__mono,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: set(B),A4: set(B)] :
          ( ! [B3: B] :
              ( aa(set(B),$o,member(B,B3),B4)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B3) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Inf_Inf(B),B4)) ) ) ).

% Inf_mono
tff(fact_5866_Inf__lower,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B,A4: set(B)] :
          ( aa(set(B),$o,member(B,X),A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),X) ) ) ).

% Inf_lower
tff(fact_5867_Inf__lower2,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [U: B,A4: set(B),V2: B] :
          ( aa(set(B),$o,member(B,U),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),V2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),V2) ) ) ) ).

% Inf_lower2
tff(fact_5868_le__Inf__iff,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B2: B,A4: set(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(set(B),B,complete_Inf_Inf(B),A4))
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),X3) ) ) ) ).

% le_Inf_iff
tff(fact_5869_Inf__greatest,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),Z: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),aa(set(B),B,complete_Inf_Inf(B),A4)) ) ) ).

% Inf_greatest
tff(fact_5870_cInf__eq,axiom,
    ! [B: $tType] :
      ( ( condit1219197933456340205attice(B)
        & no_top(B) )
     => ! [X5: set(B),A2: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),X5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2) )
         => ( ! [Y2: B] :
                ( ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),X5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X4) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),A2) )
           => ( aa(set(B),B,complete_Inf_Inf(B),X5) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_5871_cInf__eq__minimum,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Z: B,X5: set(B)] :
          ( aa(set(B),$o,member(B,Z),X5)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X2) )
           => ( aa(set(B),B,complete_Inf_Inf(B),X5) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_5872_Inf__superset__mono,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: set(B),A4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Inf_Inf(B),B4)) ) ) ).

% Inf_superset_mono
tff(fact_5873_INF__eqI,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),X: C,F: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),X),aa(B,C,F,I2)) )
         => ( ! [Y2: C] :
                ( ! [I4: B] :
                    ( aa(set(B),$o,member(B,I4),A4)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Y2),aa(B,C,F,I4)) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Y2),X) )
           => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4)) = X ) ) ) ) ).

% INF_eqI
tff(fact_5874_INF__mono,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( comple6319245703460814977attice(D)
     => ! [B4: set(B),A4: set(C),F: fun(C,D),G: fun(B,D)] :
          ( ! [M2: B] :
              ( aa(set(B),$o,member(B,M2),B4)
             => ? [X4: C] :
                  ( aa(set(C),$o,member(C,X4),A4)
                  & aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(C,D,F,X4)),aa(B,D,G,M2)) ) )
         => aa(D,$o,aa(D,fun(D,$o),ord_less_eq(D),aa(set(D),D,complete_Inf_Inf(D),aa(set(C),set(D),image2(C,D,F),A4))),aa(set(D),D,complete_Inf_Inf(D),aa(set(B),set(D),image2(B,D,G),B4))) ) ) ).

% INF_mono
tff(fact_5875_INF__lower,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I: B,A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,member(B,I),A4)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(B,C,F,I)) ) ) ).

% INF_lower
tff(fact_5876_INF__mono_H,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [F: fun(B,C),G: fun(B,C),A4: set(B)] :
          ( ! [X2: B] : aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2))
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),A4))) ) ) ).

% INF_mono'
tff(fact_5877_INF__lower2,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I: B,A4: set(B),F: fun(B,C),U: C] :
          ( aa(set(B),$o,member(B,I),A4)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I)),U)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),U) ) ) ) ).

% INF_lower2
tff(fact_5878_le__INF__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [U: B,F: fun(C,B),A4: set(C)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4)))
        <=> ! [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(C,B,F,X3)) ) ) ) ).

% le_INF_iff
tff(fact_5879_INF__greatest,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),U: C,F: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),U),aa(B,C,F,I2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),U),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ).

% INF_greatest
tff(fact_5880_less__INF__D,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Y: B,F: fun(C,B),A4: set(C),I: C] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4)))
         => ( aa(set(C),$o,member(C,I),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),aa(C,B,F,I)) ) ) ) ).

% less_INF_D
tff(fact_5881_INF__less__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [F: fun(C,B),A4: set(C),A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),A2)
        <=> ? [X3: C] :
              ( aa(set(C),$o,member(C,X3),A4)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,X3)),A2) ) ) ) ).

% INF_less_iff
tff(fact_5882_INF__absorb,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [K: B,I5: set(B),A4: fun(B,C)] :
          ( aa(set(B),$o,member(B,K),I5)
         => ( aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(B,C,A4,K)),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,A4),I5))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,A4),I5)) ) ) ) ).

% INF_absorb
tff(fact_5883_INF__inf__distrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A4: set(C),G: fun(C,B)] : aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,G),A4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_zi(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),A4)) ) ).

% INF_inf_distrib
tff(fact_5884_INT__extend__simps_I10_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,B4: fun(D,set(B)),F: fun(C,D),A4: set(C)] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,D),fun(C,set(B)),aTP_Lamp_ye(fun(D,set(B)),fun(fun(C,D),fun(C,set(B))),B4),F)),A4)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),B4),aa(set(C),set(D),image2(C,D,F),A4))) ).

% INT_extend_simps(10)
tff(fact_5885_INT__subset__iff,axiom,
    ! [B: $tType,C: $tType,B4: set(B),A4: fun(C,set(B)),I5: set(C)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5)))
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),I5)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(C,set(B),A4,X3)) ) ) ).

% INT_subset_iff
tff(fact_5886_INT__anti__mono,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(B),F: fun(B,set(C)),G: fun(B,set(C))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),F,X2)),aa(B,set(C),G,X2)) )
       => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),B4))),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),G),A4))) ) ) ).

% INT_anti_mono
tff(fact_5887_INT__greatest,axiom,
    ! [C: $tType,B: $tType,A4: set(B),C6: set(C),B4: fun(B,set(C))] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),C6),aa(B,set(C),B4,X2)) )
     => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),C6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ).

% INT_greatest
tff(fact_5888_INT__lower,axiom,
    ! [C: $tType,B: $tType,A2: B,A4: set(B),B4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,A2),A4)
     => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))),aa(B,set(C),B4,A2)) ) ).

% INT_lower
tff(fact_5889_INT__extend__simps_I5_J,axiom,
    ! [B: $tType,C: $tType,A2: B,B4: fun(C,set(B)),C6: set(C)] : aa(set(B),set(B),insert(B,A2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = 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(B)),fun(C,set(B)),aTP_Lamp_xk(B,fun(fun(C,set(B)),fun(C,set(B))),A2),B4)),C6)) ).

% INT_extend_simps(5)
tff(fact_5890_INT__insert__distrib,axiom,
    ! [C: $tType,B: $tType,U: B,A4: set(B),A2: C,B4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,U),A4)
     => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_yh(C,fun(fun(B,set(C)),fun(B,set(C))),A2),B4)),A4)) = aa(set(C),set(C),insert(C,A2),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4))) ) ) ).

% INT_insert_distrib
tff(fact_5891_INTER__UNIV__conv_I1_J,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,set(B)),A4: set(C)] :
      ( ( top_top(set(B)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) )
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),A4)
         => ( aa(C,set(B),B4,X3) = top_top(set(B)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_5892_INTER__UNIV__conv_I2_J,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,set(B)),A4: set(C)] :
      ( ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) = top_top(set(B)) )
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),A4)
         => ( aa(C,set(B),B4,X3) = top_top(set(B)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_5893_Int__Inter__image,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),B4: fun(C,set(B)),C6: set(C)] : 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(B)),fun(C,set(B)),aTP_Lamp_zj(fun(C,set(B)),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) = 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(C),set(set(B)),image2(C,set(B),A4),C6))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) ).

% Int_Inter_image
tff(fact_5894_INT__Int__distrib,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),B4: fun(C,set(B)),I5: set(C)] : 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(B)),fun(C,set(B)),aTP_Lamp_zj(fun(C,set(B)),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),I5)) = 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(C),set(set(B)),image2(C,set(B),A4),I5))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),I5))) ).

% INT_Int_distrib
tff(fact_5895_INT__absorb,axiom,
    ! [C: $tType,B: $tType,K: B,I5: set(B),A4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,K),I5)
     => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,K)),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)) ) ) ).

% INT_absorb
tff(fact_5896_INT__extend__simps_I9_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,C6: fun(D,set(B)),B4: fun(C,set(D)),A4: set(C)] : 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(D)),fun(C,set(B)),aTP_Lamp_zk(fun(D,set(B)),fun(fun(C,set(D)),fun(C,set(B))),C6),B4)),A4)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),C6),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B4),A4)))) ).

% INT_extend_simps(9)
tff(fact_5897_Int__Inter__eq_I1_J,axiom,
    ! [B: $tType,A4: set(B),B11: set(set(B))] :
      aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),B11)) = $ite(B11 = bot_bot(set(set(B))),A4,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4)),B11))) ).

% Int_Inter_eq(1)
tff(fact_5898_Int__Inter__eq_I2_J,axiom,
    ! [B: $tType,B11: set(set(B)),A4: set(B)] :
      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)),B11)),A4) = $ite(B11 = bot_bot(set(set(B))),A4,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aTP_Lamp_ya(set(B),fun(set(B),set(B)),A4)),B11))) ).

% Int_Inter_eq(2)
tff(fact_5899_INF__le__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [F: fun(C,B),A4: set(C),X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),X)
        <=> ! [Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y5)
             => ? [X3: C] :
                  ( aa(set(C),$o,member(C,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,X3)),Y5) ) ) ) ) ).

% INF_le_iff
tff(fact_5900_cINF__greatest,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),Ma: C,F: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Ma),aa(B,C,F,X2)) )
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Ma),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ) ).

% cINF_greatest
tff(fact_5901_INF__eq__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I5: set(B),F: fun(B,C),C2: C] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),I5)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),C2) )
           => ( ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),I5)) = C2 )
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),I5)
                 => ( aa(B,C,F,X3) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5902_Inf__le__Sup,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B)] :
          ( ( A4 != bot_bot(set(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ).

% Inf_le_Sup
tff(fact_5903_cInf__abs__ge,axiom,
    ! [B: $tType] :
      ( ( condit6923001295902523014norder(B)
        & linordered_idom(B) )
     => ! [S: set(B),A2: B] :
          ( ( S != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),X2)),A2) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(B),B,complete_Inf_Inf(B),S))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_5904_INF__superset__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B4: set(B),A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),B4)
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2)) )
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B4))) ) ) ) ).

% INF_superset_mono
tff(fact_5905_INF__empty,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B)] : aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),bot_bot(set(C)))) = top_top(B) ) ).

% INF_empty
tff(fact_5906_INF__constant,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [C2: B,A4: set(C)] :
          aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_yn(B,fun(C,B),C2)),A4)) = $ite(A4 = bot_bot(set(C)),top_top(B),C2) ) ).

% INF_constant
tff(fact_5907_INF__inf__const1,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I5: set(B),X: C,F: fun(B,C)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_zl(C,fun(fun(B,C),fun(B,C)),X),F)),I5)) = aa(C,C,aa(C,fun(C,C),inf_inf(C),X),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_5908_INF__inf__const2,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [I5: set(B),F: fun(B,C),X: C] :
          ( ( I5 != bot_bot(set(B)) )
         => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,aa(C,fun(B,C),aTP_Lamp_zm(fun(B,C),fun(C,fun(B,C)),F),X)),I5)) = aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),I5))),X) ) ) ) ).

% INF_inf_const2
tff(fact_5909_uminus__SUP,axiom,
    ! [B: $tType,C: $tType] :
      ( comple489889107523837845lgebra(B)
     => ! [B4: fun(C,B),A4: set(C)] : aa(B,B,uminus_uminus(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,B4),A4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_zn(fun(C,B),fun(C,B),B4)),A4)) ) ).

% uminus_SUP
tff(fact_5910_uminus__INF,axiom,
    ! [B: $tType,C: $tType] :
      ( comple489889107523837845lgebra(B)
     => ! [B4: fun(C,B),A4: set(C)] : aa(B,B,uminus_uminus(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,B4),A4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_zn(fun(C,B),fun(C,B),B4)),A4)) ) ).

% uminus_INF
tff(fact_5911_INF__insert,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A2: C,A4: set(C)] : aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),aa(set(C),set(C),insert(C,A2),A4))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(C,B,F,A2)),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))) ) ).

% INF_insert
tff(fact_5912_INT__empty,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,set(B))] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),bot_bot(set(C)))) = top_top(set(B)) ).

% INT_empty
tff(fact_5913_INT__extend__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] :
      aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = $ite(C6 = bot_bot(set(C)),A4,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(B)),fun(C,set(B)),aTP_Lamp_yk(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6))) ).

% INT_extend_simps(2)
tff(fact_5914_INT__extend__simps_I1_J,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),C6: set(C),B4: set(B)] :
      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(C),set(set(B)),image2(C,set(B),A4),C6))),B4) = $ite(C6 = bot_bot(set(C)),B4,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_yl(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6))) ).

% INT_extend_simps(1)
tff(fact_5915_UN__extend__simps_I7_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_zd(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) ).

% UN_extend_simps(7)
tff(fact_5916_INT__extend__simps_I8_J,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(set(C))] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(set(C)),set(set(B)),image2(set(C),set(B),aTP_Lamp_zo(fun(C,set(B)),fun(set(C),set(B)),B4)),A4)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),A4))) ).

% INT_extend_simps(8)
tff(fact_5917_INF__le__SUP,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ).

% INF_le_SUP
tff(fact_5918_cInf__asclose,axiom,
    ! [B: $tType] :
      ( ( condit6923001295902523014norder(B)
        & linordered_idom(B) )
     => ! [S: set(B),L: B,E2: B] :
          ( ( S != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),X2),L))),E2) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(B),B,complete_Inf_Inf(B),S)),L))),E2) ) ) ) ).

% cInf_asclose
tff(fact_5919_INT__extend__simps_I3_J,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),C6: set(C),B4: set(B)] :
      aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4) = $ite(C6 = bot_bot(set(C)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),top_top(set(B))),B4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_ym(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6))) ).

% INT_extend_simps(3)
tff(fact_5920_INT__extend__simps_I4_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] :
      aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = $ite(C6 = bot_bot(set(C)),A4,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(B)),fun(C,set(B)),aTP_Lamp_zd(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6))) ).

% INT_extend_simps(4)
tff(fact_5921_SUP__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Pa: fun(D,fun(C,B))] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_zq(fun(D,fun(C,B)),fun(C,B),Pa)),top_top(set(C)))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(C,D)),set(B),image2(fun(C,D),B,aTP_Lamp_zs(fun(D,fun(C,B)),fun(fun(C,D),B),Pa)),top_top(set(fun(C,D))))) ) ).

% SUP_INF
tff(fact_5922_INF__SUP,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Pa: fun(D,fun(C,B))] : aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aTP_Lamp_zt(fun(D,fun(C,B)),fun(C,B),Pa)),top_top(set(C)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(C,D)),set(B),image2(fun(C,D),B,aTP_Lamp_zu(fun(D,fun(C,B)),fun(fun(C,D),B),Pa)),top_top(set(fun(C,D))))) ) ).

% INF_SUP
tff(fact_5923_UN__constant__eq,axiom,
    ! [B: $tType,C: $tType,A2: B,A4: set(B),F: fun(B,set(C)),C2: set(C)] :
      ( aa(set(B),$o,member(B,A2),A4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => ( aa(B,set(C),F,X2) = C2 ) )
       => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),A4)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_5924_relChain__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ord(C)
     => ! [R: set(product_prod(B,B)),As3: fun(B,C)] :
          ( bNF_Ca3754400796208372196lChain(B,C,R,As3)
        <=> ! [I3: B,J3: B] :
              ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),I3),J3)),R)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,As3,I3)),aa(B,C,As3,J3)) ) ) ) ).

% relChain_def
tff(fact_5925_Restr__subset,axiom,
    ! [B: $tType,A4: set(B),B4: set(B),R: set(product_prod(B,B))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
     => ( aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R),product_Sigma(B,B,B4,aTP_Lamp_zv(set(B),fun(B,set(B)),B4)))),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4))) = aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4))) ) ) ).

% Restr_subset
tff(fact_5926_finite__inf__Sup,axiom,
    ! [B: $tType] :
      ( finite8700451911770168679attice(B)
     => ! [A2: B,A4: set(B)] : aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),aa(set(B),B,complete_Sup_Sup(B),A4)) = aa(set(B),B,complete_Sup_Sup(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_zw(B,fun(set(B),fun(B,$o)),A2),A4))) ) ).

% finite_inf_Sup
tff(fact_5927_INF2__I,axiom,
    ! [C: $tType,B: $tType,D: $tType,A4: set(B),B4: fun(B,fun(C,fun(D,$o))),B2: C,C2: D] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(D,$o,aa(C,fun(D,$o),aa(B,fun(C,fun(D,$o)),B4,X2),B2),C2) )
     => aa(D,$o,aa(C,fun(D,$o),aa(set(fun(C,fun(D,$o))),fun(C,fun(D,$o)),complete_Inf_Inf(fun(C,fun(D,$o))),aa(set(B),set(fun(C,fun(D,$o))),image2(B,fun(C,fun(D,$o)),B4),A4)),B2),C2) ) ).

% INF2_I
tff(fact_5928_INF1__I,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(B,fun(C,$o)),B2: C] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),A4)
         => aa(C,$o,aa(B,fun(C,$o),B4,X2),B2) )
     => aa(C,$o,aa(set(fun(C,$o)),fun(C,$o),complete_Inf_Inf(fun(C,$o)),aa(set(B),set(fun(C,$o)),image2(B,fun(C,$o),B4),A4)),B2) ) ).

% INF1_I
tff(fact_5929_subset__mset_OcINF__const,axiom,
    ! [B: $tType,C: $tType,A4: set(B),C2: multiset(C)] :
      ( ( A4 != bot_bot(set(B)) )
     => ( aa(set(multiset(C)),multiset(C),complete_Inf_Inf(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),aTP_Lamp_xl(multiset(C),fun(B,multiset(C)),C2)),A4)) = C2 ) ) ).

% subset_mset.cINF_const
tff(fact_5930_INF1__D,axiom,
    ! [C: $tType,B: $tType,B4: fun(C,fun(B,$o)),A4: set(C),B2: B,A2: C] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Inf_Inf(fun(B,$o)),aa(set(C),set(fun(B,$o)),image2(C,fun(B,$o),B4),A4)),B2)
     => ( aa(set(C),$o,member(C,A2),A4)
       => aa(B,$o,aa(C,fun(B,$o),B4,A2),B2) ) ) ).

% INF1_D
tff(fact_5931_INF1__E,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,fun(B,$o)),A4: set(C),B2: B,A2: C] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Inf_Inf(fun(B,$o)),aa(set(C),set(fun(B,$o)),image2(C,fun(B,$o),B4),A4)),B2)
     => ( ~ aa(B,$o,aa(C,fun(B,$o),B4,A2),B2)
       => ~ aa(set(C),$o,member(C,A2),A4) ) ) ).

% INF1_E
tff(fact_5932_INF2__D,axiom,
    ! [B: $tType,D: $tType,C: $tType,B4: fun(D,fun(B,fun(C,$o))),A4: set(D),B2: B,C2: C,A2: D] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),aa(set(D),set(fun(B,fun(C,$o))),image2(D,fun(B,fun(C,$o)),B4),A4)),B2),C2)
     => ( aa(set(D),$o,member(D,A2),A4)
       => aa(C,$o,aa(B,fun(C,$o),aa(D,fun(B,fun(C,$o)),B4,A2),B2),C2) ) ) ).

% INF2_D
tff(fact_5933_INF2__E,axiom,
    ! [C: $tType,B: $tType,D: $tType,B4: fun(D,fun(B,fun(C,$o))),A4: set(D),B2: B,C2: C,A2: D] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),aa(set(D),set(fun(B,fun(C,$o))),image2(D,fun(B,fun(C,$o)),B4),A4)),B2),C2)
     => ( ~ aa(C,$o,aa(B,fun(C,$o),aa(D,fun(B,fun(C,$o)),B4,A2),B2),C2)
       => ~ aa(set(D),$o,member(D,A2),A4) ) ) ).

% INF2_E
tff(fact_5934_INF__Int__eq2,axiom,
    ! [C: $tType,B: $tType,S: set(set(product_prod(B,C))),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),aa(set(set(product_prod(B,C))),set(fun(B,fun(C,$o))),image2(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o)))),S)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Inf_Inf(set(product_prod(B,C))),S)) ) ).

% INF_Int_eq2
tff(fact_5935_INF__Int__eq,axiom,
    ! [B: $tType,S: set(set(B)),X4: B] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Inf_Inf(fun(B,$o)),aa(set(set(B)),set(fun(B,$o)),image2(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o))),S)),X4)
    <=> aa(set(B),$o,member(B,X4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),S)) ) ).

% INF_Int_eq
tff(fact_5936_INF__INT__eq,axiom,
    ! [B: $tType,C: $tType,R: fun(C,set(B)),S: set(C),X4: B] :
      ( aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Inf_Inf(fun(B,$o)),aa(set(C),set(fun(B,$o)),image2(C,fun(B,$o),aTP_Lamp_xt(fun(C,set(B)),fun(C,fun(B,$o)),R)),S)),X4)
    <=> aa(set(B),$o,member(B,X4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),R),S))) ) ).

% INF_INT_eq
tff(fact_5937_INF__INT__eq2,axiom,
    ! [B: $tType,C: $tType,D: $tType,R: fun(D,set(product_prod(B,C))),S: set(D),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),aa(set(D),set(fun(B,fun(C,$o))),image2(D,fun(B,fun(C,$o)),aTP_Lamp_xs(fun(D,set(product_prod(B,C))),fun(D,fun(B,fun(C,$o))),R)),S)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Inf_Inf(set(product_prod(B,C))),aa(set(D),set(set(product_prod(B,C))),image2(D,set(product_prod(B,C)),R),S))) ) ).

% INF_INT_eq2
tff(fact_5938_Inf__set__def,axiom,
    ! [B: $tType,A4: set(set(B))] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),A4) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_zx(set(set(B)),fun(B,$o),A4)) ).

% Inf_set_def
tff(fact_5939_Inf__INT__eq2,axiom,
    ! [C: $tType,B: $tType,S: set(fun(B,fun(C,$o))),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),S),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Inf_Inf(set(product_prod(B,C))),aa(set(fun(product_prod(B,C),$o)),set(set(product_prod(B,C))),image2(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C))),aa(set(fun(B,fun(C,$o))),set(fun(product_prod(B,C),$o)),image2(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o)),S)))) ) ).

% Inf_INT_eq2
tff(fact_5940_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat))) ).

% natLess_def
tff(fact_5941_relpow__finite__bounded1,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),K: nat] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),K),R2)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_zz(set(product_prod(B,B)),fun(nat,$o),R2))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5942_mlex__eq,axiom,
    ! [B: $tType,F: fun(B,nat),R2: set(product_prod(B,B))] : mlex_prod(B,F,R2) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aTP_Lamp_aaa(fun(B,nat),fun(set(product_prod(B,B)),fun(B,fun(B,$o))),F),R2))) ).

% mlex_eq
tff(fact_5943_relpow__1,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] : aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),one_one(nat)),R2) = R2 ).

% relpow_1
tff(fact_5944_finite__relpow,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),N: nat] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
       => aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),R2)) ) ) ).

% finite_relpow
tff(fact_5945_relpow__0__E,axiom,
    ! [B: $tType,X: B,Y: B,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),zero_zero(nat)),R2))
     => ( X = Y ) ) ).

% relpow_0_E
tff(fact_5946_relpow__0__I,axiom,
    ! [B: $tType,X: B,R2: set(product_prod(B,B))] : aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),X)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),zero_zero(nat)),R2)) ).

% relpow_0_I
tff(fact_5947_relpow__add,axiom,
    ! [B: $tType,Ma: nat,N: nat,R2: set(product_prod(B,B))] : aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),R2) = relcomp(B,B,B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),Ma),R2),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),R2)) ).

% relpow_add
tff(fact_5948_relpowp__relpow__eq,axiom,
    ! [B: $tType,N: nat,R2: set(product_prod(B,B)),X4: B,Xa: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),N),aTP_Lamp_aab(set(product_prod(B,B)),fun(B,fun(B,$o)),R2)),X4),Xa)
    <=> aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X4),Xa)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),R2)) ) ).

% relpowp_relpow_eq
tff(fact_5949_relpow__E2,axiom,
    ! [B: $tType,X: B,Z: B,N: nat,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Z)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),R2))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: B,M2: nat] :
              ( ( N = aa(nat,nat,suc,M2) )
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y2)),R2)
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),M2),R2)) ) ) ) ) ).

% relpow_E2
tff(fact_5950_relpow__E,axiom,
    ! [B: $tType,X: B,Z: B,N: nat,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Z)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),R2))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: B,M2: nat] :
              ( ( N = aa(nat,nat,suc,M2) )
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y2)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),M2),R2))
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z)),R2) ) ) ) ) ).

% relpow_E
tff(fact_5951_relpow__empty,axiom,
    ! [B: $tType,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),bot_bot(set(product_prod(B,B)))) = bot_bot(set(product_prod(B,B))) ) ) ).

% relpow_empty
tff(fact_5952_relpow__fun__conv,axiom,
    ! [B: $tType,A2: B,B2: B,N: nat,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N),R2))
    <=> ? [F5: fun(nat,B)] :
          ( ( aa(nat,B,F5,zero_zero(nat)) = A2 )
          & ( aa(nat,B,F5,N) = B2 )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N)
             => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,F5,I3)),aa(nat,B,F5,aa(nat,nat,suc,I3)))),R2) ) ) ) ).

% relpow_fun_conv
tff(fact_5953_relpow__finite__bounded,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),K: nat] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),K),R2)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_aac(set(product_prod(B,B)),fun(nat,$o),R2))))) ) ).

% relpow_finite_bounded
tff(fact_5954_ntrancl__def,axiom,
    ! [B: $tType,N: nat,R2: set(product_prod(B,B))] : transitive_ntrancl(B,N,R2) = aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_aad(nat,fun(nat,$o),N)))) ).

% ntrancl_def
tff(fact_5955_mlex__leq,axiom,
    ! [B: $tType,F: fun(B,nat),X: B,Y: B,R2: set(product_prod(B,B))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,X)),aa(B,nat,F,Y))
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),R2)
       => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),mlex_prod(B,F,R2)) ) ) ).

% mlex_leq
tff(fact_5956_length__remdups__concat,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),remdups(B),concat(B,Xss))) = aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(list(B)),set(set(B)),image2(list(B),set(B),set2(B)),aa(list(list(B)),set(list(B)),set2(list(B)),Xss)))) ).

% length_remdups_concat
tff(fact_5957_length__remdups__eq,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),remdups(B),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) )
    <=> ( aa(list(B),list(B),remdups(B),Xs) = Xs ) ) ).

% length_remdups_eq
tff(fact_5958_ntrancl__Zero,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] : transitive_ntrancl(B,zero_zero(nat),R2) = R2 ).

% ntrancl_Zero
tff(fact_5959_length__remdups__leq,axiom,
    ! [B: $tType,Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),remdups(B),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_remdups_leq
tff(fact_5960_sorted__remdups,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),aa(list(B),list(B),remdups(B),Xs)) ) ) ).

% sorted_remdups
tff(fact_5961_length__remdups__card,axiom,
    ! [B: $tType,L: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),remdups(B),L)) = aa(set(B),nat,finite_card(B),aa(list(B),set(B),set2(B),L)) ).

% length_remdups_card
tff(fact_5962_length__remdups__card__conv,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),remdups(B),Xs)) = aa(set(B),nat,finite_card(B),aa(list(B),set(B),set2(B),Xs)) ).

% length_remdups_card_conv
tff(fact_5963_sorted__list__of__set__sort__remdups,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : aa(set(B),list(B),linord4507533701916653071of_set(B),aa(list(B),set(B),set2(B),Xs)) = aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),aa(list(B),list(B),remdups(B),Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_5964_remdup__sort__mergesort__remdups,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),remdups(B)),linorder_sort_key(B,B,aTP_Lamp_hr(B,B))) = mergesort_remdups(B) ) ) ).

% remdup_sort_mergesort_remdups
tff(fact_5965_trancl__finite__eq__relpow,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => ( transitive_trancl(B,R2) = aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_zz(set(product_prod(B,B)),fun(nat,$o),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5966_INF__filter__not__bot,axiom,
    ! [B: $tType,C: $tType,B4: set(B),F3: fun(B,filter(C))] :
      ( ! [X9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),B4)
         => ( aa(set(B),$o,finite_finite2(B),X9)
           => ( aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),X9)) != bot_bot(filter(C)) ) ) )
     => ( aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),B4)) != bot_bot(filter(C)) ) ) ).

% INF_filter_not_bot
tff(fact_5967_trancl__single,axiom,
    ! [B: $tType,A2: B,B2: B] : transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),bot_bot(set(product_prod(B,B))))) = aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),bot_bot(set(product_prod(B,B)))) ).

% trancl_single
tff(fact_5968_trancl__mono__mp,axiom,
    ! [B: $tType,U4: set(product_prod(B,B)),V: set(product_prod(B,B)),X: product_prod(B,B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),U4),V)
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X),transitive_trancl(B,U4))
       => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X),transitive_trancl(B,V)) ) ) ).

% trancl_mono_mp
tff(fact_5969_trancl__sub,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),transitive_trancl(B,R2)) ).

% trancl_sub
tff(fact_5970_trancl__sub__insert__trancl,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),X: product_prod(B,B)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),transitive_trancl(B,R2)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),X),R2))) ).

% trancl_sub_insert_trancl
tff(fact_5971_INF__filter__bot__base,axiom,
    ! [B: $tType,C: $tType,I5: set(B),F3: fun(B,filter(C))] :
      ( ! [I2: B] :
          ( aa(set(B),$o,member(B,I2),I5)
         => ! [J2: B] :
              ( aa(set(B),$o,member(B,J2),I5)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),I5)
                  & aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),aa(B,filter(C),F3,X4)),aa(filter(C),filter(C),aa(filter(C),fun(filter(C),filter(C)),inf_inf(filter(C)),aa(B,filter(C),F3,I2)),aa(B,filter(C),F3,J2))) ) ) )
     => ( ( aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),I5)) = bot_bot(filter(C)) )
      <=> ? [X3: B] :
            ( aa(set(B),$o,member(B,X3),I5)
            & ( aa(B,filter(C),F3,X3) = bot_bot(filter(C)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_5972_trancl__subset__Sigma,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),A4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4)))
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),transitive_trancl(B,R)),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4))) ) ).

% trancl_subset_Sigma
tff(fact_5973_trancl__power,axiom,
    ! [B: $tType,P2: product_prod(B,B),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),P2),transitive_trancl(B,R2))
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),P2),aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),N4),R2)) ) ) ).

% trancl_power
tff(fact_5974_Restr__trancl__mono,axiom,
    ! [B: $tType,V2: B,W: B,E6: set(product_prod(B,B)),U4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),E6),product_Sigma(B,B,U4,aTP_Lamp_zv(set(B),fun(B,set(B)),U4)))))
     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_trancl(B,E6)) ) ).

% Restr_trancl_mono
tff(fact_5975_trancl__set__ntrancl,axiom,
    ! [B: $tType,Xs: list(product_prod(B,B))] : transitive_trancl(B,aa(list(product_prod(B,B)),set(product_prod(B,B)),set2(product_prod(B,B)),Xs)) = transitive_ntrancl(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(B,B)),nat,finite_card(product_prod(B,B)),aa(list(product_prod(B,B)),set(product_prod(B,B)),set2(product_prod(B,B)),Xs))),one_one(nat)),aa(list(product_prod(B,B)),set(product_prod(B,B)),set2(product_prod(B,B)),Xs)) ).

% trancl_set_ntrancl
tff(fact_5976_finite__trancl__ntranl,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => ( transitive_trancl(B,R2) = transitive_ntrancl(B,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(B,B)),nat,finite_card(product_prod(B,B)),R2)),one_one(nat)),R2) ) ) ).

% finite_trancl_ntranl
tff(fact_5977_nth__step__trancl,axiom,
    ! [B: $tType,Xs: list(B),R2: set(product_prod(B,B)),N: nat,Ma: nat] :
      ( ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Xs)),one_one(nat)))
         => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,nth(B,Xs),aa(nat,nat,suc,N2))),aa(nat,B,nth(B,Xs),N2))),R2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),N)
         => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,nth(B,Xs),N)),aa(nat,B,nth(B,Xs),Ma))),transitive_trancl(B,R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5978_Id__on__def,axiom,
    ! [B: $tType,A4: set(B)] : id_on(B,A4) = aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(B),set(set(product_prod(B,B))),image2(B,set(product_prod(B,B)),aTP_Lamp_aae(B,set(product_prod(B,B)))),A4)) ).

% Id_on_def
tff(fact_5979_lex__prod__def,axiom,
    ! [B: $tType,C: $tType,Ra: set(product_prod(B,B)),Rb: set(product_prod(C,C))] : lex_prod(B,C,Ra,Rb) = aa(fun(product_prod(product_prod(B,C),product_prod(B,C)),$o),set(product_prod(product_prod(B,C),product_prod(B,C))),collect(product_prod(product_prod(B,C),product_prod(B,C))),aa(fun(product_prod(B,C),fun(product_prod(B,C),$o)),fun(product_prod(product_prod(B,C),product_prod(B,C)),$o),product_case_prod(product_prod(B,C),product_prod(B,C),$o),aa(fun(B,fun(C,fun(product_prod(B,C),$o))),fun(product_prod(B,C),fun(product_prod(B,C),$o)),product_case_prod(B,C,fun(product_prod(B,C),$o)),aa(set(product_prod(C,C)),fun(B,fun(C,fun(product_prod(B,C),$o))),aTP_Lamp_aag(set(product_prod(B,B)),fun(set(product_prod(C,C)),fun(B,fun(C,fun(product_prod(B,C),$o)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_5980_Id__on__def_H,axiom,
    ! [B: $tType,A4: fun(B,$o)] : id_on(B,aa(fun(B,$o),set(B),collect(B),A4)) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aTP_Lamp_aah(fun(B,$o),fun(B,fun(B,$o)),A4))) ).

% Id_on_def'
tff(fact_5981_Id__on__subset__Times,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),id_on(B,A4)),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4))) ).

% Id_on_subset_Times
tff(fact_5982_Id__on__set,axiom,
    ! [B: $tType,Xs: list(B)] : id_on(B,aa(list(B),set(B),set2(B),Xs)) = aa(list(product_prod(B,B)),set(product_prod(B,B)),set2(product_prod(B,B)),aa(list(B),list(product_prod(B,B)),map(B,product_prod(B,B),aTP_Lamp_sk(B,product_prod(B,B))),Xs)) ).

% Id_on_set
tff(fact_5983_list__collect__set__alt,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),L: list(C)] : list_collect_set(C,B,F,L) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aa(list(C),fun(set(B),$o),aTP_Lamp_aai(fun(C,set(B)),fun(list(C),fun(set(B),$o)),F),L))) ).

% list_collect_set_alt
tff(fact_5984_Gcd__eq__Max,axiom,
    ! [M: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M)
     => ( ( M != bot_bot(set(nat)) )
       => ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M)
         => ( gcd_Gcd(nat,M) = 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_aaj(nat,set(nat))),M))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_5985_Max__singleton,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B] : aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = X ) ).

% Max_singleton
tff(fact_5986_list__collect__set__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),A2: C] : list_collect_set(C,B,F,aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A2),nil(C))) = aa(C,set(B),F,A2) ).

% list_collect_set_simps(2)
tff(fact_5987_list__collect__set__simps_I1_J,axiom,
    ! [C: $tType,B: $tType,F: fun(C,set(B))] : list_collect_set(C,B,F,nil(C)) = bot_bot(set(B)) ).

% list_collect_set_simps(1)
tff(fact_5988_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_5989_Max_Obounded__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798349783984er_Max(B),A4)),X)
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),X) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_5990_Max__less__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,lattic643756798349783984er_Max(B),A4)),X)
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),X) ) ) ) ) ) ).

% Max_less_iff
tff(fact_5991_list__collect__set__map__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(C,set(B)),X: fun(D,C),A2: D] : list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),A2),nil(D)))) = aa(C,set(B),F,aa(D,C,X,A2)) ).

% list_collect_set_map_simps(2)
tff(fact_5992_list__collect__set__map__simps_I1_J,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(C,set(B)),X: fun(D,C)] : list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),nil(D))) = bot_bot(set(B)) ).

% list_collect_set_map_simps(1)
tff(fact_5993_Max__const,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(C)
     => ! [A4: set(B),C2: C] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(C),C,lattic643756798349783984er_Max(C),aa(set(B),set(C),image2(B,C,aTP_Lamp_ok(C,fun(B,C),C2)),A4)) = C2 ) ) ) ) ).

% Max_const
tff(fact_5994_Max__insert,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),ord_max(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),A4)) ) ) ) ) ).

% Max_insert
tff(fact_5995_Max__ge,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),A4)) ) ) ) ).

% Max_ge
tff(fact_5996_Max__eqI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [Y2: B] :
                ( aa(set(B),$o,member(B,Y2),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X) )
           => ( aa(set(B),$o,member(B,X),A4)
             => ( aa(set(B),B,lattic643756798349783984er_Max(B),A4) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_5997_Max__eq__if,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),A4)
                 => ? [Xa: B] :
                      ( aa(set(B),$o,member(B,Xa),B4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Xa) ) )
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),B4)
                   => ? [Xa: B] :
                        ( aa(set(B),$o,member(B,Xa),A4)
                        & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Xa) ) )
               => ( aa(set(B),B,lattic643756798349783984er_Max(B),A4) = aa(set(B),B,lattic643756798349783984er_Max(B),B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_5998_Max_OcoboundedI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(set(B),B,lattic643756798349783984er_Max(B),A4)) ) ) ) ).

% Max.coboundedI
tff(fact_5999_list__collect__set__as__map,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),L: list(C)] : list_collect_set(C,B,F,L) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(list(set(B)),set(set(B)),set2(set(B)),aa(list(C),list(set(B)),map(C,set(B),F),L))) ).

% list_collect_set_as_map
tff(fact_6000_Max_Oin__idem,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(B,B,aa(B,fun(B,B),ord_max(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),A4)) = aa(set(B),B,lattic643756798349783984er_Max(B),A4) ) ) ) ) ).

% Max.in_idem
tff(fact_6001_Max__in,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => aa(set(B),$o,member(B,aa(set(B),B,lattic643756798349783984er_Max(B),A4)),A4) ) ) ) ).

% Max_in
tff(fact_6002_Max__eq__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),Ma: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ( aa(set(B),B,lattic643756798349783984er_Max(B),A4) = Ma )
            <=> ( aa(set(B),$o,member(B,Ma),A4)
                & ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Ma) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6003_Max__ge__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),A4))
            <=> ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),X3) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_6004_eq__Max__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),Ma: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ( Ma = aa(set(B),B,lattic643756798349783984er_Max(B),A4) )
            <=> ( aa(set(B),$o,member(B,Ma),A4)
                & ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Ma) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6005_Max_OboundedE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798349783984er_Max(B),A4)),X)
             => ! [A8: B] :
                  ( aa(set(B),$o,member(B,A8),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A8),X) ) ) ) ) ) ).

% Max.boundedE
tff(fact_6006_Max_OboundedI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [A3: B] :
                  ( aa(set(B),$o,member(B,A3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798349783984er_Max(B),A4)),X) ) ) ) ) ).

% Max.boundedI
tff(fact_6007_Max__gr__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),A4))
            <=> ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),X3) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_6008_Max__insert2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [B3: B] :
                ( aa(set(B),$o,member(B,B3),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),A2) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),insert(B,A2),A4)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_6009_Max__Sup,axiom,
    ! [B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),A4) = aa(set(B),B,complete_Sup_Sup(B),A4) ) ) ) ) ).

% Max_Sup
tff(fact_6010_Max_Oinfinite,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic643756798349783984er_Max(B),A4) = aa(option(B),B,the2(B),none(B)) ) ) ) ).

% Max.infinite
tff(fact_6011_Sup__nat__def,axiom,
    ! [X5: set(nat)] :
      aa(set(nat),nat,complete_Sup_Sup(nat),X5) = $ite(X5 = bot_bot(set(nat)),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),X5)) ).

% Sup_nat_def
tff(fact_6012_Max_Osubset__imp,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798349783984er_Max(B),A4)),aa(set(B),B,lattic643756798349783984er_Max(B),B4)) ) ) ) ) ).

% Max.subset_imp
tff(fact_6013_Max__mono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [M: set(B),N5: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),M),N5)
         => ( ( M != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),N5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798349783984er_Max(B),M)),aa(set(B),B,lattic643756798349783984er_Max(B),N5)) ) ) ) ) ).

% Max_mono
tff(fact_6014_hom__Max__commute,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [H: fun(B,B),N5: set(B)] :
          ( ! [X2: B,Y2: B] : aa(B,B,H,aa(B,B,aa(B,fun(B,B),ord_max(B),X2),Y2)) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(B,B,H,X2)),aa(B,B,H,Y2))
         => ( aa(set(B),$o,finite_finite2(B),N5)
           => ( ( N5 != bot_bot(set(B)) )
             => ( aa(B,B,H,aa(set(B),B,lattic643756798349783984er_Max(B),N5)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),image2(B,B,H),N5)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_6015_Max_Osubset,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( B4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
             => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(set(B),B,lattic643756798349783984er_Max(B),B4)),aa(set(B),B,lattic643756798349783984er_Max(B),A4)) = aa(set(B),B,lattic643756798349783984er_Max(B),A4) ) ) ) ) ) ).

% Max.subset
tff(fact_6016_Max_Oinsert__not__elem,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ aa(set(B),$o,member(B,X),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),ord_max(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),A4)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_6017_Max_Oclosed,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [X2: B,Y2: B] : aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),ord_max(B),X2),Y2)),aa(set(B),set(B),insert(B,X2),aa(set(B),set(B),insert(B,Y2),bot_bot(set(B)))))
             => aa(set(B),$o,member(B,aa(set(B),B,lattic643756798349783984er_Max(B),A4)),A4) ) ) ) ) ).

% Max.closed
tff(fact_6018_card__le__Suc__Max,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => aa(nat,$o,aa(nat,fun(nat,$o),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_6019_Max__add__commute,axiom,
    ! [B: $tType,C: $tType] :
      ( linord4140545234300271783up_add(C)
     => ! [S: set(B),F: fun(B,C),K: C] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(C),C,lattic643756798349783984er_Max(C),aa(set(B),set(C),image2(B,C,aa(C,fun(B,C),aTP_Lamp_aak(fun(B,C),fun(C,fun(B,C)),F),K)),S)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(C),C,lattic643756798349783984er_Max(C),aa(set(B),set(C),image2(B,C,F),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_6020_divide__nat__def,axiom,
    ! [Ma: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Ma),N) = $ite(N = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aal(nat,fun(nat,fun(nat,$o)),Ma),N)))) ).

% divide_nat_def
tff(fact_6021_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,Ma: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ma),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aam(nat,fun(nat,fun(nat,$o)),N),Ma))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6022_list__collect__set__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),L: list(C)] : list_collect_set(C,B,F,L) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aa(list(C),fun(set(B),$o),aTP_Lamp_aan(fun(C,set(B)),fun(list(C),fun(set(B),$o)),F),L))) ).

% list_collect_set_def
tff(fact_6023_Max_Oinsert__remove,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),insert(B,X),A4)) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),ord_max(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ).

% Max.insert_remove
tff(fact_6024_Max_Oremove,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),A4) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),ord_max(B),X),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ) ).

% Max.remove
tff(fact_6025_sum__le__card__Max,axiom,
    ! [B: $tType,A4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(B),set(nat),image2(B,nat,F),A4)))) ) ).

% sum_le_card_Max
tff(fact_6026_dual__Min,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( lattices_Min(B,aTP_Lamp_kh(B,fun(B,$o))) = lattic643756798349783984er_Max(B) ) ) ).

% dual_Min
tff(fact_6027_lexn_Osimps_I2_J,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),N: nat] : aa(nat,set(product_prod(list(B),list(B))),lexn(B,R),aa(nat,nat,suc,N)) = aa(set(product_prod(list(B),list(B))),set(product_prod(list(B),list(B))),aa(set(product_prod(list(B),list(B))),fun(set(product_prod(list(B),list(B))),set(product_prod(list(B),list(B)))),inf_inf(set(product_prod(list(B),list(B)))),aa(set(product_prod(product_prod(B,list(B)),product_prod(B,list(B)))),set(product_prod(list(B),list(B))),image2(product_prod(product_prod(B,list(B)),product_prod(B,list(B))),product_prod(list(B),list(B)),product_map_prod(product_prod(B,list(B)),list(B),product_prod(B,list(B)),list(B),aa(fun(B,fun(list(B),list(B))),fun(product_prod(B,list(B)),list(B)),product_case_prod(B,list(B),list(B)),cons(B)),aa(fun(B,fun(list(B),list(B))),fun(product_prod(B,list(B)),list(B)),product_case_prod(B,list(B),list(B)),cons(B)))),lex_prod(B,list(B),R,aa(nat,set(product_prod(list(B),list(B))),lexn(B,R),N)))),aa(fun(product_prod(list(B),list(B)),$o),set(product_prod(list(B),list(B))),collect(product_prod(list(B),list(B))),aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_aao(nat,fun(list(B),fun(list(B),$o)),N)))) ).

% lexn.simps(2)
tff(fact_6028_map__prod__ident,axiom,
    ! [C: $tType,B: $tType,X4: product_prod(B,C)] : aa(product_prod(B,C),product_prod(B,C),product_map_prod(B,B,C,C,aTP_Lamp_il(B,B),aTP_Lamp_rv(C,C)),X4) = X4 ).

% map_prod_ident
tff(fact_6029_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_gj(int,fun(int,$o),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_6030_map__prod_Oidentity,axiom,
    ! [C: $tType,B: $tType] : product_map_prod(B,B,C,C,aTP_Lamp_il(B,B),aTP_Lamp_rv(C,C)) = id(product_prod(B,C)) ).

% map_prod.identity
tff(fact_6031_linorder_OMin_Ocong,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o))] : lattices_Min(B,Less_eq) = lattices_Min(B,Less_eq) ).

% linorder.Min.cong
tff(fact_6032_case__prod__map__prod,axiom,
    ! [D: $tType,B: $tType,C: $tType,F6: $tType,E: $tType,H: fun(C,fun(D,B)),F: fun(E,C),G: fun(F6,D),X: product_prod(E,F6)] : aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),H),aa(product_prod(E,F6),product_prod(C,D),product_map_prod(E,C,F6,D,F,G),X)) = aa(product_prod(E,F6),B,aa(fun(E,fun(F6,B)),fun(product_prod(E,F6),B),product_case_prod(E,F6,B),aa(fun(F6,D),fun(E,fun(F6,B)),aa(fun(E,C),fun(fun(F6,D),fun(E,fun(F6,B))),aTP_Lamp_aap(fun(C,fun(D,B)),fun(fun(E,C),fun(fun(F6,D),fun(E,fun(F6,B)))),H),F),G)),X) ).

% case_prod_map_prod
tff(fact_6033_map__prod__def,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,F: fun(B,D),G: fun(C,E)] : product_map_prod(B,D,C,E,F,G) = aa(fun(B,fun(C,product_prod(D,E))),fun(product_prod(B,C),product_prod(D,E)),product_case_prod(B,C,product_prod(D,E)),aa(fun(C,E),fun(B,fun(C,product_prod(D,E))),aTP_Lamp_aaq(fun(B,D),fun(fun(C,E),fun(B,fun(C,product_prod(D,E)))),F),G)) ).

% map_prod_def
tff(fact_6034_case__prod__o__map__prod,axiom,
    ! [B: $tType,E: $tType,D: $tType,F6: $tType,C: $tType,F: fun(E,fun(F6,D)),G1: fun(B,E),G22: fun(C,F6)] : aa(fun(product_prod(B,C),product_prod(E,F6)),fun(product_prod(B,C),D),aa(fun(product_prod(E,F6),D),fun(fun(product_prod(B,C),product_prod(E,F6)),fun(product_prod(B,C),D)),comp(product_prod(E,F6),D,product_prod(B,C)),aa(fun(E,fun(F6,D)),fun(product_prod(E,F6),D),product_case_prod(E,F6,D),F)),product_map_prod(B,E,C,F6,G1,G22)) = aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),aa(fun(C,F6),fun(B,fun(C,D)),aa(fun(B,E),fun(fun(C,F6),fun(B,fun(C,D))),aTP_Lamp_aar(fun(E,fun(F6,D)),fun(fun(B,E),fun(fun(C,F6),fun(B,fun(C,D)))),F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_6035_map__prod__surj__on,axiom,
    ! [E: $tType,C: $tType,B: $tType,D: $tType,F: fun(C,B),A4: set(C),A9: set(B),G: fun(E,D),B4: set(E),B12: set(D)] :
      ( ( aa(set(C),set(B),image2(C,B,F),A4) = A9 )
     => ( ( aa(set(E),set(D),image2(E,D,G),B4) = B12 )
       => ( aa(set(product_prod(C,E)),set(product_prod(B,D)),image2(product_prod(C,E),product_prod(B,D),product_map_prod(C,B,E,D,F,G)),product_Sigma(C,E,A4,aTP_Lamp_aas(set(E),fun(C,set(E)),B4))) = product_Sigma(B,D,A9,aTP_Lamp_wv(set(D),fun(B,set(D)),B12)) ) ) ) ).

% map_prod_surj_on
tff(fact_6036_gcd__is__Max__divisors__int,axiom,
    ! [N: int,Ma: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ma),N) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_aat(int,fun(int,fun(int,$o)),N),Ma))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6037_Un__set__drop__extend,axiom,
    ! [B: $tType,J: nat,L: list(set(B))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(set(B)),nat,size_size(list(set(B))),L))
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(nat,set(B),nth(set(B),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(list(set(B)),set(set(B)),set2(set(B)),drop(set(B),J,L)))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(list(set(B)),set(set(B)),set2(set(B)),drop(set(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_6038_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_6039_greaterThan__eq__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,set(B),set_ord_greaterThan(B),X) = aa(B,set(B),set_ord_greaterThan(B),Y) )
        <=> ( X = Y ) ) ) ).

% greaterThan_eq_iff
tff(fact_6040_le__sup__iff,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y)),Z)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_6041_sup_Obounded__iff,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),B2),C2)),A2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_6042_greaterThan__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,K: B] :
          ( aa(set(B),$o,member(B,I),aa(B,set(B),set_ord_greaterThan(B),K))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),I) ) ) ).

% greaterThan_iff
tff(fact_6043_Inf__greaterThan,axiom,
    ! [B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & dense_linorder(B) )
     => ! [X: B] : aa(set(B),B,complete_Inf_Inf(B),aa(B,set(B),set_ord_greaterThan(B),X)) = X ) ).

% Inf_greaterThan
tff(fact_6044_greaterThan__subset__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Y: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_greaterThan(B),X)),aa(B,set(B),set_ord_greaterThan(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% greaterThan_subset_iff
tff(fact_6045_set__mergesort__by__rel__merge,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Xs: list(B),Ys: list(B)] : aa(list(B),set(B),set2(B),merges9089515139780605204_merge(B,R2,Xs,Ys)) = 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),Ys)) ).

% set_mergesort_by_rel_merge
tff(fact_6046_list__collect__set__simps_I3_J,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),A2: C,L: list(C)] : list_collect_set(C,B,F,aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A2),L)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(C,set(B),F,A2)),list_collect_set(C,B,F,L)) ).

% list_collect_set_simps(3)
tff(fact_6047_list__collect__set__simps_I4_J,axiom,
    ! [B: $tType,C: $tType,F: fun(C,set(B)),L: list(C),L3: list(C)] : list_collect_set(C,B,F,aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),list_collect_set(C,B,F,L)),list_collect_set(C,B,F,L3)) ).

% list_collect_set_simps(4)
tff(fact_6048_Compl__atMost,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [K: B] : aa(set(B),set(B),uminus_uminus(set(B)),aa(B,set(B),set_ord_atMost(B),K)) = aa(B,set(B),set_ord_greaterThan(B),K) ) ).

% Compl_atMost
tff(fact_6049_Compl__greaterThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [K: B] : aa(set(B),set(B),uminus_uminus(set(B)),aa(B,set(B),set_ord_greaterThan(B),K)) = aa(B,set(B),set_ord_atMost(B),K) ) ).

% Compl_greaterThan
tff(fact_6050_if__image__distrib,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,$o),F: fun(C,B),G: fun(C,B),S: set(C)] : aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aa(fun(C,B),fun(fun(C,B),fun(C,B)),aTP_Lamp_aau(fun(C,$o),fun(fun(C,B),fun(fun(C,B),fun(C,B))),Pa),F),G)),S) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(C),set(B),image2(C,B,F),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),S),aa(fun(C,$o),set(C),collect(C),Pa)))),aa(set(C),set(B),image2(C,B,G),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),S),aa(fun(C,$o),set(C),collect(C),aTP_Lamp_aav(fun(C,$o),fun(C,$o),Pa))))) ).

% if_image_distrib
tff(fact_6051_UN__Un,axiom,
    ! [B: $tType,C: $tType,M: fun(C,set(B)),A4: set(C),B4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),M),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),A4),B4))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),M),A4))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),M),B4))) ).

% UN_Un
tff(fact_6052_Sup__greaterThanAtLeast,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),top_top(B))
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(B,set(B),set_ord_greaterThan(B),X)) = top_top(B) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6053_image__uminus__lessThan,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),aa(B,set(B),set_ord_lessThan(B),X)) = aa(B,set(B),set_ord_greaterThan(B),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_lessThan
tff(fact_6054_image__uminus__greaterThan,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),aa(B,set(B),set_ord_greaterThan(B),X)) = aa(B,set(B),set_ord_lessThan(B),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_greaterThan
tff(fact_6055_list__collect__set__map__simps_I3_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(C,set(B)),X: fun(D,C),A2: D,L: list(D)] : list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),A2),L))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(C,set(B),F,aa(D,C,X,A2))),list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),L))) ).

% list_collect_set_map_simps(3)
tff(fact_6056_list__collect__set__map__simps_I4_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(C,set(B)),X: fun(D,C),L: list(D),L3: list(D)] : list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),aa(list(D),list(D),aa(list(D),fun(list(D),list(D)),append(D),L),L3))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),L))),list_collect_set(C,B,F,aa(list(D),list(C),map(D,C,X),L3))) ).

% list_collect_set_map_simps(4)
tff(fact_6057_UN__simps_I2_J,axiom,
    ! [C: $tType,B: $tType,A4: fun(C,set(B)),B4: set(B),C6: set(C)] :
      aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_aaw(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6)) = $ite(C6 = bot_bot(set(C)),bot_bot(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4)) ).

% UN_simps(2)
tff(fact_6058_UN__simps_I3_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] :
      aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_aax(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) = $ite(C6 = bot_bot(set(C)),bot_bot(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6)))) ).

% UN_simps(3)
tff(fact_6059_UN__insert,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A2: C,A4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),aa(set(C),set(C),insert(C,A2),A4))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(C,set(B),B4,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4))) ).

% UN_insert
tff(fact_6060_insert__Times__insert,axiom,
    ! [B: $tType,C: $tType,A2: B,A4: set(B),B2: C,B4: set(C)] : product_Sigma(B,C,aa(set(B),set(B),insert(B,A2),A4),aa(set(C),fun(B,set(C)),aTP_Lamp_aay(C,fun(set(C),fun(B,set(C))),B2),B4)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),insert(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),sup_sup(set(product_prod(B,C))),product_Sigma(B,C,A4,aa(set(C),fun(B,set(C)),aTP_Lamp_aay(C,fun(set(C),fun(B,set(C))),B2),B4))),product_Sigma(B,C,aa(set(B),set(B),insert(B,A2),A4),aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))) ).

% insert_Times_insert
tff(fact_6061_sup__Inf,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A2: B,B4: set(B)] : aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),aa(set(B),B,complete_Inf_Inf(B),B4)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),sup_sup(B),A2)),B4)) ) ).

% sup_Inf
tff(fact_6062_SUP__union,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [M: fun(C,B),A4: set(C),B4: set(C)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,M),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),A4),B4))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,M),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,M),B4))) ) ).

% SUP_union
tff(fact_6063_set__union__code,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] : 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),Ys)) = aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) ).

% set_union_code
tff(fact_6064_distrib__inf__le,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [X: B,Y: B,Z: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y)),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Z))),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),Y),Z))) ) ).

% distrib_inf_le
tff(fact_6065_distrib__sup__le,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [X: B,Y: B,Z: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(B,B,aa(B,fun(B,B),inf_inf(B),Y),Z))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y)),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Z))) ) ).

% distrib_sup_le
tff(fact_6066_ivl__disj__un__two_I6_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or3652927894154168847AtMost(B,L,Ma)),set_or3652927894154168847AtMost(B,Ma,U)) = set_or3652927894154168847AtMost(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_6067_inf__sup__ord_I4_J,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [Y: B,X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y)) ) ).

% inf_sup_ord(4)
tff(fact_6068_inf__sup__ord_I3_J,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y)) ) ).

% inf_sup_ord(3)
tff(fact_6069_le__supE,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,B2: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)),X)
         => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),X) ) ) ) ).

% le_supE
tff(fact_6070_le__supI,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,X: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),X)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)),X) ) ) ) ).

% le_supI
tff(fact_6071_sup__ge1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B,Y: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y)) ) ).

% sup_ge1
tff(fact_6072_sup__ge2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [Y: B,X: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y)) ) ).

% sup_ge2
tff(fact_6073_le__supI1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ) ).

% le_supI1
tff(fact_6074_le__supI2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ) ).

% le_supI2
tff(fact_6075_sup_Omono,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [C2: B,A2: B,D2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),D2),B2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),C2),D2)),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_6076_sup__mono,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),D2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)),aa(B,B,aa(B,fun(B,B),sup_sup(B),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_6077_sup__least,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [Y: B,X: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z),X)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),Y),Z)),X) ) ) ) ).

% sup_least
tff(fact_6078_le__iff__sup,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
        <=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_6079_sup_OorderE,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( A2 = aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_6080_sup_OorderI,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,B2: B] :
          ( ( A2 = aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2) ) ) ).

% sup.orderI
tff(fact_6081_sup__unique,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [F: fun(B,fun(B,B)),X: B,Y: B] :
          ( ! [X2: B,Y2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),aa(B,B,aa(B,fun(B,B),F,X2),Y2))
         => ( ! [X2: B,Y2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),aa(B,B,aa(B,fun(B,B),F,X2),Y2))
           => ( ! [X2: B,Y2: B,Z2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X2)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z2),X2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),F,Y2),Z2)),X2) ) )
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y) = aa(B,B,aa(B,fun(B,B),F,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_6082_sup_Oabsorb1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_6083_sup_Oabsorb2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_6084_sup__absorb1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_6085_sup__absorb2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_6086_sup_OboundedE,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,C2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),B2),C2)),A2)
         => ~ ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
             => ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2) ) ) ) ).

% sup.boundedE
tff(fact_6087_sup_OboundedI,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,A2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),B2),C2)),A2) ) ) ) ).

% sup.boundedI
tff(fact_6088_sup_Oorder__iff,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( A2 = aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_6089_sup_Ocobounded1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ).

% sup.cobounded1
tff(fact_6090_sup_Ocobounded2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,A2: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ).

% sup.cobounded2
tff(fact_6091_sup_Oabsorb__iff1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
        <=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_6092_sup_Oabsorb__iff2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
        <=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_6093_sup_OcoboundedI1,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),A2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_6094_sup_OcoboundedI2,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [C2: B,B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),B2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_6095_ivl__disj__un__one_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or3652927894154168847AtMost(B,L,U)),aa(B,set(B),set_ord_greaterThan(B),U)) = aa(B,set(B),set_ord_greaterThan(B),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_6096_ivl__disj__un__two_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or7035219750837199246ssThan(B,L,Ma)),set_or7035219750837199246ssThan(B,Ma,U)) = set_or7035219750837199246ssThan(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_6097_ivl__disj__un__two__touch_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or1337092689740270186AtMost(B,L,Ma)),set_or1337092689740270186AtMost(B,Ma,U)) = set_or1337092689740270186AtMost(B,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_6098_Times__Un__distrib1,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(B),C6: set(C)] : product_Sigma(B,C,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4),aTP_Lamp_vu(set(C),fun(B,set(C)),C6)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),sup_sup(set(product_prod(B,C))),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))),product_Sigma(B,C,B4,aTP_Lamp_vu(set(C),fun(B,set(C)),C6))) ).

% Times_Un_distrib1
tff(fact_6099_Sigma__Un__distrib2,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C)),B4: fun(B,set(C))] : product_Sigma(B,C,I5,aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_aaz(fun(B,set(C)),fun(fun(B,set(C)),fun(B,set(C))),A4),B4)) = aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),sup_sup(set(product_prod(B,C))),product_Sigma(B,C,I5,A4)),product_Sigma(B,C,I5,B4)) ).

% Sigma_Un_distrib2
tff(fact_6100_Collect__imp__eq,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] : aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_aba(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,$o),set(B),collect(B),Pa))),aa(fun(B,$o),set(B),collect(B),Q)) ).

% Collect_imp_eq
tff(fact_6101_set__diff__diff__left,axiom,
    ! [B: $tType,A4: set(B),B4: set(B),C6: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4)),C6) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),C6)) ).

% set_diff_diff_left
tff(fact_6102_insert__def,axiom,
    ! [B: $tType,A2: B,B4: set(B)] : aa(set(B),set(B),insert(B,A2),B4) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_af(B,fun(B,$o),A2))),B4) ).

% insert_def
tff(fact_6103_greaterThan__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [L: B] : aa(B,set(B),set_ord_greaterThan(B),L) = aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),ord_less(B),L)) ) ).

% greaterThan_def
tff(fact_6104_Collect__disj__eq,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o)] : aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ge(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,$o),set(B),collect(B),Pa)),aa(fun(B,$o),set(B),collect(B),Q)) ).

% Collect_disj_eq
tff(fact_6105_Un__def,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_abb(set(B),fun(set(B),fun(B,$o)),A4),B4)) ).

% Un_def
tff(fact_6106_sup__option__def,axiom,
    ! [B: $tType] :
      ( sup(B)
     => ! [X: option(B),Y: option(B)] : aa(option(B),option(B),aa(option(B),fun(option(B),option(B)),sup_sup(option(B)),X),Y) = case_option(option(B),B,Y,aa(option(B),fun(B,option(B)),aTP_Lamp_abd(option(B),fun(option(B),fun(B,option(B))),X),Y),X) ) ).

% sup_option_def
tff(fact_6107_infinite__Ioi,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [A2: B] : ~ aa(set(B),$o,finite_finite2(B),aa(B,set(B),set_ord_greaterThan(B),A2)) ) ).

% infinite_Ioi
tff(fact_6108_greaterThan__non__empty,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [X: B] : aa(B,set(B),set_ord_greaterThan(B),X) != bot_bot(set(B)) ) ).

% greaterThan_non_empty
tff(fact_6109_foldl__un__empty__eq,axiom,
    ! [B: $tType,I: set(B),Ww: list(set(B))] : aa(list(set(B)),set(B),aa(set(B),fun(list(set(B)),set(B)),foldl(set(B),set(B),sup_sup(set(B))),I),Ww) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),I),aa(list(set(B)),set(B),aa(set(B),fun(list(set(B)),set(B)),foldl(set(B),set(B),sup_sup(set(B))),bot_bot(set(B))),Ww)) ).

% foldl_un_empty_eq
tff(fact_6110_Inf__sup__eq__top__iff,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [B4: set(B),A2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Inf_Inf(B),B4)),A2) = top_top(B) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),B4)
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X3),A2) = top_top(B) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_6111_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A4: set(C),G: fun(C,B)] : aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G),A4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_abe(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),A4)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_6112_SUP__absorb,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [K: B,I5: set(B),A4: fun(B,C)] :
          ( aa(set(B),$o,member(B,K),I5)
         => ( aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(B,C,A4,K)),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,A4),I5))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,A4),I5)) ) ) ) ).

% SUP_absorb
tff(fact_6113_INF__sup__distrib2,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [F: fun(C,B),A4: set(C),G: fun(D,B),B4: set(D)] : aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,G),B4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(set(D),fun(C,B),aa(fun(D,B),fun(set(D),fun(C,B)),aTP_Lamp_abg(fun(C,B),fun(fun(D,B),fun(set(D),fun(C,B))),F),G),B4)),A4)) ) ).

% INF_sup_distrib2
tff(fact_6114_sup__INF,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A2: B,F: fun(C,B),B4: set(C)] : aa(B,B,aa(B,fun(B,B),sup_sup(B),A2),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),B4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_abh(B,fun(fun(C,B),fun(C,B)),A2),F)),B4)) ) ).

% sup_INF
tff(fact_6115_Inf__sup,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [B4: set(B),A2: B] : aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Inf_Inf(B),B4)),A2) = aa(set(B),B,complete_Inf_Inf(B),aa(set(B),set(B),image2(B,B,aTP_Lamp_abi(B,fun(B,B),A2)),B4)) ) ).

% Inf_sup
tff(fact_6116_INF__sup,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [F: fun(C,B),B4: set(C),A2: B] : aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),B4))),A2) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(B,fun(C,B),aTP_Lamp_abj(fun(C,B),fun(B,fun(C,B)),F),A2)),B4)) ) ).

% INF_sup
tff(fact_6117_Un__Union__image,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),B4: fun(C,set(B)),C6: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_abk(fun(C,set(B)),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) ).

% Un_Union_image
tff(fact_6118_UN__Un__distrib,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),B4: fun(C,set(B)),I5: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_abk(fun(C,set(B)),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),I5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),I5))) ).

% UN_Un_distrib
tff(fact_6119_UN__absorb,axiom,
    ! [C: $tType,B: $tType,K: B,I5: set(B),A4: fun(B,set(C))] :
      ( aa(set(B),$o,member(B,K),I5)
     => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),aa(B,set(C),A4,K)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)) ) ) ).

% UN_absorb
tff(fact_6120_INT__extend__simps_I7_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = 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(B)),fun(C,set(B)),aTP_Lamp_aax(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6)) ).

% INT_extend_simps(7)
tff(fact_6121_INT__extend__simps_I6_J,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),C6: set(C),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_aaw(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6)) ).

% INT_extend_simps(6)
tff(fact_6122_Un__INT__distrib,axiom,
    ! [B: $tType,C: $tType,B4: set(B),A4: fun(C,set(B)),I5: set(C)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))) = 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(B)),fun(C,set(B)),aTP_Lamp_aax(set(B),fun(fun(C,set(B)),fun(C,set(B))),B4),A4)),I5)) ).

% Un_INT_distrib
tff(fact_6123_Un__INT__distrib2,axiom,
    ! [D: $tType,B: $tType,C: $tType,A4: fun(C,set(B)),I5: set(C),B4: fun(D,set(B)),J4: set(D)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),B4),J4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(D),fun(C,set(B)),aa(fun(D,set(B)),fun(set(D),fun(C,set(B))),aTP_Lamp_abm(fun(C,set(B)),fun(fun(D,set(B)),fun(set(D),fun(C,set(B)))),A4),B4),J4)),I5)) ).

% Un_INT_distrib2
tff(fact_6124_Union__take__drop__id,axiom,
    ! [B: $tType,N: nat,L: list(set(B))] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(list(set(B)),set(set(B)),set2(set(B)),drop(set(B),N,L)))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(list(set(B)),set(set(B)),set2(set(B)),take(set(B),N,L)))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(list(set(B)),set(set(B)),set2(set(B)),L)) ).

% Union_take_drop_id
tff(fact_6125_lessThan__Int__lessThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_greaterThan(B),A2)),aa(B,set(B),set_ord_greaterThan(B),B2)) = aa(B,set(B),set_ord_greaterThan(B),aa(B,B,aa(B,fun(B,B),ord_max(B),A2),B2)) ) ).

% lessThan_Int_lessThan
tff(fact_6126_Un__Inter,axiom,
    ! [B: $tType,A4: set(B),B4: set(set(B))] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),B4)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4)),B4)) ).

% Un_Inter
tff(fact_6127_sup__shunt,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),Y) = top_top(B) )
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,uminus_uminus(B),X)),Y) ) ) ).

% sup_shunt
tff(fact_6128_sup__neg__inf,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [P2: B,Q2: B,R: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),P2),aa(B,B,aa(B,fun(B,B),sup_sup(B),Q2),R))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),P2),aa(B,B,uminus_uminus(B),Q2))),R) ) ) ).

% sup_neg_inf
tff(fact_6129_shunt2,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(B,B,uminus_uminus(B),Y))),Z)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),Y),Z)) ) ) ).

% shunt2
tff(fact_6130_shunt1,axiom,
    ! [B: $tType] :
      ( boolea8198339166811842893lgebra(B)
     => ! [X: B,Y: B,Z: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),X),Y)),Z)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(B,B,uminus_uminus(B),Y)),Z)) ) ) ).

% shunt1
tff(fact_6131_less__eq__Inf__inter,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B),B4: set(B)] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Inf_Inf(B),B4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ).

% less_eq_Inf_inter
tff(fact_6132_ivl__disj__un__two_I7_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or7035219750837199246ssThan(B,L,Ma)),set_or1337092689740270186AtMost(B,Ma,U)) = set_or1337092689740270186AtMost(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_6133_card__Un__le,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),B4))) ).

% card_Un_le
tff(fact_6134_ivl__disj__un__one_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(B,set(B),set_ord_lessThan(B),L)),set_or7035219750837199246ssThan(B,L,U)) = aa(B,set(B),set_ord_lessThan(B),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_6135_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_6136_ivl__disj__un__two_I8_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or1337092689740270186AtMost(B,L,Ma)),set_or3652927894154168847AtMost(B,Ma,U)) = set_or1337092689740270186AtMost(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_6137_ivl__disj__un__one_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(B,set(B),set_ord_atMost(B),L)),set_or3652927894154168847AtMost(B,L,U)) = aa(B,set(B),set_ord_atMost(B),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_6138_SUP__insert,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(C,B),A2: C,A4: set(C)] : aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),aa(set(C),set(C),insert(C,A2),A4))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(C,B,F,A2)),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))) ) ).

% SUP_insert
tff(fact_6139_Un__interval,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(B)
     => ! [B13: B,B23: B,B32: B,F: fun(B,C)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B13),B23)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B23),B32)
           => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),aa(fun(C,$o),set(C),collect(C),aa(fun(B,C),fun(C,$o),aa(B,fun(fun(B,C),fun(C,$o)),aTP_Lamp_abn(B,fun(B,fun(fun(B,C),fun(C,$o))),B13),B23),F))),aa(fun(C,$o),set(C),collect(C),aa(fun(B,C),fun(C,$o),aa(B,fun(fun(B,C),fun(C,$o)),aTP_Lamp_abn(B,fun(B,fun(fun(B,C),fun(C,$o))),B23),B32),F))) = aa(fun(C,$o),set(C),collect(C),aa(fun(B,C),fun(C,$o),aa(B,fun(fun(B,C),fun(C,$o)),aTP_Lamp_abn(B,fun(B,fun(fun(B,C),fun(C,$o))),B13),B32),F)) ) ) ) ) ).

% Un_interval
tff(fact_6140_INF__union,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [M: fun(C,B),A4: set(C),B4: set(C)] : aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,M),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),A4),B4))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,M),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,M),B4))) ) ).

% INF_union
tff(fact_6141_UN__extend__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),C6: set(C),B4: set(B)] :
      aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),C6))),B4) = $ite(C6 = bot_bot(set(C)),B4,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_aaw(fun(C,set(B)),fun(set(B),fun(C,set(B))),A4),B4)),C6))) ).

% UN_extend_simps(2)
tff(fact_6142_UN__extend__simps_I3_J,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: fun(C,set(B)),C6: set(C)] :
      aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),C6))) = $ite(C6 = bot_bot(set(C)),A4,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_aax(set(B),fun(fun(C,set(B)),fun(C,set(B))),A4),B4)),C6))) ).

% UN_extend_simps(3)
tff(fact_6143_INT__Un,axiom,
    ! [B: $tType,C: $tType,M: fun(C,set(B)),A4: set(C),B4: set(C)] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),M),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),A4),B4))) = 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(C),set(set(B)),image2(C,set(B),M),A4))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),M),B4))) ).

% INT_Un
tff(fact_6144_ivl__disj__int__one_I7_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or1337092689740270186AtMost(B,L,U)),aa(B,set(B),set_ord_greaterThan(B),U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(7)
tff(fact_6145_ivl__disj__int__one_I5_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or3652927894154168847AtMost(B,L,U)),aa(B,set(B),set_ord_greaterThan(B),U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(5)
tff(fact_6146_foldl__set,axiom,
    ! [B: $tType,L: list(set(B))] : aa(list(set(B)),set(B),aa(set(B),fun(list(set(B)),set(B)),foldl(set(B),set(B),sup_sup(set(B))),bot_bot(set(B))),L) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_abo(list(set(B)),fun(set(B),$o),L))) ).

% foldl_set
tff(fact_6147_greaterThanAtMost__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [L: B,U: B] : set_or3652927894154168847AtMost(B,L,U) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_greaterThan(B),L)),aa(B,set(B),set_ord_atMost(B),U)) ) ).

% greaterThanAtMost_def
tff(fact_6148_ivl__disj__un__two__touch_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or1337092689740270186AtMost(B,L,Ma)),set_or7035219750837199246ssThan(B,Ma,U)) = set_or7035219750837199246ssThan(B,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_6149_sum_Ounion__inter,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),B4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),B4)) ) ) ) ) ).

% sum.union_inter
tff(fact_6150_greaterThanLessThan__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [L: B,U: B] : set_or5935395276787703475ssThan(B,L,U) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_greaterThan(B),L)),aa(B,set(B),set_ord_lessThan(B),U)) ) ).

% greaterThanLessThan_def
tff(fact_6151_greaterThanLessThan__eq,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [A2: B,B2: B] : set_or5935395276787703475ssThan(B,A2,B2) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_greaterThan(B),A2)),aa(B,set(B),set_ord_lessThan(B),B2)) ) ).

% greaterThanLessThan_eq
tff(fact_6152_card__Un__Int,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4))),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ).

% card_Un_Int
tff(fact_6153_ivl__disj__un__two__touch_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or3652927894154168847AtMost(B,L,Ma)),set_or1337092689740270186AtMost(B,Ma,U)) = set_or3652927894154168847AtMost(B,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6154_ivl__disj__un__two_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or5935395276787703475ssThan(B,L,Ma)),set_or7035219750837199246ssThan(B,Ma,U)) = set_or5935395276787703475ssThan(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6155_ivl__disj__un__singleton_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(B,set(B),set_ord_lessThan(B),U)),aa(set(B),set(B),insert(B,U),bot_bot(set(B)))) = aa(B,set(B),set_ord_atMost(B),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_6156_ivl__disj__un__one_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(B,set(B),set_ord_lessThan(B),L)),set_or1337092689740270186AtMost(B,L,U)) = aa(B,set(B),set_ord_atMost(B),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6157_Max_Ounion,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => ( ( B4 != bot_bot(set(B)) )
               => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(set(B),B,lattic643756798349783984er_Max(B),A4)),aa(set(B),B,lattic643756798349783984er_Max(B),B4)) ) ) ) ) ) ) ).

% Max.union
tff(fact_6158_ivl__disj__un__two_I2_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or3652927894154168847AtMost(B,L,Ma)),set_or5935395276787703475ssThan(B,Ma,U)) = set_or5935395276787703475ssThan(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6159_SUP__UNIV__bool__expand,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: fun($o,B)] : aa(set(B),B,complete_Sup_Sup(B),aa(set($o),set(B),image2($o,B,A4),top_top(set($o)))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa($o,B,A4,$true)),aa($o,B,A4,$false)) ) ).

% SUP_UNIV_bool_expand
tff(fact_6160_ivl__disj__un__one_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(B,set(B),set_ord_atMost(B),L)),set_or5935395276787703475ssThan(B,L,U)) = aa(B,set(B),set_ord_lessThan(B),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_6161_ivl__disj__un__two__touch_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or3652927894154168847AtMost(B,L,Ma)),set_or7035219750837199246ssThan(B,Ma,U)) = set_or5935395276787703475ssThan(B,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6162_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [B: $tType] :
      ( bounde4967611905675639751up_bot(B)
     => semila1105856199041335345_order(B,sup_sup(B),bot_bot(B),aTP_Lamp_abp(B,fun(B,$o)),aTP_Lamp_abq(B,fun(B,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6163_trancl__insert2,axiom,
    ! [B: $tType,A2: B,B2: B,R: set(product_prod(B,B))] : transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),R)) = aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),transitive_trancl(B,R)),aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aa(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o))),aTP_Lamp_abr(B,fun(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o)))),A2),B2),R)))) ).

% trancl_insert2
tff(fact_6164_UN__bool__eq,axiom,
    ! [B: $tType,A4: fun($o,set(B))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set($o),set(set(B)),image2($o,set(B),A4),top_top(set($o)))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa($o,set(B),A4,$true)),aa($o,set(B),A4,$false)) ).

% UN_bool_eq
tff(fact_6165_Un__eq__UN,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set($o),set(set(B)),image2($o,set(B),aa(set(B),fun($o,set(B)),aTP_Lamp_abs(set(B),fun(set(B),fun($o,set(B))),A4),B4)),top_top(set($o)))) ).

% Un_eq_UN
tff(fact_6166_sum_Ounion__inter__neutral,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),B4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))
                 => ( aa(B,C,G,X2) = zero_zero(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6167_sum__Un,axiom,
    ! [C: $tType,B: $tType] :
      ( ab_group_add(C)
     => ! [A4: set(B),B4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ) ).

% sum_Un
tff(fact_6168_sum_Ounion__disjoint,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),B4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = bot_bot(set(B)) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),B4)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6169_prod_Ounion__inter__neutral,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(B),B4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))
                 => ( aa(B,C,G,X2) = one_one(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6170_ivl__disj__un__singleton_I6_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or7035219750837199246ssThan(B,L,U)),aa(set(B),set(B),insert(B,U),bot_bot(set(B)))) = set_or1337092689740270186AtMost(B,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_6171_sum_Ounion__diff2,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),B4: set(B),G: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4)))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ) ).

% sum.union_diff2
tff(fact_6172_sum__Un2,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(B),B4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4))
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4)))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ).

% sum_Un2
tff(fact_6173_card__Un__disjoint,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = bot_bot(set(B)) )
         => ( aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6174_ivl__disj__un__singleton_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),insert(B,L),bot_bot(set(B)))),set_or3652927894154168847AtMost(B,L,U)) = set_or1337092689740270186AtMost(B,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_6175_ivl__disj__un__singleton_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),insert(B,L),bot_bot(set(B)))),set_or5935395276787703475ssThan(B,L,U)) = set_or7035219750837199246ssThan(B,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6176_ivl__disj__un__two_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or1337092689740270186AtMost(B,L,Ma)),set_or5935395276787703475ssThan(B,Ma,U)) = set_or7035219750837199246ssThan(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6177_sum__Un__nat,axiom,
    ! [B: $tType,A4: set(B),B4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),B4))),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_6178_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_6179_ivl__disj__un__singleton_I4_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or5935395276787703475ssThan(B,L,U)),aa(set(B),set(B),insert(B,U),bot_bot(set(B)))) = set_or3652927894154168847AtMost(B,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6180_ivl__disj__un__two_I5_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,Ma: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),Ma)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),U)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or5935395276787703475ssThan(B,L,Ma)),set_or1337092689740270186AtMost(B,Ma,U)) = set_or3652927894154168847AtMost(B,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6181_merge__correct,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L1: list(B),L22: list(B)] :
          ( ( distinct(B,L1)
            & sorted_wrt(B,ord_less_eq(B),L1) )
         => ( ( distinct(B,L22)
              & sorted_wrt(B,ord_less_eq(B),L22) )
           => ( distinct(B,merge(B,L1,L22))
              & sorted_wrt(B,ord_less_eq(B),merge(B,L1,L22))
              & ( aa(list(B),set(B),set2(B),merge(B,L1,L22)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),L1)),aa(list(B),set(B),set2(B),L22)) ) ) ) ) ) ).

% merge_correct
tff(fact_6182_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_6183_prod__Un,axiom,
    ! [C: $tType,B: $tType] :
      ( field(C)
     => ! [A4: set(B),B4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),B4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))
                 => ( aa(B,C,F,X2) != zero_zero(C) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),B4))),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_6184_UN__le__eq__Un0,axiom,
    ! [B: $tType,M: fun(nat,set(B)),N: nat] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),M),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),M),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(B),M,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_6185_interval__cases,axiom,
    ! [B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [S: set(B)] :
          ( ! [A3: B,B3: B,X2: B] :
              ( aa(set(B),$o,member(B,A3),S)
             => ( aa(set(B),$o,member(B,B3),S)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X2)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B3)
                   => aa(set(B),$o,member(B,X2),S) ) ) ) )
         => ? [A3: B,B3: B] :
              ( ( S = bot_bot(set(B)) )
              | ( S = top_top(set(B)) )
              | ( S = aa(B,set(B),set_ord_lessThan(B),B3) )
              | ( S = aa(B,set(B),set_ord_atMost(B),B3) )
              | ( S = aa(B,set(B),set_ord_greaterThan(B),A3) )
              | ( S = aa(B,set(B),set_ord_atLeast(B),A3) )
              | ( S = set_or5935395276787703475ssThan(B,A3,B3) )
              | ( S = set_or3652927894154168847AtMost(B,A3,B3) )
              | ( S = set_or7035219750837199246ssThan(B,A3,B3) )
              | ( S = set_or1337092689740270186AtMost(B,A3,B3) ) ) ) ) ).

% interval_cases
tff(fact_6186_brk__rel__def,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C))] : brk_rel(B,C,R2) = aa(set(product_prod(product_prod($o,B),product_prod($o,C))),set(product_prod(product_prod($o,B),product_prod($o,C))),aa(set(product_prod(product_prod($o,B),product_prod($o,C))),fun(set(product_prod(product_prod($o,B),product_prod($o,C))),set(product_prod(product_prod($o,B),product_prod($o,C)))),sup_sup(set(product_prod(product_prod($o,B),product_prod($o,C)))),aa(fun(product_prod(product_prod($o,B),product_prod($o,C)),$o),set(product_prod(product_prod($o,B),product_prod($o,C))),collect(product_prod(product_prod($o,B),product_prod($o,C))),aTP_Lamp_abt(set(product_prod(B,C)),fun(product_prod(product_prod($o,B),product_prod($o,C)),$o),R2))),aa(fun(product_prod(product_prod($o,B),product_prod($o,C)),$o),set(product_prod(product_prod($o,B),product_prod($o,C))),collect(product_prod(product_prod($o,B),product_prod($o,C))),aTP_Lamp_abu(product_prod(product_prod($o,B),product_prod($o,C)),$o))) ).

% brk_rel_def
tff(fact_6187_atLeast__eq__iff,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [X: B,Y: B] :
          ( ( aa(B,set(B),set_ord_atLeast(B),X) = aa(B,set(B),set_ord_atLeast(B),Y) )
        <=> ( X = Y ) ) ) ).

% atLeast_eq_iff
tff(fact_6188_atLeast__iff,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [I: B,K: B] :
          ( aa(set(B),$o,member(B,I),aa(B,set(B),set_ord_atLeast(B),K))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K),I) ) ) ).

% atLeast_iff
tff(fact_6189_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_6190_Inf__atLeast,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B] : aa(set(B),B,complete_Inf_Inf(B),aa(B,set(B),set_ord_atLeast(B),X)) = X ) ).

% Inf_atLeast
tff(fact_6191_atLeast__empty__triv,axiom,
    ! [B: $tType] : aa(set(B),set(set(B)),set_ord_atLeast(set(B)),bot_bot(set(B))) = top_top(set(set(B))) ).

% atLeast_empty_triv
tff(fact_6192_atLeast__subset__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [X: B,Y: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atLeast(B),X)),aa(B,set(B),set_ord_atLeast(B),Y))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ).

% atLeast_subset_iff
tff(fact_6193_image__add__atLeast,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [K: B,I: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),K)),aa(B,set(B),set_ord_atLeast(B),I)) = aa(B,set(B),set_ord_atLeast(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),K),I)) ) ).

% image_add_atLeast
tff(fact_6194_Sup__atLeast,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X: B] : aa(set(B),B,complete_Sup_Sup(B),aa(B,set(B),set_ord_atLeast(B),X)) = top_top(B) ) ).

% Sup_atLeast
tff(fact_6195_Compl__atLeast,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [K: B] : aa(set(B),set(B),uminus_uminus(set(B)),aa(B,set(B),set_ord_atLeast(B),K)) = aa(B,set(B),set_ord_lessThan(B),K) ) ).

% Compl_atLeast
tff(fact_6196_Compl__lessThan,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [K: B] : aa(set(B),set(B),uminus_uminus(set(B)),aa(B,set(B),set_ord_lessThan(B),K)) = aa(B,set(B),set_ord_atLeast(B),K) ) ).

% Compl_lessThan
tff(fact_6197_Icc__subset__Ici__iff,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [L: B,H: B,L3: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),set_or1337092689740270186AtMost(B,L,H)),aa(B,set(B),set_ord_atLeast(B),L3))
        <=> ( ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),H)
            | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L3),L) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_6198_image__minus__const__atLeast,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,A2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),minus_minus(B),C2)),aa(B,set(B),set_ord_atLeast(B),A2)) = aa(B,set(B),set_ord_atMost(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),A2)) ) ).

% image_minus_const_atLeast
tff(fact_6199_image__minus__const__AtMost,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [C2: B,B2: B] : aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),minus_minus(B),C2)),aa(B,set(B),set_ord_atMost(B),B2)) = aa(B,set(B),set_ord_atLeast(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),C2),B2)) ) ).

% image_minus_const_AtMost
tff(fact_6200_image__uminus__atMost,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),aa(B,set(B),set_ord_atMost(B),X)) = aa(B,set(B),set_ord_atLeast(B),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_atMost
tff(fact_6201_image__uminus__atLeast,axiom,
    ! [B: $tType] :
      ( ordered_ab_group_add(B)
     => ! [X: B] : aa(set(B),set(B),image2(B,B,uminus_uminus(B)),aa(B,set(B),set_ord_atLeast(B),X)) = aa(B,set(B),set_ord_atMost(B),aa(B,B,uminus_uminus(B),X)) ) ).

% image_uminus_atLeast
tff(fact_6202_Int__atLeastAtMostR2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,C2: B,D2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atLeast(B),A2)),set_or1337092689740270186AtMost(B,C2,D2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),ord_max(B),A2),C2),D2) ) ).

% Int_atLeastAtMostR2
tff(fact_6203_Int__atLeastAtMostL2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B,C2: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or1337092689740270186AtMost(B,A2,B2)),aa(B,set(B),set_ord_atLeast(B),C2)) = set_or1337092689740270186AtMost(B,aa(B,B,aa(B,fun(B,B),ord_max(B),A2),C2),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_6204_sup__Un__eq2,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(B,C)),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(C,$o)),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),sup_sup(fun(B,fun(C,$o))),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),R2)),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),S)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),sup_sup(set(product_prod(B,C))),R2),S)) ) ).

% sup_Un_eq2
tff(fact_6205_sup__set__def,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),sup_sup(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),B4))) ).

% sup_set_def
tff(fact_6206_sup__Un__eq,axiom,
    ! [B: $tType,R2: set(B),S: set(B),X4: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),sup_sup(fun(B,$o)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),R2)),aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),S)),X4)
    <=> aa(set(B),$o,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),R2),S)) ) ).

% sup_Un_eq
tff(fact_6207_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6208_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_6209_not__UNIV__le__Ici,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [L: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),top_top(set(B))),aa(B,set(B),set_ord_atLeast(B),L)) ) ).

% not_UNIV_le_Ici
tff(fact_6210_not__empty__eq__Ici__eq__empty,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [L: B] : bot_bot(set(B)) != aa(B,set(B),set_ord_atLeast(B),L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_6211_infinite__Ici,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [A2: B] : ~ aa(set(B),$o,finite_finite2(B),aa(B,set(B),set_ord_atLeast(B),A2)) ) ).

% infinite_Ici
tff(fact_6212_not__Iic__eq__Ici,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [H: B,L3: B] : aa(B,set(B),set_ord_atMost(B),H) != aa(B,set(B),set_ord_atLeast(B),L3) ) ).

% not_Iic_eq_Ici
tff(fact_6213_not__Ici__eq__Icc,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [L3: B,L: B,H: B] : aa(B,set(B),set_ord_atLeast(B),L3) != set_or1337092689740270186AtMost(B,L,H) ) ).

% not_Ici_eq_Icc
tff(fact_6214_not__UNIV__eq__Ici,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [L3: B] : top_top(set(B)) != aa(B,set(B),set_ord_atLeast(B),L3) ) ).

% not_UNIV_eq_Ici
tff(fact_6215_atLeast__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [L: B] : aa(B,set(B),set_ord_atLeast(B),L) = aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),ord_less_eq(B),L)) ) ).

% atLeast_def
tff(fact_6216_atLeast__eq__UNIV__iff,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [X: B] :
          ( ( aa(B,set(B),set_ord_atLeast(B),X) = top_top(set(B)) )
        <=> ( X = bot_bot(B) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_6217_not__Ici__le__Icc,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [L: B,L3: B,H4: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atLeast(B),L)),set_or1337092689740270186AtMost(B,L3,H4)) ) ).

% not_Ici_le_Icc
tff(fact_6218_not__Iic__le__Ici,axiom,
    ! [B: $tType] :
      ( no_bot(B)
     => ! [H: B,L3: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atMost(B),H)),aa(B,set(B),set_ord_atLeast(B),L3)) ) ).

% not_Iic_le_Ici
tff(fact_6219_not__Ici__le__Iic,axiom,
    ! [B: $tType] :
      ( no_top(B)
     => ! [L: B,H4: B] : ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atLeast(B),L)),aa(B,set(B),set_ord_atMost(B),H4)) ) ).

% not_Ici_le_Iic
tff(fact_6220_Ioi__le__Ico,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A2: B] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_greaterThan(B),A2)),aa(B,set(B),set_ord_atLeast(B),A2)) ) ).

% Ioi_le_Ico
tff(fact_6221_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_greaterThan(nat),K) ).

% atLeast_Suc_greaterThan
tff(fact_6222_ivl__disj__un__one_I8_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or7035219750837199246ssThan(B,L,U)),aa(B,set(B),set_ord_atLeast(B),U)) = aa(B,set(B),set_ord_atLeast(B),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_6223_ivl__disj__int__one_I8_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or7035219750837199246ssThan(B,L,U)),aa(B,set(B),set_ord_atLeast(B),U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(8)
tff(fact_6224_atLeastAtMost__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [L: B,U: B] : set_or1337092689740270186AtMost(B,L,U) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atLeast(B),L)),aa(B,set(B),set_ord_atMost(B),U)) ) ).

% atLeastAtMost_def
tff(fact_6225_atLeastLessThan__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [L: B,U: B] : set_or7035219750837199246ssThan(B,L,U) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atLeast(B),L)),aa(B,set(B),set_ord_lessThan(B),U)) ) ).

% atLeastLessThan_def
tff(fact_6226_Ici__subset__Ioi__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(B,set(B),set_ord_atLeast(B),A2)),aa(B,set(B),set_ord_greaterThan(B),B2))
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),A2) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6227_ivl__disj__int__one_I6_J,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [L: B,U: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),set_or5935395276787703475ssThan(B,L,U)),aa(B,set(B),set_ord_atLeast(B),U)) = bot_bot(set(B)) ) ).

% ivl_disj_int_one(6)
tff(fact_6228_atMost__Int__atLeast,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [N: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(B,set(B),set_ord_atMost(B),N)),aa(B,set(B),set_ord_atLeast(B),N)) = aa(set(B),set(B),insert(B,N),bot_bot(set(B))) ) ).

% atMost_Int_atLeast
tff(fact_6229_ivl__disj__un__singleton_I1_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),insert(B,L),bot_bot(set(B)))),aa(B,set(B),set_ord_greaterThan(B),L)) = aa(B,set(B),set_ord_atLeast(B),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_6230_ivl__disj__un__one_I7_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or1337092689740270186AtMost(B,L,U)),aa(B,set(B),set_ord_greaterThan(B),U)) = aa(B,set(B),set_ord_atLeast(B),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_6231_ivl__disj__un__one_I6_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [L: B,U: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),U)
         => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set_or5935395276787703475ssThan(B,L,U)),aa(B,set(B),set_ord_atLeast(B),U)) = aa(B,set(B),set_ord_greaterThan(B),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_6232_atLeast__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),insert(nat,K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_6233_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_6234_insert__relcomp__union__fold,axiom,
    ! [D: $tType,C: $tType,B: $tType,S: set(product_prod(B,C)),X: product_prod(D,B),X5: set(product_prod(D,C))] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S)
     => ( aa(set(product_prod(D,C)),set(product_prod(D,C)),aa(set(product_prod(D,C)),fun(set(product_prod(D,C)),set(product_prod(D,C))),sup_sup(set(product_prod(D,C))),relcomp(D,B,C,aa(set(product_prod(D,B)),set(product_prod(D,B)),insert(product_prod(D,B),X),bot_bot(set(product_prod(D,B)))),S)),X5) = finite_fold(product_prod(B,C),set(product_prod(D,C)),aa(fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),fun(product_prod(B,C),fun(set(product_prod(D,C)),set(product_prod(D,C)))),product_case_prod(B,C,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aTP_Lamp_abv(product_prod(D,B),fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),X)),X5,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_6235_fold__union__pair,axiom,
    ! [C: $tType,B: $tType,B4: set(B),X: C,A4: set(product_prod(C,B))] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => ( 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))),aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(B),set(set(product_prod(C,B))),image2(B,set(product_prod(C,B)),aTP_Lamp_abw(C,fun(B,set(product_prod(C,B))),X)),B4))),A4) = finite_fold(B,set(product_prod(C,B)),aTP_Lamp_abx(C,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),X),A4,B4) ) ) ).

% fold_union_pair
tff(fact_6236_sum_Oeq__fold,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4) = finite_fold(C,B,aa(fun(C,B),fun(C,fun(B,B)),aa(fun(B,fun(B,B)),fun(fun(C,B),fun(C,fun(B,B))),comp(B,fun(B,B),C),plus_plus(B)),G),zero_zero(B),A4) ) ).

% sum.eq_fold
tff(fact_6237_prod_Oeq__fold,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,B),A4: set(C)] : aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),G),A4) = finite_fold(C,B,aa(fun(C,B),fun(C,fun(B,B)),aa(fun(B,fun(B,B)),fun(fun(C,B),fun(C,fun(B,B))),comp(B,fun(B,B),C),times_times(B)),G),one_one(B),A4) ) ).

% prod.eq_fold
tff(fact_6238_Max_Oeq__fold,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),insert(B,X),A4)) = finite_fold(B,B,ord_max(B),X,A4) ) ) ) ).

% Max.eq_fold
tff(fact_6239_image__fold__insert,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),set(C),image2(B,C,F),A4) = finite_fold(B,set(C),aTP_Lamp_aby(fun(B,C),fun(B,fun(set(C),set(C))),F),bot_bot(set(C)),A4) ) ) ).

% image_fold_insert
tff(fact_6240_product__fold,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => ( product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)) = finite_fold(B,set(product_prod(B,C)),aTP_Lamp_aca(set(C),fun(B,fun(set(product_prod(B,C)),set(product_prod(B,C)))),B4),bot_bot(set(product_prod(B,C))),A4) ) ) ) ).

% product_fold
tff(fact_6241_relcomp__fold,axiom,
    ! [D: $tType,C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(C,D))] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),R2)
     => ( aa(set(product_prod(C,D)),$o,finite_finite2(product_prod(C,D)),S)
       => ( relcomp(B,C,D,R2,S) = finite_fold(product_prod(B,C),set(product_prod(B,D)),aa(fun(B,fun(C,fun(set(product_prod(B,D)),set(product_prod(B,D))))),fun(product_prod(B,C),fun(set(product_prod(B,D)),set(product_prod(B,D)))),product_case_prod(B,C,fun(set(product_prod(B,D)),set(product_prod(B,D)))),aTP_Lamp_acc(set(product_prod(C,D)),fun(B,fun(C,fun(set(product_prod(B,D)),set(product_prod(B,D))))),S)),bot_bot(set(product_prod(B,D))),R2) ) ) ) ).

% relcomp_fold
tff(fact_6242_Id__on__fold,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( id_on(B,A4) = finite_fold(B,set(product_prod(B,B)),aTP_Lamp_acd(B,fun(set(product_prod(B,B)),set(product_prod(B,B)))),bot_bot(set(product_prod(B,B))),A4) ) ) ).

% Id_on_fold
tff(fact_6243_insert__relcomp__fold,axiom,
    ! [D: $tType,C: $tType,B: $tType,S: set(product_prod(B,C)),X: product_prod(D,B),R2: set(product_prod(D,B))] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S)
     => ( relcomp(D,B,C,aa(set(product_prod(D,B)),set(product_prod(D,B)),insert(product_prod(D,B),X),R2),S) = finite_fold(product_prod(B,C),set(product_prod(D,C)),aa(fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),fun(product_prod(B,C),fun(set(product_prod(D,C)),set(product_prod(D,C)))),product_case_prod(B,C,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aTP_Lamp_abv(product_prod(D,B),fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),X)),relcomp(D,B,C,R2,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_6244_Set__filter__fold,axiom,
    ! [B: $tType,A4: set(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( filter3(B,Pa,A4) = finite_fold(B,set(B),aTP_Lamp_ace(fun(B,$o),fun(B,fun(set(B),set(B))),Pa),bot_bot(set(B)),A4) ) ) ).

% Set_filter_fold
tff(fact_6245_comp__fun__commute__relcomp__fold,axiom,
    ! [D: $tType,C: $tType,B: $tType,S: set(product_prod(B,C))] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S)
     => finite6289374366891150609ommute(product_prod(D,B),set(product_prod(D,C)),aa(fun(D,fun(B,fun(set(product_prod(D,C)),set(product_prod(D,C))))),fun(product_prod(D,B),fun(set(product_prod(D,C)),set(product_prod(D,C)))),product_case_prod(D,B,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aTP_Lamp_acg(set(product_prod(B,C)),fun(D,fun(B,fun(set(product_prod(D,C)),set(product_prod(D,C))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_6246_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [C: $tType,B: $tType,F: fun(B,fun(C,C)),G: fun(B,nat)] :
      ( finite6289374366891150609ommute(B,C,F)
     => finite6289374366891150609ommute(B,C,aa(fun(B,nat),fun(B,fun(C,C)),aTP_Lamp_ach(fun(B,fun(C,C)),fun(fun(B,nat),fun(B,fun(C,C))),F),G)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_6247_Set_Ofilter__def,axiom,
    ! [B: $tType,Pa: fun(B,$o),A4: set(B)] : filter3(B,Pa,A4) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_aci(fun(B,$o),fun(set(B),fun(B,$o)),Pa),A4)) ).

% Set.filter_def
tff(fact_6248_comp__fun__commute__const,axiom,
    ! [B: $tType,C: $tType,F: fun(C,C)] : finite6289374366891150609ommute(B,C,aTP_Lamp_acj(fun(C,C),fun(B,fun(C,C)),F)) ).

% comp_fun_commute_const
tff(fact_6249_comp__fun__commute__filter__fold,axiom,
    ! [B: $tType,Pa: fun(B,$o)] : finite6289374366891150609ommute(B,set(B),aTP_Lamp_ace(fun(B,$o),fun(B,fun(set(B),set(B))),Pa)) ).

% comp_fun_commute_filter_fold
tff(fact_6250_comp__fun__commute_Ofoldl__f__commute,axiom,
    ! [C: $tType,B: $tType,F: fun(B,fun(C,C)),A2: B,B2: C,Xs: list(B)] :
      ( finite6289374366891150609ommute(B,C,F)
     => ( aa(C,C,aa(B,fun(C,C),F,A2),aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,aTP_Lamp_ack(fun(B,fun(C,C)),fun(C,fun(B,C)),F)),B2),Xs)) = aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,aTP_Lamp_ack(fun(B,fun(C,C)),fun(C,fun(B,C)),F)),aa(C,C,aa(B,fun(C,C),F,A2),B2)),Xs) ) ) ).

% comp_fun_commute.foldl_f_commute
tff(fact_6251_card_Oeq__fold,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(B),nat,finite_card(B),A4) = finite_fold(B,nat,aTP_Lamp_ar(B,fun(nat,nat)),zero_zero(nat),A4) ).

% card.eq_fold
tff(fact_6252_comp__fun__commute_Ofoldr__conv__foldl,axiom,
    ! [C: $tType,B: $tType,F: fun(B,fun(C,C)),Xs: list(B),A2: C] :
      ( finite6289374366891150609ommute(B,C,F)
     => ( aa(C,C,foldr(B,C,F,Xs),A2) = aa(list(B),C,aa(C,fun(list(B),C),foldl(C,B,aTP_Lamp_ack(fun(B,fun(C,C)),fun(C,fun(B,C)),F)),A2),Xs) ) ) ).

% comp_fun_commute.foldr_conv_foldl
tff(fact_6253_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] : aa(set(B),list(B),linord4507533701916653071of_set(B),A4) = finite_fold(B,list(B),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),nil(B),A4) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_6254_comp__fun__commute__Image__fold,axiom,
    ! [C: $tType,B: $tType,S: set(B)] : finite6289374366891150609ommute(product_prod(B,C),set(C),aa(fun(B,fun(C,fun(set(C),set(C)))),fun(product_prod(B,C),fun(set(C),set(C))),product_case_prod(B,C,fun(set(C),set(C))),aTP_Lamp_acl(set(B),fun(B,fun(C,fun(set(C),set(C)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_6255_inter__Set__filter,axiom,
    ! [B: $tType,B4: set(B),A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = filter3(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4),B4) ) ) ).

% inter_Set_filter
tff(fact_6256_comp__fun__commute__product__fold,axiom,
    ! [C: $tType,B: $tType,B4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),B4)
     => finite6289374366891150609ommute(C,set(product_prod(C,B)),aTP_Lamp_acm(set(B),fun(C,fun(set(product_prod(C,B)),set(product_prod(C,B)))),B4)) ) ).

% comp_fun_commute_product_fold
tff(fact_6257_Max_Oeq__fold_H,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] : aa(set(B),B,lattic643756798349783984er_Max(B),A4) = aa(option(B),B,the2(B),finite_fold(B,option(B),aTP_Lamp_acn(B,fun(option(B),option(B))),none(B),A4)) ) ).

% Max.eq_fold'
tff(fact_6258_range__abs__Nats,axiom,
    aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_6259_same__fst__def,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),R2: fun(B,set(product_prod(C,C)))] : same_fst(B,C,Pa,R2) = aa(fun(product_prod(product_prod(B,C),product_prod(B,C)),$o),set(product_prod(product_prod(B,C),product_prod(B,C))),collect(product_prod(product_prod(B,C),product_prod(B,C))),aa(fun(product_prod(B,C),fun(product_prod(B,C),$o)),fun(product_prod(product_prod(B,C),product_prod(B,C)),$o),product_case_prod(product_prod(B,C),product_prod(B,C),$o),aa(fun(B,fun(C,fun(product_prod(B,C),$o))),fun(product_prod(B,C),fun(product_prod(B,C),$o)),product_case_prod(B,C,fun(product_prod(B,C),$o)),aa(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(product_prod(B,C),$o))),aTP_Lamp_acp(fun(B,$o),fun(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(product_prod(B,C),$o)))),Pa),R2)))) ).

% same_fst_def
tff(fact_6260_same__fst__trancl,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),R2: fun(B,set(product_prod(C,C)))] : transitive_trancl(product_prod(B,C),same_fst(B,C,Pa,R2)) = same_fst(B,C,Pa,aTP_Lamp_acq(fun(B,set(product_prod(C,C))),fun(B,set(product_prod(C,C))),R2)) ).

% same_fst_trancl
tff(fact_6261_comp__fun__commute__plus__mset,axiom,
    ! [B: $tType] : finite6289374366891150609ommute(multiset(B),multiset(B),plus_plus(multiset(B))) ).

% comp_fun_commute_plus_mset
tff(fact_6262_comp__fun__commute__insort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => finite6289374366891150609ommute(B,list(B),linorder_insort_key(B,B,aTP_Lamp_hr(B,B))) ) ).

% comp_fun_commute_insort
tff(fact_6263_Nats__cases,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [X: B] :
          ( aa(set(B),$o,member(B,X),semiring_1_Nats(B))
         => ~ ! [N2: nat] : X != aa(nat,B,semiring_1_of_nat(B),N2) ) ) ).

% Nats_cases
tff(fact_6264_Nats__induct,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [X: B,Pa: fun(B,$o)] :
          ( aa(set(B),$o,member(B,X),semiring_1_Nats(B))
         => ( ! [N2: nat] : aa(B,$o,Pa,aa(nat,B,semiring_1_of_nat(B),N2))
           => aa(B,$o,Pa,X) ) ) ) ).

% Nats_induct
tff(fact_6265_of__nat__in__Nats,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] : aa(set(B),$o,member(B,aa(nat,B,semiring_1_of_nat(B),N)),semiring_1_Nats(B)) ) ).

% of_nat_in_Nats
tff(fact_6266_Nats__0,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => aa(set(B),$o,member(B,zero_zero(B)),semiring_1_Nats(B)) ) ).

% Nats_0
tff(fact_6267_Nats__numeral,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [W: num] : aa(set(B),$o,member(B,aa(num,B,numeral_numeral(B),W)),semiring_1_Nats(B)) ) ).

% Nats_numeral
tff(fact_6268_Nats__add,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),semiring_1_Nats(B))
         => ( aa(set(B),$o,member(B,B2),semiring_1_Nats(B))
           => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),semiring_1_Nats(B)) ) ) ) ).

% Nats_add
tff(fact_6269_Nats__mult,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),semiring_1_Nats(B))
         => ( aa(set(B),$o,member(B,B2),semiring_1_Nats(B))
           => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),semiring_1_Nats(B)) ) ) ) ).

% Nats_mult
tff(fact_6270_Nats__1,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => aa(set(B),$o,member(B,one_one(B)),semiring_1_Nats(B)) ) ).

% Nats_1
tff(fact_6271_Nats__diff,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),semiring_1_Nats(B))
         => ( aa(set(B),$o,member(B,B2),semiring_1_Nats(B))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),A2)
             => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),semiring_1_Nats(B)) ) ) ) ) ).

% Nats_diff
tff(fact_6272_fold__atLeastAtMost__nat,axiom,
    ! [B: $tType,F: fun(nat,fun(B,B)),A2: nat,B2: nat,Acc: B] :
      ( finite6289374366891150609ommute(nat,B,F)
     => ( set_fo6178422350223883121st_nat(B,F,A2,B2,Acc) = finite_fold(nat,B,F,Acc,set_or1337092689740270186AtMost(nat,A2,B2)) ) ) ).

% fold_atLeastAtMost_nat
tff(fact_6273_Nats__subset__Ints,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),semiring_1_Nats(B)),ring_1_Ints(B)) ) ).

% Nats_subset_Ints
tff(fact_6274_comp__fun__commute__Pow__fold,axiom,
    ! [B: $tType] : finite6289374366891150609ommute(B,set(set(B)),aTP_Lamp_acr(B,fun(set(set(B)),set(set(B))))) ).

% comp_fun_commute_Pow_fold
tff(fact_6275_Nats__def,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( semiring_1_Nats(B) = aa(set(nat),set(B),image2(nat,B,semiring_1_of_nat(B)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_6276_Nats__altdef2,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ( semiring_1_Nats(B) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_acs(B,$o)) ) ) ).

% Nats_altdef2
tff(fact_6277_Nats__altdef1,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ( semiring_1_Nats(B) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_act(B,$o)) ) ) ).

% Nats_altdef1
tff(fact_6278_Sup__fin_Oeq__fold_H,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B)] : aa(set(B),B,lattic5882676163264333800up_fin(B),A4) = aa(option(B),B,the2(B),finite_fold(B,option(B),aTP_Lamp_acu(B,fun(option(B),option(B))),none(B),A4)) ) ).

% Sup_fin.eq_fold'
tff(fact_6279_Inf__fin_Oeq__fold_H,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B)] : aa(set(B),B,lattic7752659483105999362nf_fin(B),A4) = aa(option(B),B,the2(B),finite_fold(B,option(B),aTP_Lamp_acv(B,fun(option(B),option(B))),none(B),A4)) ) ).

% Inf_fin.eq_fold'
tff(fact_6280_Sup__fin_Osingleton,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [X: B] : aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = X ) ).

% Sup_fin.singleton
tff(fact_6281_Inf__fin_Osingleton,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [X: B] : aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = X ) ).

% Inf_fin.singleton
tff(fact_6282_inf__Sup__absorb,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),A2),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) = A2 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_6283_sup__Inf__absorb,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)),A2) = A2 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_6284_Inf__fin_Oinsert,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_6285_Sup__fin_Oinsert,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_6286_Inf__fin_Oin__idem,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) = aa(set(B),B,lattic7752659483105999362nf_fin(B),A4) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_6287_Sup__fin_Oin__idem,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) = aa(set(B),B,lattic5882676163264333800up_fin(B),A4) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_6288_Sup__fin__Max,axiom,
    ! [B: $tType] :
      ( ( semilattice_sup(B)
        & linorder(B) )
     => ( lattic5882676163264333800up_fin(B) = lattic643756798349783984er_Max(B) ) ) ).

% Sup_fin_Max
tff(fact_6289_Inf__fin__le__Sup__fin,axiom,
    ! [B: $tType] :
      ( lattice(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_6290_Inf__fin_OcoboundedI,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_6291_Sup__fin_OcoboundedI,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_6292_Sup__fin_Obounded__iff,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),X)
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),X) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6293_Inf__fin_Obounded__iff,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4))
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),X3) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6294_Sup__fin_OboundedI,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [A3: B] :
                  ( aa(set(B),$o,member(B,A3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),X) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6295_Sup__fin_OboundedE,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),X)
             => ! [A8: B] :
                  ( aa(set(B),$o,member(B,A8),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A8),X) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6296_Inf__fin_OboundedI,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [A3: B] :
                  ( aa(set(B),$o,member(B,A3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A3) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6297_Inf__fin_OboundedE,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4))
             => ! [A8: B] :
                  ( aa(set(B),$o,member(B,A8),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A8) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6298_Sup__fin__Sup,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic5882676163264333800up_fin(B),A4) = aa(set(B),B,complete_Sup_Sup(B),A4) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_6299_Inf__fin__Inf,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),A4) = aa(set(B),B,complete_Inf_Inf(B),A4) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_6300_Inf__fin_Oinfinite,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),A4) = aa(option(B),B,the2(B),none(B)) ) ) ) ).

% Inf_fin.infinite
tff(fact_6301_Sup__fin_Oinfinite,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic5882676163264333800up_fin(B),A4) = aa(option(B),B,the2(B),none(B)) ) ) ) ).

% Sup_fin.infinite
tff(fact_6302_Inf__fin_Osubset__imp,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),B4)),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_6303_Sup__fin_Osubset__imp,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),aa(set(B),B,lattic5882676163264333800up_fin(B),B4)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_6304_Inf__fin_Ohom__commute,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [H: fun(B,B),N5: set(B)] :
          ( ! [X2: B,Y2: B] : aa(B,B,H,aa(B,B,aa(B,fun(B,B),inf_inf(B),X2),Y2)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(B,B,H,X2)),aa(B,B,H,Y2))
         => ( aa(set(B),$o,finite_finite2(B),N5)
           => ( ( N5 != bot_bot(set(B)) )
             => ( aa(B,B,H,aa(set(B),B,lattic7752659483105999362nf_fin(B),N5)) = aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),image2(B,B,H),N5)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_6305_Sup__fin_Ohom__commute,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [H: fun(B,B),N5: set(B)] :
          ( ! [X2: B,Y2: B] : aa(B,B,H,aa(B,B,aa(B,fun(B,B),sup_sup(B),X2),Y2)) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(B,B,H,X2)),aa(B,B,H,Y2))
         => ( aa(set(B),$o,finite_finite2(B),N5)
           => ( ( N5 != bot_bot(set(B)) )
             => ( aa(B,B,H,aa(set(B),B,lattic5882676163264333800up_fin(B),N5)) = aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),image2(B,B,H),N5)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_6306_Inf__fin_Osubset,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( B4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),B4)),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) = aa(set(B),B,lattic7752659483105999362nf_fin(B),A4) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_6307_Sup__fin_Osubset,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( B4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,lattic5882676163264333800up_fin(B),B4)),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) = aa(set(B),B,lattic5882676163264333800up_fin(B),A4) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_6308_Inf__fin_Oclosed,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [X2: B,Y2: B] : aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),inf_inf(B),X2),Y2)),aa(set(B),set(B),insert(B,X2),aa(set(B),set(B),insert(B,Y2),bot_bot(set(B)))))
             => aa(set(B),$o,member(B,aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)),A4) ) ) ) ) ).

% Inf_fin.closed
tff(fact_6309_Inf__fin_Oinsert__not__elem,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ aa(set(B),$o,member(B,X),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_6310_Sup__fin_Oinsert__not__elem,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ aa(set(B),$o,member(B,X),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => ( aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_6311_Sup__fin_Oclosed,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [X2: B,Y2: B] : aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),sup_sup(B),X2),Y2)),aa(set(B),set(B),insert(B,X2),aa(set(B),set(B),insert(B,Y2),bot_bot(set(B)))))
             => aa(set(B),$o,member(B,aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),A4) ) ) ) ) ).

% Sup_fin.closed
tff(fact_6312_Inf__fin_Ounion,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => ( ( B4 != bot_bot(set(B)) )
               => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)),aa(set(B),B,lattic7752659483105999362nf_fin(B),B4)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_6313_Sup__fin_Ounion,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => ( ( B4 != bot_bot(set(B)) )
               => ( aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),aa(set(B),B,lattic5882676163264333800up_fin(B),B4)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_6314_Inf__fin_Oeq__fold,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),insert(B,X),A4)) = finite_fold(B,B,inf_inf(B),X,A4) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_6315_Sup__fin_Oeq__fold,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),insert(B,X),A4)) = finite_fold(B,B,sup_sup(B),X,A4) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_6316_inf__Sup2__distrib,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => ( ( B4 != bot_bot(set(B)) )
               => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)),aa(set(B),B,lattic5882676163264333800up_fin(B),B4)) = aa(set(B),B,lattic5882676163264333800up_fin(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_acw(set(B),fun(set(B),fun(B,$o)),A4),B4))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_6317_inf__Sup1__distrib,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,lattic5882676163264333800up_fin(B),A4)) = aa(set(B),B,lattic5882676163264333800up_fin(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_acx(set(B),fun(B,fun(B,$o)),A4),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_6318_sup__Inf2__distrib,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => ( ( B4 != bot_bot(set(B)) )
               => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)),aa(set(B),B,lattic7752659483105999362nf_fin(B),B4)) = aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_acy(set(B),fun(set(B),fun(B,$o)),A4),B4))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_6319_sup__Inf1__distrib,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),A4)) = aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_acz(set(B),fun(B,fun(B,$o)),A4),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_6320_Inf__fin_Oremove,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),A4) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_6321_Inf__fin_Oinsert__remove,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),insert(B,X),A4)) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,lattic7752659483105999362nf_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_6322_Sup__fin_Oinsert__remove,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),insert(B,X),A4)) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_6323_Sup__fin_Oremove,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),B,lattic5882676163264333800up_fin(B),A4) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),sup_sup(B),X),aa(set(B),B,lattic5882676163264333800up_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_6324_Pow__fold,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( pow2(B,A4) = finite_fold(B,set(set(B)),aTP_Lamp_acr(B,fun(set(set(B)),set(set(B)))),aa(set(set(B)),set(set(B)),insert(set(B),bot_bot(set(B))),bot_bot(set(set(B)))),A4) ) ) ).

% Pow_fold
tff(fact_6325_Min_Oeq__fold_H,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] : aa(set(B),B,lattic643756798350308766er_Min(B),A4) = aa(option(B),B,the2(B),finite_fold(B,option(B),aTP_Lamp_ada(B,fun(option(B),option(B))),none(B),A4)) ) ).

% Min.eq_fold'
tff(fact_6326_Min__singleton,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B] : aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = X ) ).

% Min_singleton
tff(fact_6327_Min_Obounded__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4))
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),X3) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6328_Min__gr__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4))
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),X3) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6329_Min__const,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(C)
     => ! [A4: set(B),C2: C] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(C),C,lattic643756798350308766er_Min(C),aa(set(B),set(C),image2(B,C,aTP_Lamp_ok(C,fun(B,C),C2)),A4)) = C2 ) ) ) ) ).

% Min_const
tff(fact_6330_Min__insert,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),ord_min(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4)) ) ) ) ) ).

% Min_insert
tff(fact_6331_minus__Min__eq__Max,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [S: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ( aa(B,B,uminus_uminus(B),aa(set(B),B,lattic643756798350308766er_Min(B),S)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(B),set(B),image2(B,B,uminus_uminus(B)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_6332_minus__Max__eq__Min,axiom,
    ! [B: $tType] :
      ( linord5086331880401160121up_add(B)
     => ! [S: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ( aa(B,B,uminus_uminus(B),aa(set(B),B,lattic643756798349783984er_Max(B),S)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),image2(B,B,uminus_uminus(B)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_6333_Inf__fin__Min,axiom,
    ! [B: $tType] :
      ( ( semilattice_inf(B)
        & linorder(B) )
     => ( lattic7752659483105999362nf_fin(B) = lattic643756798350308766er_Min(B) ) ) ).

% Inf_fin_Min
tff(fact_6334_Pow__Compl,axiom,
    ! [B: $tType,A4: set(B)] : pow2(B,aa(set(B),set(B),uminus_uminus(set(B)),A4)) = aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_adb(set(B),fun(set(B),$o),A4)) ).

% Pow_Compl
tff(fact_6335_Pow__def,axiom,
    ! [B: $tType,A4: set(B)] : pow2(B,A4) = aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_gc(set(B),fun(set(B),$o),A4)) ).

% Pow_def
tff(fact_6336_Min_OcoboundedI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,A2),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798350308766er_Min(B),A4)),A2) ) ) ) ).

% Min.coboundedI
tff(fact_6337_Min__eqI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [Y2: B] :
                ( aa(set(B),$o,member(B,Y2),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2) )
           => ( aa(set(B),$o,member(B,X),A4)
             => ( aa(set(B),B,lattic643756798350308766er_Min(B),A4) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_6338_Min__le,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798350308766er_Min(B),A4)),X) ) ) ) ).

% Min_le
tff(fact_6339_Min_Oin__idem,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(B,B,aa(B,fun(B,B),ord_min(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4)) = aa(set(B),B,lattic643756798350308766er_Min(B),A4) ) ) ) ) ).

% Min.in_idem
tff(fact_6340_Min__in,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => aa(set(B),$o,member(B,aa(set(B),B,lattic643756798350308766er_Min(B),A4)),A4) ) ) ) ).

% Min_in
tff(fact_6341_Pow__INT__eq,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C)] : pow2(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4))) = aa(set(set(set(B))),set(set(B)),complete_Inf_Inf(set(set(B))),aa(set(C),set(set(set(B))),image2(C,set(set(B)),aTP_Lamp_adc(fun(C,set(B)),fun(C,set(set(B))),B4)),A4)) ).

% Pow_INT_eq
tff(fact_6342_Min__eq__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),Ma: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ( aa(set(B),B,lattic643756798350308766er_Min(B),A4) = Ma )
            <=> ( aa(set(B),$o,member(B,Ma),A4)
                & ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),X3) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6343_Min__le__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798350308766er_Min(B),A4)),X)
            <=> ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),X) ) ) ) ) ) ).

% Min_le_iff
tff(fact_6344_eq__Min__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),Ma: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ( Ma = aa(set(B),B,lattic643756798350308766er_Min(B),A4) )
            <=> ( aa(set(B),$o,member(B,Ma),A4)
                & ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),X3) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6345_Min_OboundedE,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4))
             => ! [A8: B] :
                  ( aa(set(B),$o,member(B,A8),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A8) ) ) ) ) ) ).

% Min.boundedE
tff(fact_6346_Min_OboundedI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [A3: B] :
                  ( aa(set(B),$o,member(B,A3),A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),A3) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4)) ) ) ) ) ).

% Min.boundedI
tff(fact_6347_Min__less__iff,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,lattic643756798350308766er_Min(B),A4)),X)
            <=> ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),X) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6348_Min__insert2,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [B3: B] :
                ( aa(set(B),$o,member(B,B3),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B3) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),insert(B,A2),A4)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_6349_Min__Inf,axiom,
    ! [B: $tType] :
      ( comple5582772986160207858norder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),A4) = aa(set(B),B,complete_Inf_Inf(B),A4) ) ) ) ) ).

% Min_Inf
tff(fact_6350_Min_Oinfinite,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic643756798350308766er_Min(B),A4) = aa(option(B),B,the2(B),none(B)) ) ) ) ).

% Min.infinite
tff(fact_6351_UN__Pow__subset,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C)] : aa(set(set(B)),$o,aa(set(set(B)),fun(set(set(B)),$o),ord_less_eq(set(set(B))),aa(set(set(set(B))),set(set(B)),complete_Sup_Sup(set(set(B))),aa(set(C),set(set(set(B))),image2(C,set(set(B)),aTP_Lamp_adc(fun(C,set(B)),fun(C,set(set(B))),B4)),A4))),pow2(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)))) ).

% UN_Pow_subset
tff(fact_6352_Min_Osubset__imp,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798350308766er_Min(B),B4)),aa(set(B),B,lattic643756798350308766er_Min(B),A4)) ) ) ) ) ).

% Min.subset_imp
tff(fact_6353_Min__antimono,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [M: set(B),N5: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),M),N5)
         => ( ( M != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),N5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,lattic643756798350308766er_Min(B),N5)),aa(set(B),B,lattic643756798350308766er_Min(B),M)) ) ) ) ) ).

% Min_antimono
tff(fact_6354_hom__Min__commute,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [H: fun(B,B),N5: set(B)] :
          ( ! [X2: B,Y2: B] : aa(B,B,H,aa(B,B,aa(B,fun(B,B),ord_min(B),X2),Y2)) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(B,B,H,X2)),aa(B,B,H,Y2))
         => ( aa(set(B),$o,finite_finite2(B),N5)
           => ( ( N5 != bot_bot(set(B)) )
             => ( aa(B,B,H,aa(set(B),B,lattic643756798350308766er_Min(B),N5)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),image2(B,B,H),N5)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_6355_Min_Osubset,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( B4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
             => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(set(B),B,lattic643756798350308766er_Min(B),B4)),aa(set(B),B,lattic643756798350308766er_Min(B),A4)) = aa(set(B),B,lattic643756798350308766er_Min(B),A4) ) ) ) ) ) ).

% Min.subset
tff(fact_6356_Min_Oclosed,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( ! [X2: B,Y2: B] : aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),ord_min(B),X2),Y2)),aa(set(B),set(B),insert(B,X2),aa(set(B),set(B),insert(B,Y2),bot_bot(set(B)))))
             => aa(set(B),$o,member(B,aa(set(B),B,lattic643756798350308766er_Min(B),A4)),A4) ) ) ) ) ).

% Min.closed
tff(fact_6357_Min_Oinsert__not__elem,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ aa(set(B),$o,member(B,X),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),insert(B,X),A4)) = aa(B,B,aa(B,fun(B,B),ord_min(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),A4)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_6358_Min_Ounion,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),B4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( A4 != bot_bot(set(B)) )
           => ( aa(set(B),$o,finite_finite2(B),B4)
             => ( ( B4 != bot_bot(set(B)) )
               => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),ord_min(B),aa(set(B),B,lattic643756798350308766er_Min(B),A4)),aa(set(B),B,lattic643756798350308766er_Min(B),B4)) ) ) ) ) ) ) ).

% Min.union
tff(fact_6359_Min_Oeq__fold,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),insert(B,X),A4)) = finite_fold(B,B,ord_min(B),X,A4) ) ) ) ).

% Min.eq_fold
tff(fact_6360_Min__add__commute,axiom,
    ! [B: $tType,C: $tType] :
      ( linord4140545234300271783up_add(C)
     => ! [S: set(B),F: fun(B,C),K: C] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(C),C,lattic643756798350308766er_Min(C),aa(set(B),set(C),image2(B,C,aa(C,fun(B,C),aTP_Lamp_aak(fun(B,C),fun(C,fun(B,C)),F),K)),S)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(set(C),C,lattic643756798350308766er_Min(C),aa(set(B),set(C),image2(B,C,F),S))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_6361_binomial__def,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(N),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_add(nat,fun(nat,fun(set(nat),$o)),N),K))) ).

% binomial_def
tff(fact_6362_Min_Oinsert__remove,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),insert(B,X),A4)) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),ord_min(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ).

% Min.insert_remove
tff(fact_6363_Min_Oremove,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B),X: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),A4) = $ite(aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) = bot_bot(set(B)),X,aa(B,B,aa(B,fun(B,B),ord_min(B),X),aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ) ) ) ) ).

% Min.remove
tff(fact_6364_arg__min__SOME__Min,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [S: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( lattic7623131987881927897min_on(B,C,F,S) = fChoice(B,aa(fun(B,C),fun(B,$o),aTP_Lamp_ade(set(B),fun(fun(B,C),fun(B,$o)),S),F)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_6365_sorted__find__Min,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Pa: fun(B,$o)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => ( ? [X4: B] :
                ( aa(set(B),$o,member(B,X4),aa(list(B),set(B),set2(B),Xs))
                & aa(B,$o,Pa,X4) )
           => ( find(B,Pa,Xs) = aa(B,option(B),some(B),aa(set(B),B,lattic643756798350308766er_Min(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_adf(list(B),fun(fun(B,$o),fun(B,$o)),Xs),Pa)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_6366_Pow__set_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      pow2(B,aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))) = $let(
        a3: set(set(B)),
        a3:= pow2(B,aa(list(B),set(B),set2(B),Xs)),
        aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),a3),aa(set(set(B)),set(set(B)),image2(set(B),set(B),insert(B,X)),a3)) ) ).

% Pow_set(2)
tff(fact_6367_card__Min__le__sum,axiom,
    ! [B: $tType,A4: set(B),F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(B),set(nat),image2(B,nat,F),A4)))),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A4)) ) ).

% card_Min_le_sum
tff(fact_6368_dual__Max,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ( lattices_Max(B,aTP_Lamp_kh(B,fun(B,$o))) = lattic643756798350308766er_Min(B) ) ) ).

% dual_Max
tff(fact_6369_rtrancl__finite__eq__relpow,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => ( transitive_rtrancl(B,R2) = aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_aac(set(product_prod(B,B)),fun(nat,$o),R2)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_6370_rtrancl__sub__insert__rtrancl,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),X: product_prod(B,B)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),transitive_rtrancl(B,R2)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),X),R2))) ).

% rtrancl_sub_insert_rtrancl
tff(fact_6371_in__rtrancl__insert,axiom,
    ! [B: $tType,X: product_prod(B,B),R2: set(product_prod(B,B)),R: product_prod(B,B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X),transitive_rtrancl(B,R2))
     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),R),R2))) ) ).

% in_rtrancl_insert
tff(fact_6372_linorder_OMax_Ocong,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o))] : lattices_Max(B,Less_eq) = lattices_Max(B,Less_eq) ).

% linorder.Max.cong
tff(fact_6373_rtrancl__mono__rightI,axiom,
    ! [B: $tType,S: set(product_prod(B,B)),S5: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),S),S5)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),S),transitive_rtrancl(B,S5)) ) ).

% rtrancl_mono_rightI
tff(fact_6374_rtrancl__mono__mp,axiom,
    ! [B: $tType,U4: set(product_prod(B,B)),V: set(product_prod(B,B)),X: product_prod(B,B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),U4),V)
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X),transitive_rtrancl(B,U4))
       => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X),transitive_rtrancl(B,V)) ) ) ).

% rtrancl_mono_mp
tff(fact_6375_r__le__rtrancl,axiom,
    ! [B: $tType,S: set(product_prod(B,B))] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),S),transitive_rtrancl(B,S)) ).

% r_le_rtrancl
tff(fact_6376_converse__rtranclE_H,axiom,
    ! [B: $tType,U: B,V2: B,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_rtrancl(B,R2))
     => ( ( U != V2 )
       => ~ ! [Vh: B] :
              ( ( U != Vh )
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),Vh)),R2)
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Vh),V2)),transitive_rtrancl(B,R2)) ) ) ) ) ).

% converse_rtranclE'
tff(fact_6377_trancl__union__outside,axiom,
    ! [B: $tType,V2: B,W: B,E6: set(product_prod(B,B)),U4: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),E6),U4)))
     => ( ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_trancl(B,E6))
       => ? [X2: B,Y2: B] :
            ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),X2)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),E6),U4)))
            & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X2),Y2)),U4)
            & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),W)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),E6),U4))) ) ) ) ).

% trancl_union_outside
tff(fact_6378_trancl__over__edgeE,axiom,
    ! [B: $tType,U: B,W: B,V1: B,V22: B,E6: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),W)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V1),V22)),E6)))
     => ( ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),W)),transitive_trancl(B,E6))
       => ~ ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V1)),transitive_rtrancl(B,E6))
           => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V22),W)),transitive_rtrancl(B,E6)) ) ) ) ).

% trancl_over_edgeE
tff(fact_6379_rtrancl__listrel1__eq__len,axiom,
    ! [B: $tType,X: list(B),Y: list(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),X),Y)),transitive_rtrancl(list(B),listrel1(B,R)))
     => ( aa(list(B),nat,size_size(list(B)),X) = aa(list(B),nat,size_size(list(B)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_6380_trancl__subset__Sigma__aux,axiom,
    ! [B: $tType,A2: B,B2: B,R: set(product_prod(B,B)),A4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_rtrancl(B,R))
     => ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4)))
       => ( ( A2 = B2 )
          | aa(set(B),$o,member(B,A2),A4) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_6381_Restr__rtrancl__mono,axiom,
    ! [B: $tType,V2: B,W: B,E6: set(product_prod(B,B)),U4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),E6),product_Sigma(B,B,U4,aTP_Lamp_zv(set(B),fun(B,set(B)),U4)))))
     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_rtrancl(B,E6)) ) ).

% Restr_rtrancl_mono
tff(fact_6382_rtrancl__mapI,axiom,
    ! [C: $tType,B: $tType,A2: B,B2: B,E6: set(product_prod(B,B)),F: fun(B,C)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_rtrancl(B,E6))
     => aa(set(product_prod(C,C)),$o,member(product_prod(C,C),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),aa(B,C,F,A2)),aa(B,C,F,B2))),transitive_rtrancl(C,aa(set(product_prod(B,B)),set(product_prod(C,C)),image2(product_prod(B,B),product_prod(C,C),pairself(B,C,F)),E6))) ) ).

% rtrancl_mapI
tff(fact_6383_rtrancl__last__touch,axiom,
    ! [B: $tType,Q2: B,Q5: B,R2: set(product_prod(B,B)),S: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Q5)),transitive_rtrancl(B,R2))
     => ( aa(set(B),$o,member(B,Q2),S)
       => ~ ! [Qt: B] :
              ( aa(set(B),$o,member(B,Qt),S)
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Qt)),transitive_rtrancl(B,R2))
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Qt),Q5)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_zv(set(B),fun(B,set(B)),S))))) ) ) ) ) ).

% rtrancl_last_touch
tff(fact_6384_rtrancl__last__visit_H,axiom,
    ! [B: $tType,Q2: B,Q5: B,R2: set(product_prod(B,B)),S: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Q5)),transitive_rtrancl(B,R2))
     => ( ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Q5)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_zv(set(B),fun(B,set(B)),S)))))
       => ~ ! [Qt: B] :
              ( aa(set(B),$o,member(B,Qt),S)
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Qt)),transitive_rtrancl(B,R2))
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Qt),Q5)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_zv(set(B),fun(B,set(B)),S))))) ) ) ) ) ).

% rtrancl_last_visit'
tff(fact_6385_rtrancl__insert,axiom,
    ! [B: $tType,A2: B,B2: B,R: set(product_prod(B,B))] : transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),R)) = aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),transitive_rtrancl(B,R)),aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aa(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o))),aTP_Lamp_adg(B,fun(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o)))),A2),B2),R)))) ).

% rtrancl_insert
tff(fact_6386_rtrancl__imp__UN__relpow,axiom,
    ! [B: $tType,P2: product_prod(B,B),R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),P2),transitive_rtrancl(B,R2))
     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),P2),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),top_top(set(nat))))) ) ).

% rtrancl_imp_UN_relpow
tff(fact_6387_rtrancl__is__UN__relpow,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] : transitive_rtrancl(B,R2) = aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(nat),set(set(product_prod(B,B))),image2(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),R2)),top_top(set(nat)))) ).

% rtrancl_is_UN_relpow
tff(fact_6388_rtrancl__last__visit,axiom,
    ! [B: $tType,Q2: B,Q5: B,R2: set(product_prod(B,B)),S: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Q5)),transitive_rtrancl(B,R2))
     => ( ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Q5)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_zv(set(B),fun(B,set(B)),S)))))
       => ~ ! [Qt: B] :
              ( aa(set(B),$o,member(B,Qt),S)
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),Qt)),transitive_trancl(B,R2))
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Qt),Q5)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_zv(set(B),fun(B,set(B)),S))))) ) ) ) ) ).

% rtrancl_last_visit
tff(fact_6389_trancl__insert,axiom,
    ! [B: $tType,Y: B,X: B,R: set(product_prod(B,B))] : transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y),X)),R)) = aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),transitive_trancl(B,R)),aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aa(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o))),aTP_Lamp_adg(B,fun(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o)))),Y),X),R)))) ).

% trancl_insert
tff(fact_6390_trancl__multi__insert2,axiom,
    ! [B: $tType,A2: B,B2: B,R: set(product_prod(B,B)),Ma: B,X5: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),R),product_Sigma(B,B,aa(set(B),set(B),insert(B,Ma),bot_bot(set(B))),aTP_Lamp_zv(set(B),fun(B,set(B)),X5)))))
     => ( ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_trancl(B,R))
       => ~ ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),X5)
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),Ma)),transitive_rtrancl(B,R))
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X2),B2)),transitive_rtrancl(B,R)) ) ) ) ) ).

% trancl_multi_insert2
tff(fact_6391_trancl__multi__insert,axiom,
    ! [B: $tType,A2: B,B2: B,R: set(product_prod(B,B)),X5: set(B),Ma: B] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),sup_sup(set(product_prod(B,B))),R),product_Sigma(B,B,X5,aTP_Lamp_adh(B,fun(B,set(B)),Ma)))))
     => ( ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_trancl(B,R))
       => ~ ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),X5)
             => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),X2)),transitive_rtrancl(B,R))
               => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Ma),B2)),transitive_rtrancl(B,R)) ) ) ) ) ).

% trancl_multi_insert
tff(fact_6392_rtrancl__last__visit__node,axiom,
    ! [B: $tType,S2: B,S4: B,R2: set(product_prod(B,B)),Sh: B] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),S2),S4)),transitive_rtrancl(B,R2))
     => ( ( ( S2 != Sh )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),S2),S4)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_adi(B,fun(B,set(B)),Sh))))) )
        | ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),S2),Sh)),transitive_rtrancl(B,R2))
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Sh),S4)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R2),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_adi(B,fun(B,set(B)),Sh))))) ) ) ) ).

% rtrancl_last_visit_node
tff(fact_6393_finite__psubset__def,axiom,
    ! [B: $tType] : finite_psubset(B) = aa(fun(product_prod(set(B),set(B)),$o),set(product_prod(set(B),set(B))),collect(product_prod(set(B),set(B))),aa(fun(set(B),fun(set(B),$o)),fun(product_prod(set(B),set(B)),$o),product_case_prod(set(B),set(B),$o),aTP_Lamp_adj(set(B),fun(set(B),$o)))) ).

% finite_psubset_def
tff(fact_6394_Gr__incl,axiom,
    ! [B: $tType,C: $tType,A4: set(B),F: fun(B,C),B4: set(C)] :
      ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),bNF_Gr(B,C,A4,F)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
    <=> aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F),A4)),B4) ) ).

% Gr_incl
tff(fact_6395_GrD2,axiom,
    ! [B: $tType,C: $tType,X: B,Fx: C,A4: set(B),F: fun(B,C)] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Fx)),bNF_Gr(B,C,A4,F))
     => ( aa(B,C,F,X) = Fx ) ) ).

% GrD2
tff(fact_6396_GrD1,axiom,
    ! [C: $tType,B: $tType,X: B,Fx: C,A4: set(B),F: fun(B,C)] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Fx)),bNF_Gr(B,C,A4,F))
     => aa(set(B),$o,member(B,X),A4) ) ).

% GrD1
tff(fact_6397_Id__on__Gr,axiom,
    ! [B: $tType,A4: set(B)] : id_on(B,A4) = bNF_Gr(B,B,A4,id(B)) ).

% Id_on_Gr
tff(fact_6398_Gr__def,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C)] : bNF_Gr(B,C,A4,F) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,C),fun(product_prod(B,C),$o),aTP_Lamp_adk(set(B),fun(fun(B,C),fun(product_prod(B,C),$o)),A4),F)) ).

% Gr_def
tff(fact_6399_set__list__bind,axiom,
    ! [B: $tType,C: $tType,Xs: list(C),F: fun(C,list(B))] : aa(list(B),set(B),set2(B),bind(C,B,Xs,F)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_adl(fun(C,list(B)),fun(C,set(B)),F)),aa(list(C),set(C),set2(C),Xs))) ).

% set_list_bind
tff(fact_6400_pred__nat__def,axiom,
    pred_nat = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_adm(nat,fun(nat,$o)))) ).

% pred_nat_def
tff(fact_6401_pred__nat__trancl__eq__le,axiom,
    ! [Ma: nat,N: nat] :
      ( aa(set(product_prod(nat,nat)),$o,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Ma),N)),transitive_rtrancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ).

% pred_nat_trancl_eq_le
tff(fact_6402_rtrancl__restrictI,axiom,
    ! [B: $tType,U: B,V2: B,E6: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_rtrancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),minus_minus(set(product_prod(B,B))),E6),product_Sigma(B,B,top_top(set(B)),aTP_Lamp_zv(set(B),fun(B,set(B)),R2)))))
     => ( ~ aa(set(B),$o,member(B,U),R2)
       => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_rtrancl(B,rel_restrict(B,E6,R2))) ) ) ).

% rtrancl_restrictI
tff(fact_6403_image__split__eq__Sigma,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(D,B),G: fun(D,C),A4: set(D)] : aa(set(D),set(product_prod(B,C)),image2(D,product_prod(B,C),aa(fun(D,C),fun(D,product_prod(B,C)),aTP_Lamp_adn(fun(D,B),fun(fun(D,C),fun(D,product_prod(B,C))),F),G)),A4) = product_Sigma(B,C,aa(set(D),set(B),image2(D,B,F),A4),aa(set(D),fun(B,set(C)),aa(fun(D,C),fun(set(D),fun(B,set(C))),aTP_Lamp_ado(fun(D,B),fun(fun(D,C),fun(set(D),fun(B,set(C)))),F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_6404_vimage__Collect__eq,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),Pa: fun(C,$o)] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(fun(C,$o),set(C),collect(C),Pa)) = aa(fun(B,$o),set(B),collect(B),aa(fun(C,$o),fun(B,$o),aTP_Lamp_adp(fun(B,C),fun(fun(C,$o),fun(B,$o)),F),Pa)) ).

% vimage_Collect_eq
tff(fact_6405_vimage__ident,axiom,
    ! [B: $tType,Y4: set(B)] : aa(set(B),set(B),aa(fun(B,B),fun(set(B),set(B)),vimage(B,B),aTP_Lamp_il(B,B)),Y4) = Y4 ).

% vimage_ident
tff(fact_6406_rel__restrict__empty,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] : rel_restrict(B,R2,bot_bot(set(B))) = R2 ).

% rel_restrict_empty
tff(fact_6407_vimage__const,axiom,
    ! [C: $tType,B: $tType,C2: C,A4: set(C)] :
      aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),aTP_Lamp_iu(C,fun(B,C),C2)),A4) = $ite(aa(set(C),$o,member(C,C2),A4),top_top(set(B)),bot_bot(set(B))) ).

% vimage_const
tff(fact_6408_vimage__if,axiom,
    ! [C: $tType,B: $tType,B4: set(B),C2: C,D2: C,A4: set(C)] :
      aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),aa(C,fun(B,C),aa(C,fun(C,fun(B,C)),aTP_Lamp_adq(set(B),fun(C,fun(C,fun(B,C))),B4),C2),D2)),A4) = $ite(
        aa(set(C),$o,member(C,C2),A4),
        $ite(aa(set(C),$o,member(C,D2),A4),top_top(set(B)),B4),
        $ite(aa(set(C),$o,member(C,D2),A4),aa(set(B),set(B),uminus_uminus(set(B)),B4),bot_bot(set(B))) ) ).

% vimage_if
tff(fact_6409_vimage__Union,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),A4: set(set(C))] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),A4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(C)),set(set(B)),image2(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F)),A4)) ).

% vimage_Union
tff(fact_6410_vimage__INT,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(B,C),B4: fun(D,set(C)),A4: set(D)] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_adr(fun(B,C),fun(fun(D,set(C)),fun(D,set(B))),F),B4)),A4)) ).

% vimage_INT
tff(fact_6411_vimage__UN,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(B,C),B4: fun(D,set(C)),A4: set(D)] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_adr(fun(B,C),fun(fun(D,set(C)),fun(D,set(B))),F),B4)),A4)) ).

% vimage_UN
tff(fact_6412_rel__restrict__union,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),A4: set(B),B4: set(B)] : rel_restrict(B,R2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = rel_restrict(B,rel_restrict(B,R2,A4),B4) ).

% rel_restrict_union
tff(fact_6413_rel__restrict__notR_I2_J,axiom,
    ! [B: $tType,X: B,Y: B,A4: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),rel_restrict(B,A4,R2))
     => ~ aa(set(B),$o,member(B,Y),R2) ) ).

% rel_restrict_notR(2)
tff(fact_6414_rel__restrict__notR_I1_J,axiom,
    ! [B: $tType,X: B,Y: B,A4: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),rel_restrict(B,A4,R2))
     => ~ aa(set(B),$o,member(B,X),R2) ) ).

% rel_restrict_notR(1)
tff(fact_6415_rel__restrictI,axiom,
    ! [B: $tType,X: B,R2: set(B),Y: B,E6: set(product_prod(B,B))] :
      ( ~ aa(set(B),$o,member(B,X),R2)
     => ( ~ aa(set(B),$o,member(B,Y),R2)
       => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),E6)
         => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),rel_restrict(B,E6,R2)) ) ) ) ).

% rel_restrictI
tff(fact_6416_rel__restrict__lift,axiom,
    ! [B: $tType,X: B,Y: B,E6: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),rel_restrict(B,E6,R2))
     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),E6) ) ).

% rel_restrict_lift
tff(fact_6417_rel__restrict__sub,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),A4: set(B)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),rel_restrict(B,R2,A4)),R2) ).

% rel_restrict_sub
tff(fact_6418_rel__restrict__mono,axiom,
    ! [B: $tType,A4: set(product_prod(B,B)),B4: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),A4),B4)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),rel_restrict(B,A4,R2)),rel_restrict(B,B4,R2)) ) ).

% rel_restrict_mono
tff(fact_6419_vimage__def,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),B4: set(C)] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),B4) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_ads(fun(B,C),fun(set(C),fun(B,$o)),F),B4)) ).

% vimage_def
tff(fact_6420_finite__rel__restrict,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),A4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),rel_restrict(B,R2,A4)) ) ).

% finite_rel_restrict
tff(fact_6421_set_Oidentity,axiom,
    ! [B: $tType] : aa(fun(B,B),fun(set(B),set(B)),vimage(B,B),aTP_Lamp_il(B,B)) = id(set(B)) ).

% set.identity
tff(fact_6422_vimage__Times,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(B,product_prod(C,D)),A4: set(C),B4: set(D)] : aa(set(product_prod(C,D)),set(B),aa(fun(B,product_prod(C,D)),fun(set(product_prod(C,D)),set(B)),vimage(B,product_prod(C,D)),F),product_Sigma(C,D,A4,aTP_Lamp_wu(set(D),fun(C,set(D)),B4))) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),aa(fun(B,product_prod(C,D)),fun(B,C),aa(fun(product_prod(C,D),C),fun(fun(B,product_prod(C,D)),fun(B,C)),comp(product_prod(C,D),C,B),product_fst(C,D)),F)),A4)),aa(set(D),set(B),aa(fun(B,D),fun(set(D),set(B)),vimage(B,D),aa(fun(B,product_prod(C,D)),fun(B,D),aa(fun(product_prod(C,D),D),fun(fun(B,product_prod(C,D)),fun(B,D)),comp(product_prod(C,D),D,B),product_snd(C,D)),F)),B4)) ).

% vimage_Times
tff(fact_6423_rel__restrict__trancl__mem,axiom,
    ! [B: $tType,A2: B,B2: B,A4: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),transitive_trancl(B,rel_restrict(B,A4,R2)))
     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)),rel_restrict(B,transitive_trancl(B,A4),R2)) ) ).

% rel_restrict_trancl_mem
tff(fact_6424_rel__restrict__trancl__notR_I1_J,axiom,
    ! [B: $tType,V2: B,W: B,E6: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_trancl(B,rel_restrict(B,E6,R2)))
     => ~ aa(set(B),$o,member(B,V2),R2) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_6425_rel__restrict__trancl__notR_I2_J,axiom,
    ! [B: $tType,V2: B,W: B,E6: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V2),W)),transitive_trancl(B,rel_restrict(B,E6,R2)))
     => ~ aa(set(B),$o,member(B,W),R2) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_6426_rel__restrict__mono2,axiom,
    ! [B: $tType,R2: set(B),S: set(B),A4: set(product_prod(B,B))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),R2),S)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),rel_restrict(B,A4,S)),rel_restrict(B,A4,R2)) ) ).

% rel_restrict_mono2
tff(fact_6427_rel__restrict__trancl__sub,axiom,
    ! [B: $tType,A4: set(product_prod(B,B)),R2: set(B)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),transitive_trancl(B,rel_restrict(B,A4,R2))),rel_restrict(B,transitive_trancl(B,A4),R2)) ).

% rel_restrict_trancl_sub
tff(fact_6428_vimage__Suc__insert__0,axiom,
    ! [A4: set(nat)] : aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),vimage(nat,nat),suc),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),A4)) = aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),vimage(nat,nat),suc),A4) ).

% vimage_Suc_insert_0
tff(fact_6429_rel__restrict__def,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),A4: set(B)] : rel_restrict(B,R2,A4) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(set(B),fun(B,fun(B,$o)),aTP_Lamp_adt(set(product_prod(B,B)),fun(set(B),fun(B,fun(B,$o))),R2),A4))) ).

% rel_restrict_def
tff(fact_6430_vimage__fst,axiom,
    ! [C: $tType,B: $tType,A4: set(B)] : aa(set(B),set(product_prod(B,C)),aa(fun(product_prod(B,C),B),fun(set(B),set(product_prod(B,C))),vimage(product_prod(B,C),B),product_fst(B,C)),A4) = product_Sigma(B,C,A4,aTP_Lamp_vv(B,set(C))) ).

% vimage_fst
tff(fact_6431_vimage__snd,axiom,
    ! [B: $tType,C: $tType,A4: set(C)] : aa(set(C),set(product_prod(B,C)),aa(fun(product_prod(B,C),C),fun(set(C),set(product_prod(B,C))),vimage(product_prod(B,C),C),product_snd(B,C)),A4) = product_Sigma(B,C,top_top(set(B)),aTP_Lamp_vu(set(C),fun(B,set(C)),A4)) ).

% vimage_snd
tff(fact_6432_vimage__eq__UN,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),B4: set(C)] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),B4) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_adu(fun(B,C),fun(C,set(B)),F)),B4)) ).

% vimage_eq_UN
tff(fact_6433_rel__restrict__compl,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),A4: set(B)] : aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),rel_restrict(B,R2,A4)),rel_restrict(B,R2,aa(set(B),set(B),uminus_uminus(set(B)),A4))) = bot_bot(set(product_prod(B,B))) ).

% rel_restrict_compl
tff(fact_6434_homo__rel__restrict__mono,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),B4: set(B),A4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),product_Sigma(B,B,B4,aTP_Lamp_zv(set(B),fun(B,set(B)),B4)))
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),rel_restrict(B,R2,A4)),product_Sigma(B,B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B4),A4),aa(set(B),fun(B,set(B)),aTP_Lamp_adv(set(B),fun(set(B),fun(B,set(B))),B4),A4))) ) ).

% homo_rel_restrict_mono
tff(fact_6435_rel__restrict__alt__def,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),A4: set(B)] : rel_restrict(B,R2,A4) = aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R2),product_Sigma(B,B,aa(set(B),set(B),uminus_uminus(set(B)),A4),aTP_Lamp_adw(set(B),fun(B,set(B)),A4))) ).

% rel_restrict_alt_def
tff(fact_6436_rel__restrict__Sigma__sub,axiom,
    ! [B: $tType,A4: set(B),R2: set(B)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),rel_restrict(B,transitive_trancl(B,product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4))),R2)),transitive_trancl(B,product_Sigma(B,B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),R2),aa(set(B),fun(B,set(B)),aTP_Lamp_adv(set(B),fun(set(B),fun(B,set(B))),A4),R2)))) ).

% rel_restrict_Sigma_sub
tff(fact_6437_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B)] :
          ( aa(set(B),$o,finite_finite2(B),aa(set(nat),set(B),image2(nat,B,F),top_top(set(nat))))
         => ( aa(fun(nat,B),$o,order_mono(nat,B),F)
           => ( ! [N2: nat] :
                  ( ( aa(nat,B,F,N2) = aa(nat,B,F,aa(nat,nat,suc,N2)) )
                 => ( aa(nat,B,F,aa(nat,nat,suc,N2)) = aa(nat,B,F,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
             => ? [N9: nat] :
                  ( ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N9)
                     => ! [M5: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N9)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N8)
                           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(nat,B,F,M5)),aa(nat,B,F,N8)) ) ) )
                  & ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N8)
                     => ( aa(nat,B,F,N9) = aa(nat,B,F,N8) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_6438_finite__def,axiom,
    ! [B: $tType] : finite_finite2(B) = complete_lattice_lfp(fun(set(B),$o),aTP_Lamp_adx(fun(set(B),$o),fun(set(B),$o))) ).

% finite_def
tff(fact_6439_lfp__Kleene__iter,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),K: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( ( aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,suc,K)),F),bot_bot(B)) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),K),F),bot_bot(B)) )
           => ( complete_lattice_lfp(B,F) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),K),F),bot_bot(B)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_6440_mono__funpow,axiom,
    ! [B: $tType] :
      ( ( lattice(B)
        & order_bot(B) )
     => ! [Q: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),Q)
         => aa(fun(nat,B),$o,order_mono(nat,B),aTP_Lamp_ady(fun(B,B),fun(nat,B),Q)) ) ) ).

% mono_funpow
tff(fact_6441_lfp__funpow,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),N: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( complete_lattice_lfp(B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,suc,N)),F)) = complete_lattice_lfp(B,F) ) ) ) ).

% lfp_funpow
tff(fact_6442_mono__pow,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),N: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => aa(fun(B,B),$o,order_mono(B,B),aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F)) ) ) ).

% mono_pow
tff(fact_6443_mono__strict__invE,axiom,
    ! [C: $tType,B: $tType] :
      ( ( linorder(B)
        & order(C) )
     => ! [F: fun(B,C),X: B,Y: B] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X)),aa(B,C,F,Y))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y) ) ) ) ).

% mono_strict_invE
tff(fact_6444_mono__Suc,axiom,
    aa(fun(nat,nat),$o,order_mono(nat,nat),suc) ).

% mono_Suc
tff(fact_6445_def__lfp__unfold,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [H: B,F: fun(B,B)] :
          ( ( H = complete_lattice_lfp(B,F) )
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => ( H = aa(B,B,F,H) ) ) ) ) ).

% def_lfp_unfold
tff(fact_6446_lfp__fixpoint,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,B,F,complete_lattice_lfp(B,F)) = complete_lattice_lfp(B,F) ) ) ) ).

% lfp_fixpoint
tff(fact_6447_lfp__unfold,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( complete_lattice_lfp(B,F) = aa(B,B,F,complete_lattice_lfp(B,F)) ) ) ) ).

% lfp_unfold
tff(fact_6448_mono__add,axiom,
    ! [B: $tType] :
      ( ordere6658533253407199908up_add(B)
     => ! [A2: B] : aa(fun(B,B),$o,order_mono(B,B),aa(B,fun(B,B),plus_plus(B),A2)) ) ).

% mono_add
tff(fact_6449_lfp__const,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ta: B] : complete_lattice_lfp(B,aTP_Lamp_adz(B,fun(B,B),Ta)) = Ta ) ).

% lfp_const
tff(fact_6450_lfp__rolling,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(C)
        & comple6319245703460814977attice(B) )
     => ! [G: fun(B,C),F: fun(C,B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),G)
         => ( aa(fun(C,B),$o,order_mono(C,B),F)
           => ( aa(B,C,G,complete_lattice_lfp(B,aa(fun(C,B),fun(B,B),aTP_Lamp_aea(fun(B,C),fun(fun(C,B),fun(B,B)),G),F))) = complete_lattice_lfp(C,aa(fun(C,B),fun(C,C),aTP_Lamp_aeb(fun(B,C),fun(fun(C,B),fun(C,C)),G),F)) ) ) ) ) ).

% lfp_rolling
tff(fact_6451_mono__invE,axiom,
    ! [C: $tType,B: $tType] :
      ( ( linorder(B)
        & order(C) )
     => ! [F: fun(B,C),X: B,Y: B] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X)),aa(B,C,F,Y))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y) ) ) ) ).

% mono_invE
tff(fact_6452_mono__iff__le__Suc,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B)] :
          ( aa(fun(nat,B),$o,order_mono(nat,B),F)
        <=> ! [N4: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,N4)),aa(nat,B,F,aa(nat,nat,suc,N4))) ) ) ).

% mono_iff_le_Suc
tff(fact_6453_lfp__lfp,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,fun(B,B))] :
          ( ! [X2: B,Y2: B,W2: B,Z2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),W2),Z2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),F,X2),W2)),aa(B,B,aa(B,fun(B,B),F,Y2),Z2)) ) )
         => ( complete_lattice_lfp(B,aTP_Lamp_aec(fun(B,fun(B,B)),fun(B,B),F)) = complete_lattice_lfp(B,aTP_Lamp_aed(fun(B,fun(B,B)),fun(B,B),F)) ) ) ) ).

% lfp_lfp
tff(fact_6454_mono__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
        <=> ! [X3: B,Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y5)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X3)),aa(B,C,F,Y5)) ) ) ) ).

% mono_def
tff(fact_6455_monoI,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C)] :
          ( ! [X2: B,Y2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) )
         => aa(fun(B,C),$o,order_mono(B,C),F) ) ) ).

% monoI
tff(fact_6456_monoE,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C),X: B,Y: B] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,Y)) ) ) ) ).

% monoE
tff(fact_6457_monoD,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C),X: B,Y: B] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,Y)) ) ) ) ).

% monoD
tff(fact_6458_lfp__eqI,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(B,B),X: B] :
          ( aa(fun(B,B),$o,order_mono(B,B),F3)
         => ( ( aa(B,B,F3,X) = X )
           => ( ! [Z2: B] :
                  ( ( aa(B,B,F3,Z2) = Z2 )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Z2) )
             => ( complete_lattice_lfp(B,F3) = X ) ) ) ) ) ).

% lfp_eqI
tff(fact_6459_lfp__greatest,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),A4: B] :
          ( ! [U3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,U3)),U3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A4),U3) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A4),complete_lattice_lfp(B,F)) ) ) ).

% lfp_greatest
tff(fact_6460_lfp__lowerbound,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),A4: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,A4)),A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_lfp(B,F)),A4) ) ) ).

% lfp_lowerbound
tff(fact_6461_lfp__mono,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),G: fun(B,B)] :
          ( ! [Z6: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,Z6)),aa(B,B,G,Z6))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_lfp(B,F)),complete_lattice_lfp(B,G)) ) ) ).

% lfp_mono
tff(fact_6462_funpow__mono,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(B,B),A4: B,B4: B,N: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A4),B4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),A4)),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),B4)) ) ) ) ).

% funpow_mono
tff(fact_6463_def__lfp__induct,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: B,F: fun(B,B),Pa: B] :
          ( ( A4 = complete_lattice_lfp(B,F) )
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,aa(B,B,aa(B,fun(B,B),inf_inf(B),A4),Pa))),Pa)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A4),Pa) ) ) ) ) ).

% def_lfp_induct
tff(fact_6464_lfp__induct,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),Pa: B] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,aa(B,B,aa(B,fun(B,B),inf_inf(B),complete_lattice_lfp(B,F)),Pa))),Pa)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_lfp(B,F)),Pa) ) ) ) ).

% lfp_induct
tff(fact_6465_mono__inf,axiom,
    ! [C: $tType,B: $tType] :
      ( ( semilattice_inf(B)
        & semilattice_inf(C) )
     => ! [F: fun(B,C),A4: B,B4: B] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,aa(B,B,aa(B,fun(B,B),inf_inf(B),A4),B4))),aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(B,C,F,A4)),aa(B,C,F,B4))) ) ) ).

% mono_inf
tff(fact_6466_mono__sup,axiom,
    ! [C: $tType,B: $tType] :
      ( ( semilattice_sup(B)
        & semilattice_sup(C) )
     => ! [F: fun(B,C),A4: B,B4: B] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(B,C,F,A4)),aa(B,C,F,B4))),aa(B,C,F,aa(B,B,aa(B,fun(B,B),sup_sup(B),A4),B4))) ) ) ).

% mono_sup
tff(fact_6467_lfp__ordinal__induct,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),Pa: fun(B,$o)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( ! [S6: B] :
                ( aa(B,$o,Pa,S6)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),S6),complete_lattice_lfp(B,F))
                 => aa(B,$o,Pa,aa(B,B,F,S6)) ) )
           => ( ! [M8: set(B)] :
                  ( ! [X4: B] :
                      ( aa(set(B),$o,member(B,X4),M8)
                     => aa(B,$o,Pa,X4) )
                 => aa(B,$o,Pa,aa(set(B),B,complete_Sup_Sup(B),M8)) )
             => aa(B,$o,Pa,complete_lattice_lfp(B,F)) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_6468_Rings_Omono__mult,axiom,
    ! [B: $tType] :
      ( ordered_semiring(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),A2)
         => aa(fun(B,B),$o,order_mono(B,B),aa(B,fun(B,B),times_times(B),A2)) ) ) ).

% Rings.mono_mult
tff(fact_6469_mono__times__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
     => aa(fun(nat,nat),$o,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N)) ) ).

% mono_times_nat
tff(fact_6470_mono__image__least,axiom,
    ! [B: $tType,C: $tType] :
      ( ( order(C)
        & order(B) )
     => ! [F: fun(B,C),Ma: B,N: B,M3: C,N3: C] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( ( aa(set(B),set(C),image2(B,C,F),set_or7035219750837199246ssThan(B,Ma,N)) = set_or7035219750837199246ssThan(C,M3,N3) )
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ma),N)
             => ( aa(B,C,F,Ma) = M3 ) ) ) ) ) ).

% mono_image_least
tff(fact_6471_Kleene__iter__lpfp,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [F: fun(B,B),P2: B,K: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,P2)),P2)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),K),F),bot_bot(B))),P2) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_6472_Kleene__iter__gpfp,axiom,
    ! [B: $tType] :
      ( order_top(B)
     => ! [F: fun(B,B),P2: B,K: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),P2),aa(B,B,F,P2))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),P2),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),K),F),top_top(B))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_6473_funpow__mono2,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(B,B),I: nat,J: nat,X: B,Y: B] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(B,B,F,X))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),I),F),X)),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),J),F),Y)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_6474_lfp__def,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] : complete_lattice_lfp(B,F) = aa(set(B),B,complete_Inf_Inf(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_aee(fun(B,B),fun(B,$o),F))) ) ).

% lfp_def
tff(fact_6475_mono__SUP,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(C) )
     => ! [F: fun(B,C),A4: fun(D,B),I5: set(D)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(D),set(C),image2(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aef(fun(B,C),fun(fun(D,B),fun(D,C)),F),A4)),I5))),aa(B,C,F,aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,A4),I5)))) ) ) ).

% mono_SUP
tff(fact_6476_mono__Sup,axiom,
    ! [C: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(B,C,F,aa(set(B),B,complete_Sup_Sup(B),A4))) ) ) ).

% mono_Sup
tff(fact_6477_mono__INF,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( ( comple6319245703460814977attice(C)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(B,C),A4: fun(D,B),I5: set(D)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,A4),I5)))),aa(set(C),C,complete_Inf_Inf(C),aa(set(D),set(C),image2(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aef(fun(B,C),fun(fun(D,B),fun(D,C)),F),A4)),I5))) ) ) ).

% mono_INF
tff(fact_6478_mono__Inf,axiom,
    ! [C: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,aa(set(B),B,complete_Inf_Inf(B),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ).

% mono_Inf
tff(fact_6479_funpow__decreasing,axiom,
    ! [B: $tType] :
      ( ( lattice(B)
        & order_bot(B) )
     => ! [Ma: nat,N: nat,F: fun(B,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Ma),F),bot_bot(B))),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),bot_bot(B))) ) ) ) ).

% funpow_decreasing
tff(fact_6480_funpow__increasing,axiom,
    ! [B: $tType] :
      ( ( lattice(B)
        & order_top(B) )
     => ! [Ma: nat,N: nat,F: fun(B,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),top_top(B))),aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Ma),F),top_top(B))) ) ) ) ).

% funpow_increasing
tff(fact_6481_mono__Max__commute,axiom,
    ! [C: $tType,B: $tType] :
      ( ( linorder(B)
        & linorder(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( aa(set(B),$o,finite_finite2(B),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => ( aa(B,C,F,aa(set(B),B,lattic643756798349783984er_Max(B),A4)) = aa(set(C),C,lattic643756798349783984er_Max(C),aa(set(B),set(C),image2(B,C,F),A4)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_6482_mono__Min__commute,axiom,
    ! [C: $tType,B: $tType] :
      ( ( linorder(B)
        & linorder(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( aa(set(B),$o,finite_finite2(B),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => ( aa(B,C,F,aa(set(B),B,lattic643756798350308766er_Min(B),A4)) = aa(set(C),C,lattic643756798350308766er_Min(C),aa(set(B),set(C),image2(B,C,F),A4)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_6483_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(fun(nat,nat),$o,order_mono(nat,nat),aTP_Lamp_aeg(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_6484_remdups__adj__altdef,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( remdups_adj(B,Xs) = Ys )
    <=> ? [F5: fun(nat,nat)] :
          ( aa(fun(nat,nat),$o,order_mono(nat,nat),F5)
          & ( aa(set(nat),set(nat),image2(nat,nat,F5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Ys)) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
             => ( aa(nat,B,nth(B,Xs),I3) = aa(nat,B,nth(B,Ys),aa(nat,nat,F5,I3)) ) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(list(B),nat,size_size(list(B)),Xs))
             => ( ( aa(nat,B,nth(B,Xs),I3) = aa(nat,B,nth(B,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I3) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_6485_INF__principal__finite,axiom,
    ! [C: $tType,B: $tType,X5: set(B),F: fun(B,set(C))] :
      ( aa(set(B),$o,finite_finite2(B),X5)
     => ( aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),aTP_Lamp_aeh(fun(B,set(C)),fun(B,filter(C)),F)),X5)) = principal(C,aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),X5))) ) ) ).

% INF_principal_finite
tff(fact_6486_SUP__principal,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),I5: set(C)] : aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),aTP_Lamp_aei(fun(C,set(B)),fun(C,filter(B)),A4)),I5)) = principal(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5))) ).

% SUP_principal
tff(fact_6487_lfp__induct2,axiom,
    ! [B: $tType,C: $tType,A2: B,B2: C,F: fun(set(product_prod(B,C)),set(product_prod(B,C))),Pa: fun(B,fun(C,$o))] :
      ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),complete_lattice_lfp(set(product_prod(B,C)),F))
     => ( aa(fun(set(product_prod(B,C)),set(product_prod(B,C))),$o,order_mono(set(product_prod(B,C)),set(product_prod(B,C))),F)
       => ( ! [A3: B,B3: C] :
              ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)),aa(set(product_prod(B,C)),set(product_prod(B,C)),F,aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),complete_lattice_lfp(set(product_prod(B,C)),F)),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Pa)))))
             => aa(C,$o,aa(B,fun(C,$o),Pa,A3),B3) )
         => aa(C,$o,aa(B,fun(C,$o),Pa,A2),B2) ) ) ) ).

% lfp_induct2
tff(fact_6488_lfp__induct__set,axiom,
    ! [B: $tType,A2: B,F: fun(set(B),set(B)),Pa: fun(B,$o)] :
      ( aa(set(B),$o,member(B,A2),complete_lattice_lfp(set(B),F))
     => ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),complete_lattice_lfp(set(B),F)),aa(fun(B,$o),set(B),collect(B),Pa))))
             => aa(B,$o,Pa,X2) )
         => aa(B,$o,Pa,A2) ) ) ) ).

% lfp_induct_set
tff(fact_6489_def__lfp__induct__set,axiom,
    ! [B: $tType,A4: set(B),F: fun(set(B),set(B)),A2: B,Pa: fun(B,$o)] :
      ( ( A4 = complete_lattice_lfp(set(B),F) )
     => ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
       => ( aa(set(B),$o,member(B,A2),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),aa(set(B),set(B),F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(fun(B,$o),set(B),collect(B),Pa))))
               => aa(B,$o,Pa,X2) )
           => aa(B,$o,Pa,A2) ) ) ) ) ).

% def_lfp_induct_set
tff(fact_6490_coinduct3__mono__lemma,axiom,
    ! [B: $tType,C: $tType] :
      ( order(B)
     => ! [F: fun(B,set(C)),X5: set(C),B4: set(C)] :
          ( aa(fun(B,set(C)),$o,order_mono(B,set(C)),F)
         => aa(fun(B,set(C)),$o,order_mono(B,set(C)),aa(set(C),fun(B,set(C)),aa(set(C),fun(set(C),fun(B,set(C))),aTP_Lamp_aej(fun(B,set(C)),fun(set(C),fun(set(C),fun(B,set(C)))),F),X5),B4)) ) ) ).

% coinduct3_mono_lemma
tff(fact_6491_tranclp_Omono,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o))] : aa(fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),$o,order_mono(fun(B,fun(B,$o)),fun(B,fun(B,$o))),aTP_Lamp_aek(fun(B,fun(B,$o)),fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),R)) ).

% tranclp.mono
tff(fact_6492_rtranclp_Omono,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o))] : aa(fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),$o,order_mono(fun(B,fun(B,$o)),fun(B,fun(B,$o))),aTP_Lamp_ael(fun(B,fun(B,$o)),fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),R)) ).

% rtranclp.mono
tff(fact_6493_lfp__ordinal__induct__set,axiom,
    ! [B: $tType,F: fun(set(B),set(B)),Pa: fun(set(B),$o)] :
      ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
     => ( ! [S6: set(B)] :
            ( aa(set(B),$o,Pa,S6)
           => aa(set(B),$o,Pa,aa(set(B),set(B),F,S6)) )
       => ( ! [M8: set(set(B))] :
              ( ! [X4: set(B)] :
                  ( aa(set(set(B)),$o,member(set(B),X4),M8)
                 => aa(set(B),$o,Pa,X4) )
             => aa(set(B),$o,Pa,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),M8)) )
         => aa(set(B),$o,Pa,complete_lattice_lfp(set(B),F)) ) ) ) ).

% lfp_ordinal_induct_set
tff(fact_6494_sorted__remdups__adj,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),remdups_adj(B,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_6495_remdups__adj__length,axiom,
    ! [B: $tType,Xs: list(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),remdups_adj(B,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% remdups_adj_length
tff(fact_6496_remdups__adj__Cons_H,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : remdups_adj(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),remdups_adj(B,dropWhile(B,aTP_Lamp_af(B,fun(B,$o),X),Xs))) ).

% remdups_adj_Cons'
tff(fact_6497_remdups__adj__Cons,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : remdups_adj(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),case_list(list(B),B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)),aTP_Lamp_aem(B,fun(B,fun(list(B),list(B))),X)),remdups_adj(B,Xs)) ).

% remdups_adj_Cons
tff(fact_6498_remdups__adj__adjacent,axiom,
    ! [B: $tType,I: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),aa(list(B),nat,size_size(list(B)),remdups_adj(B,Xs)))
     => ( aa(nat,B,nth(B,remdups_adj(B,Xs)),I) != aa(nat,B,nth(B,remdups_adj(B,Xs)),aa(nat,nat,suc,I)) ) ) ).

% remdups_adj_adjacent
tff(fact_6499_ord_Olexordp_Omono,axiom,
    ! [B: $tType,Less: fun(B,fun(B,$o))] : aa(fun(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),$o,order_mono(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),aTP_Lamp_aen(fun(B,fun(B,$o)),fun(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),Less)) ).

% ord.lexordp.mono
tff(fact_6500_remdups__adj__replicate,axiom,
    ! [B: $tType,N: nat,X: B] :
      remdups_adj(B,replicate(B,N,X)) = $ite(N = zero_zero(nat),nil(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ).

% remdups_adj_replicate
tff(fact_6501_remdups__adj__singleton,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( ( remdups_adj(B,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)) )
     => ( Xs = replicate(B,aa(list(B),nat,size_size(list(B)),Xs),X) ) ) ).

% remdups_adj_singleton
tff(fact_6502_finite_Omono,axiom,
    ! [B: $tType] : aa(fun(fun(set(B),$o),fun(set(B),$o)),$o,order_mono(fun(set(B),$o),fun(set(B),$o)),aTP_Lamp_adx(fun(set(B),$o),fun(set(B),$o))) ).

% finite.mono
tff(fact_6503_remdups__adj__append__dropWhile,axiom,
    ! [B: $tType,Xs: list(B),Y: B,Ys: list(B)] : remdups_adj(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),remdups_adj(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),nil(B))))),remdups_adj(B,dropWhile(B,aTP_Lamp_af(B,fun(B,$o),Y),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_6504_remdups__adj__append_H_H,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( Xs != nil(B) )
     => ( remdups_adj(B,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),remdups_adj(B,Xs)),remdups_adj(B,dropWhile(B,aTP_Lamp_aeo(list(B),fun(B,$o),Xs),Ys))) ) ) ).

% remdups_adj_append''
tff(fact_6505_tl__remdups__adj,axiom,
    ! [B: $tType,Ys: list(B)] :
      ( ( Ys != nil(B) )
     => ( aa(list(B),list(B),tl(B),remdups_adj(B,Ys)) = remdups_adj(B,dropWhile(B,aTP_Lamp_aep(list(B),fun(B,$o),Ys),aa(list(B),list(B),tl(B),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_6506_lexordp_Omono,axiom,
    ! [B: $tType] :
      ( ord(B)
     => aa(fun(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),$o,order_mono(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),aTP_Lamp_aeq(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o)))) ) ).

% lexordp.mono
tff(fact_6507_remdups__adj__length__ge1,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(B),nat,size_size(list(B)),remdups_adj(B,Xs))) ) ).

% remdups_adj_length_ge1
tff(fact_6508_remdups__adj__singleton__iff,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( aa(list(B),nat,size_size(list(B)),remdups_adj(B,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xs != nil(B) )
        & ( Xs = replicate(B,aa(list(B),nat,size_size(list(B)),Xs),aa(list(B),B,hd(B),Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_6509_at__bot__def,axiom,
    ! [B: $tType] :
      ( order(B)
     => ( at_bot(B) = aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(B),set(filter(B)),image2(B,filter(B),aTP_Lamp_aer(B,filter(B))),top_top(set(B)))) ) ) ).

% at_bot_def
tff(fact_6510_at__top__def,axiom,
    ! [B: $tType] :
      ( order(B)
     => ( at_top(B) = aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(B),set(filter(B)),image2(B,filter(B),aTP_Lamp_aes(B,filter(B))),top_top(set(B)))) ) ) ).

% at_top_def
tff(fact_6511_mono__compose,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType] :
      ( ( order(D)
        & order(B) )
     => ! [Q: fun(B,fun(C,D)),F: fun(E,C)] :
          ( aa(fun(B,fun(C,D)),$o,order_mono(B,fun(C,D)),Q)
         => aa(fun(B,fun(E,D)),$o,order_mono(B,fun(E,D)),aa(fun(E,C),fun(B,fun(E,D)),aTP_Lamp_aet(fun(B,fun(C,D)),fun(fun(E,C),fun(B,fun(E,D))),Q),F)) ) ) ).

% mono_compose
tff(fact_6512_at__top__sub,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B] : at_top(B) = aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(B),set(filter(B)),image2(B,filter(B),aTP_Lamp_aeu(B,filter(B))),aa(B,set(B),set_ord_atLeast(B),C2))) ) ).

% at_top_sub
tff(fact_6513_at__bot__sub,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B] : at_bot(B) = aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(B),set(filter(B)),image2(B,filter(B),aTP_Lamp_aev(B,filter(B))),aa(B,set(B),set_ord_atMost(B),C2))) ) ).

% at_bot_sub
tff(fact_6514_ord__class_Olexordp__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ( ord_lexordp(B) = complete_lattice_lfp(fun(list(B),fun(list(B),$o)),aTP_Lamp_aeq(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_6515_image__o__collect,axiom,
    ! [C: $tType,D: $tType,B: $tType,G: fun(D,C),F3: set(fun(B,set(D)))] : bNF_collect(B,C,aa(set(fun(B,set(D))),set(fun(B,set(C))),image2(fun(B,set(D)),fun(B,set(C)),aa(fun(set(D),set(C)),fun(fun(B,set(D)),fun(B,set(C))),comp(set(D),set(C),B),image2(D,C,G))),F3)) = aa(fun(B,set(D)),fun(B,set(C)),aa(fun(set(D),set(C)),fun(fun(B,set(D)),fun(B,set(C))),comp(set(D),set(C),B),image2(D,C,G)),bNF_collect(B,D,F3)) ).

% image_o_collect
tff(fact_6516_collect__def,axiom,
    ! [B: $tType,C: $tType,F3: set(fun(C,set(B))),X: C] : aa(C,set(B),bNF_collect(C,B,F3),X) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(fun(C,set(B))),set(set(B)),image2(fun(C,set(B)),set(B),aTP_Lamp_aew(C,fun(fun(C,set(B)),set(B)),X)),F3)) ).

% collect_def
tff(fact_6517_collect__comp,axiom,
    ! [B: $tType,C: $tType,D: $tType,F3: set(fun(D,set(C))),G: fun(B,D)] : aa(fun(B,D),fun(B,set(C)),aa(fun(D,set(C)),fun(fun(B,D),fun(B,set(C))),comp(D,set(C),B),bNF_collect(D,C,F3)),G) = bNF_collect(B,C,aa(set(fun(D,set(C))),set(fun(B,set(C))),image2(fun(D,set(C)),fun(B,set(C)),aTP_Lamp_aex(fun(B,D),fun(fun(D,set(C)),fun(B,set(C))),G)),F3)) ).

% collect_comp
tff(fact_6518_lexordp__conv__lexord,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),Ys: list(B)] :
          ( aa(list(B),$o,aa(list(B),fun(list(B),$o),ord_lexordp(B),Xs),Ys)
        <=> aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Xs),Ys)),lexord(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),ord_less(B))))) ) ) ).

% lexordp_conv_lexord
tff(fact_6519_ord_Olexordp__def,axiom,
    ! [B: $tType,Less: fun(B,fun(B,$o))] : lexordp2(B,Less) = complete_lattice_lfp(fun(list(B),fun(list(B),$o)),aTP_Lamp_aen(fun(B,fun(B,$o)),fun(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),Less)) ).

% ord.lexordp_def
tff(fact_6520_bijective__Empty,axiom,
    ! [C: $tType,B: $tType] : bijective(B,C,bot_bot(set(product_prod(B,C)))) ).

% bijective_Empty
tff(fact_6521_bijective__def,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C))] :
      ( bijective(B,C,R2)
    <=> ( ! [X3: B,Y5: C,Z5: C] :
            ( ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y5)),R2)
              & aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Z5)),R2) )
           => ( Y5 = Z5 ) )
        & ! [X3: B,Y5: B,Z5: C] :
            ( ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Z5)),R2)
              & aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Y5),Z5)),R2) )
           => ( X3 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_6522_map__of__mapk__SomeI,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(B,C),Ta: list(product_prod(B,D)),K: B,X: D] :
      ( inj_on(B,C,F,top_top(set(B)))
     => ( ( aa(B,option(D),map_of(B,D,Ta),K) = aa(D,option(D),some(D),X) )
       => ( aa(C,option(D),map_of(C,D,aa(list(product_prod(B,D)),list(product_prod(C,D)),map(product_prod(B,D),product_prod(C,D),aa(fun(B,fun(D,product_prod(C,D))),fun(product_prod(B,D),product_prod(C,D)),product_case_prod(B,D,product_prod(C,D)),aTP_Lamp_aey(fun(B,C),fun(B,fun(D,product_prod(C,D))),F))),Ta)),aa(B,C,F,K)) = aa(D,option(D),some(D),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_6523_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_aez(rat,fun(product_prod(int,int),$o),X)) ).

% quotient_of_def
tff(fact_6524_coprime__mult__left__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2),C2)
        <=> ( algebr8660921524188924756oprime(B,A2,C2)
            & algebr8660921524188924756oprime(B,B2,C2) ) ) ) ).

% coprime_mult_left_iff
tff(fact_6525_coprime__mult__right__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [C2: B,A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,C2,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))
        <=> ( algebr8660921524188924756oprime(B,C2,A2)
            & algebr8660921524188924756oprime(B,C2,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_6526_coprime__minus__right__iff,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,aa(B,B,uminus_uminus(B),B2))
        <=> algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% coprime_minus_right_iff
tff(fact_6527_coprime__minus__left__iff,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,aa(B,B,uminus_uminus(B),A2),B2)
        <=> algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% coprime_minus_left_iff
tff(fact_6528_coprime__self,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] :
          ( algebr8660921524188924756oprime(B,A2,A2)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B)) ) ) ).

% coprime_self
tff(fact_6529_coprime__mod__right__iff,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( algebr8660921524188924756oprime(B,A2,aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),A2))
          <=> algebr8660921524188924756oprime(B,A2,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_6530_coprime__mod__left__iff,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2),B2)
          <=> algebr8660921524188924756oprime(B,A2,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_6531_coprime__power__left__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,N: nat,B2: B] :
          ( algebr8660921524188924756oprime(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N),B2)
        <=> ( algebr8660921524188924756oprime(B,A2,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_6532_coprime__power__right__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,N: nat] :
          ( algebr8660921524188924756oprime(B,A2,aa(nat,B,aa(B,fun(nat,B),power_power(B),B2),N))
        <=> ( algebr8660921524188924756oprime(B,A2,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_6533_coprime__imp__gcd__eq__1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,B2)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = one_one(B) ) ) ) ).

% coprime_imp_gcd_eq_1
tff(fact_6534_inj__mult__left,axiom,
    ! [B: $tType] :
      ( idom(B)
     => ! [A2: B] :
          ( inj_on(B,B,aa(B,fun(B,B),times_times(B),A2),top_top(set(B)))
        <=> ( A2 != zero_zero(B) ) ) ) ).

% inj_mult_left
tff(fact_6535_coprime__0__left__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] :
          ( algebr8660921524188924756oprime(B,zero_zero(B),A2)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B)) ) ) ).

% coprime_0_left_iff
tff(fact_6536_coprime__0__right__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] :
          ( algebr8660921524188924756oprime(B,A2,zero_zero(B))
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B)) ) ) ).

% coprime_0_right_iff
tff(fact_6537_coprime__mult__self__right__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,C2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B))
            & algebr8660921524188924756oprime(B,A2,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_6538_coprime__mult__self__left__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B))
            & algebr8660921524188924756oprime(B,A2,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_6539_is__unit__gcd,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),one_one(B))
        <=> algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% is_unit_gcd
tff(fact_6540_inj__divide__right,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [A2: B] :
          ( inj_on(B,B,aTP_Lamp_afa(B,fun(B,B),A2),top_top(set(B)))
        <=> ( A2 != zero_zero(B) ) ) ) ).

% inj_divide_right
tff(fact_6541_inj__on__apsnd,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(C,D),A4: set(C)] :
      ( inj_on(product_prod(B,C),product_prod(B,D),aa(fun(C,D),fun(product_prod(B,C),product_prod(B,D)),product_apsnd(C,D,B),F),product_Sigma(B,C,top_top(set(B)),aTP_Lamp_vu(set(C),fun(B,set(C)),A4)))
    <=> inj_on(C,D,F,A4) ) ).

% inj_on_apsnd
tff(fact_6542_normalize__stable,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2)
     => ( algebr8660921524188924756oprime(int,P2,Q2)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) ) ) ) ).

% normalize_stable
tff(fact_6543_subset__image__inj,axiom,
    ! [B: $tType,C: $tType,S: set(B),F: fun(C,B),T4: set(C)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S),aa(set(C),set(B),image2(C,B,F),T4))
    <=> ? [U5: set(C)] :
          ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),U5),T4)
          & inj_on(C,B,F,U5)
          & ( S = aa(set(C),set(B),image2(C,B,F),U5) ) ) ) ).

% subset_image_inj
tff(fact_6544_prod_Oimage__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [G: fun(B,C),A4: set(B)] :
          ( inj_on(B,C,G,A4)
         => ( aa(set(C),C,aa(fun(C,C),fun(set(C),C),groups7121269368397514597t_prod(C,C),aTP_Lamp_afb(C,C)),aa(set(B),set(C),image2(B,C,G),A4)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),A4) ) ) ) ).

% prod.image_eq
tff(fact_6545_sum_Oimage__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [G: fun(B,C),A4: set(B)] :
          ( inj_on(B,C,G,A4)
         => ( aa(set(C),C,aa(fun(C,C),fun(set(C),C),groups7311177749621191930dd_sum(C,C),aTP_Lamp_afc(C,C)),aa(set(B),set(C),image2(B,C,G),A4)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),A4) ) ) ) ).

% sum.image_eq
tff(fact_6546_inj__Pair_I1_J,axiom,
    ! [C: $tType,B: $tType,C2: fun(B,C),S: set(B)] : inj_on(B,product_prod(B,C),aTP_Lamp_rx(fun(B,C),fun(B,product_prod(B,C)),C2),S) ).

% inj_Pair(1)
tff(fact_6547_inj__Pair_I2_J,axiom,
    ! [C: $tType,B: $tType,C2: fun(B,C),S: set(B)] : inj_on(B,product_prod(C,B),aTP_Lamp_afd(fun(B,C),fun(B,product_prod(C,B)),C2),S) ).

% inj_Pair(2)
tff(fact_6548_inj__on__convol__ident,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),X5: set(B)] : inj_on(B,product_prod(B,C),aTP_Lamp_rx(fun(B,C),fun(B,product_prod(B,C)),F),X5) ).

% inj_on_convol_ident
tff(fact_6549_inj__on__diff__nat,axiom,
    ! [N5: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( aa(set(nat),$o,member(nat,N2),N5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2) )
     => inj_on(nat,nat,aTP_Lamp_jn(nat,fun(nat,nat),K),N5) ) ).

% inj_on_diff_nat
tff(fact_6550_linorder__inj__onI,axiom,
    ! [C: $tType,B: $tType] :
      ( order(B)
     => ! [A4: set(B),F: fun(B,C)] :
          ( ! [X2: B,Y2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
             => ( aa(set(B),$o,member(B,X2),A4)
               => ( aa(set(B),$o,member(B,Y2),A4)
                 => ( aa(B,C,F,X2) != aa(B,C,F,Y2) ) ) ) )
         => ( ! [X2: B,Y2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => ( aa(set(B),$o,member(B,Y2),A4)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                    | aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),X2) ) ) )
           => inj_on(B,C,F,A4) ) ) ) ).

% linorder_inj_onI
tff(fact_6551_inj__of__nat,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => inj_on(nat,B,semiring_1_of_nat(B),top_top(set(nat))) ) ).

% inj_of_nat
tff(fact_6552_sorted__list__of__set_Oinj__on,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => inj_on(B,B,aTP_Lamp_hr(B,B),top_top(set(B))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6553_inj__fun,axiom,
    ! [C: $tType,D: $tType,B: $tType,F: fun(B,C)] :
      ( inj_on(B,C,F,top_top(set(B)))
     => inj_on(B,fun(D,C),aTP_Lamp_afe(fun(B,C),fun(B,fun(D,C)),F),top_top(set(B))) ) ).

% inj_fun
tff(fact_6554_inj__add__left,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [A2: B] : inj_on(B,B,aa(B,fun(B,B),plus_plus(B),A2),top_top(set(B))) ) ).

% inj_add_left
tff(fact_6555_inj__on__fst__map__to__set,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : inj_on(product_prod(B,C),B,product_fst(B,C),map_to_set(B,C,Ma)) ).

% inj_on_fst_map_to_set
tff(fact_6556_inj__diff__right,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [A2: B] : inj_on(B,B,aTP_Lamp_jj(B,fun(B,B),A2),top_top(set(B))) ) ).

% inj_diff_right
tff(fact_6557_finite__Collect,axiom,
    ! [B: $tType,C: $tType,S: set(B),F: fun(C,B)] :
      ( aa(set(B),$o,finite_finite2(B),S)
     => ( inj_on(C,B,F,top_top(set(C)))
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(C,B),fun(C,$o),aTP_Lamp_aff(set(B),fun(fun(C,B),fun(C,$o)),S),F))) ) ) ).

% finite_Collect
tff(fact_6558_finite__inverse__image,axiom,
    ! [B: $tType,C: $tType,A4: set(B),F: fun(C,B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( inj_on(C,B,F,top_top(set(C)))
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(C,B),fun(C,$o),aTP_Lamp_aff(set(B),fun(fun(C,B),fun(C,$o)),A4),F))) ) ) ).

% finite_inverse_image
tff(fact_6559_prod__coprime__right,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_gcd(C)
     => ! [A4: set(B),A2: C,F: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => algebr8660921524188924756oprime(C,A2,aa(B,C,F,I2)) )
         => algebr8660921524188924756oprime(C,A2,aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4)) ) ) ).

% prod_coprime_right
tff(fact_6560_prod__coprime__left,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_gcd(C)
     => ! [A4: set(B),F: fun(B,C),A2: C] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),A4)
             => algebr8660921524188924756oprime(C,aa(B,C,F,I2),A2) )
         => algebr8660921524188924756oprime(C,aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),F),A4),A2) ) ) ).

% prod_coprime_left
tff(fact_6561_coprime__diff__one__left,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B] : algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B)),A2) ) ).

% coprime_diff_one_left
tff(fact_6562_coprime__doff__one__right,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B] : algebr8660921524188924756oprime(B,A2,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))) ) ).

% coprime_doff_one_right
tff(fact_6563_gcd__mult__left__left__cancel,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,C2: B,A2: B] :
          ( algebr8660921524188924756oprime(B,B2,C2)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)),B2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_6564_gcd__mult__left__right__cancel,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,C2: B,A2: B] :
          ( algebr8660921524188924756oprime(B,B2,C2)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),B2) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_6565_gcd__mult__right__left__cancel,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,C2)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_6566_gcd__mult__right__right__cancel,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,C2)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_6567_gcd__eq__1__imp__coprime,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = one_one(B) )
         => algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% gcd_eq_1_imp_coprime
tff(fact_6568_coprime__iff__gcd__eq__1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,B2)
        <=> ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) = one_one(B) ) ) ) ).

% coprime_iff_gcd_eq_1
tff(fact_6569_coprime__crossproduct__int,axiom,
    ! [A2: int,D2: int,B2: int,C2: int] :
      ( algebr8660921524188924756oprime(int,A2,D2)
     => ( algebr8660921524188924756oprime(int,B2,C2)
       => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),C2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D2)) )
        <=> ( ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) )
            & ( aa(int,int,abs_abs(int),C2) = aa(int,int,abs_abs(int),D2) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_6570_coprimeI,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( ! [C4: B] :
              ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C4),A2)
             => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C4),B2)
               => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C4),one_one(B)) ) )
         => algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% coprimeI
tff(fact_6571_coprime__def,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,B2)
        <=> ! [C5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C5),A2)
             => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C5),B2)
               => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C5),one_one(B)) ) ) ) ) ).

% coprime_def
tff(fact_6572_not__coprimeE,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( ~ algebr8660921524188924756oprime(B,A2,B2)
         => ~ ! [C4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C4),A2)
               => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C4),B2)
                 => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C4),one_one(B)) ) ) ) ) ).

% not_coprimeE
tff(fact_6573_not__coprimeI,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),A2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),B2)
           => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B))
             => ~ algebr8660921524188924756oprime(B,A2,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_6574_coprime__absorb__left,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [X: B,Y: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),Y)
         => ( algebr8660921524188924756oprime(B,X,Y)
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X),one_one(B)) ) ) ) ).

% coprime_absorb_left
tff(fact_6575_coprime__imp__coprime,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [C2: B,D2: B,A2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,C2,D2)
         => ( ! [E4: B] :
                ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),one_one(B))
               => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),A2)
                 => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),B2)
                   => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),C2) ) ) )
           => ( ! [E4: B] :
                  ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),one_one(B))
                 => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),A2)
                   => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),B2)
                     => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),E4),D2) ) ) )
             => algebr8660921524188924756oprime(B,A2,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_6576_coprime__absorb__right,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [Y: B,X: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Y),X)
         => ( algebr8660921524188924756oprime(B,X,Y)
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Y),one_one(B)) ) ) ) ).

% coprime_absorb_right
tff(fact_6577_coprime__common__divisor,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B,C2: B] :
          ( algebr8660921524188924756oprime(B,A2,B2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),A2)
           => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),B2)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),C2),one_one(B)) ) ) ) ) ).

% coprime_common_divisor
tff(fact_6578_is__unit__left__imp__coprime,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_6579_is__unit__right__imp__coprime,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [B2: B,A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B))
         => algebr8660921524188924756oprime(B,A2,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_6580_divides__mult,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2)
           => ( algebr8660921524188924756oprime(B,A2,B2)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2) ) ) ) ) ).

% divides_mult
tff(fact_6581_coprime__dvd__mult__left__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_6582_coprime__dvd__mult__right__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( algebr8660921524188924756oprime(B,A2,C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(B,B,aa(B,fun(B,B),times_times(B),C2),B2))
          <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_6583_inj__on__mult,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B,A4: set(B)] :
          ( ( A2 != zero_zero(B) )
         => inj_on(B,B,aa(B,fun(B,B),times_times(B),A2),A4) ) ) ).

% inj_on_mult
tff(fact_6584_inj__on__of__nat,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N5: set(nat)] : inj_on(nat,B,semiring_1_of_nat(B),N5) ) ).

% inj_on_of_nat
tff(fact_6585_inj__on__id2,axiom,
    ! [B: $tType,A4: set(B)] : inj_on(B,B,aTP_Lamp_il(B,B),A4) ).

% inj_on_id2
tff(fact_6586_inj__on__add_H,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [A2: B,A4: set(B)] : inj_on(B,B,aTP_Lamp_afg(B,fun(B,B),A2),A4) ) ).

% inj_on_add'
tff(fact_6587_inj__on__add,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [A2: B,A4: set(B)] : inj_on(B,B,aa(B,fun(B,B),plus_plus(B),A2),A4) ) ).

% inj_on_add
tff(fact_6588_inj__Suc,axiom,
    ! [N5: set(nat)] : inj_on(nat,nat,suc,N5) ).

% inj_Suc
tff(fact_6589_coprime__1__left,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] : algebr8660921524188924756oprime(B,one_one(B),A2) ) ).

% coprime_1_left
tff(fact_6590_coprime__1__right,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A2: B] : algebr8660921524188924756oprime(B,A2,one_one(B)) ) ).

% coprime_1_right
tff(fact_6591_coprime__add__one__right,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B] : algebr8660921524188924756oprime(B,A2,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B))) ) ).

% coprime_add_one_right
tff(fact_6592_coprime__add__one__left,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B] : algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),one_one(B)),A2) ) ).

% coprime_add_one_left
tff(fact_6593_mult__mod__cancel__right,axiom,
    ! [B: $tType] :
      ( ( euclid8851590272496341667cancel(B)
        & semiring_gcd(B) )
     => ! [A2: B,N: B,Ma: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),N)),Ma) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),N)),Ma) )
         => ( algebr8660921524188924756oprime(B,Ma,N)
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),Ma) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),Ma) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_6594_mult__mod__cancel__left,axiom,
    ! [B: $tType] :
      ( ( euclid8851590272496341667cancel(B)
        & semiring_gcd(B) )
     => ! [N: B,A2: B,Ma: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),N),A2)),Ma) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),N),B2)),Ma) )
         => ( algebr8660921524188924756oprime(B,Ma,N)
           => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),Ma) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),B2),Ma) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_6595_finite__inverse__image__gen,axiom,
    ! [B: $tType,C: $tType,A4: set(B),F: fun(C,B),D3: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( inj_on(C,B,F,D3)
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(set(C),fun(C,$o),aa(fun(C,B),fun(set(C),fun(C,$o)),aTP_Lamp_afh(set(B),fun(fun(C,B),fun(set(C),fun(C,$o))),A4),F),D3))) ) ) ).

% finite_inverse_image_gen
tff(fact_6596_inj__fn,axiom,
    ! [B: $tType,F: fun(B,B),N: nat] :
      ( inj_on(B,B,F,top_top(set(B)))
     => inj_on(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),top_top(set(B))) ) ).

% inj_fn
tff(fact_6597_map__prod__inj__on,axiom,
    ! [E: $tType,C: $tType,B: $tType,D: $tType,F: fun(B,C),A4: set(B),G: fun(D,E),B4: set(D)] :
      ( inj_on(B,C,F,A4)
     => ( inj_on(D,E,G,B4)
       => inj_on(product_prod(B,D),product_prod(C,E),product_map_prod(B,C,D,E,F,G),product_Sigma(B,D,A4,aTP_Lamp_wv(set(D),fun(B,set(D)),B4))) ) ) ).

% map_prod_inj_on
tff(fact_6598_inj__singleton,axiom,
    ! [B: $tType,A4: set(B)] : inj_on(B,set(B),aTP_Lamp_xj(B,set(B)),A4) ).

% inj_singleton
tff(fact_6599_swap__inj__on,axiom,
    ! [C: $tType,B: $tType,A4: set(product_prod(B,C))] : inj_on(product_prod(B,C),product_prod(C,B),aa(fun(B,fun(C,product_prod(C,B))),fun(product_prod(B,C),product_prod(C,B)),product_case_prod(B,C,product_prod(C,B)),aTP_Lamp_te(B,fun(C,product_prod(C,B)))),A4) ).

% swap_inj_on
tff(fact_6600_arg__min__inj__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [F: fun(B,C),Pa: fun(B,$o),A2: B] :
          ( inj_on(B,C,F,aa(fun(B,$o),set(B),collect(B),Pa))
         => ( aa(B,$o,Pa,A2)
           => ( ! [Y2: B] :
                  ( aa(B,$o,Pa,Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,A2)),aa(B,C,F,Y2)) )
             => ( lattices_ord_arg_min(B,C,F,Pa) = A2 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_6601_inj__split__Cons,axiom,
    ! [B: $tType,X5: set(product_prod(list(B),B))] : inj_on(product_prod(list(B),B),list(B),aa(fun(list(B),fun(B,list(B))),fun(product_prod(list(B),B),list(B)),product_case_prod(list(B),B,list(B)),aTP_Lamp_qj(list(B),fun(B,list(B)))),X5) ).

% inj_split_Cons
tff(fact_6602_inj__on__iff__surj,axiom,
    ! [C: $tType,B: $tType,A4: set(B),A9: set(C)] :
      ( ( A4 != bot_bot(set(B)) )
     => ( ? [F5: fun(B,C)] :
            ( inj_on(B,C,F5,A4)
            & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F5),A4)),A9) )
      <=> ? [G3: fun(C,B)] : aa(set(C),set(B),image2(C,B,G3),A9) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_6603_invertible__coprime,axiom,
    ! [B: $tType] :
      ( euclid8851590272496341667cancel(B)
     => ! [A2: B,B2: B,C2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),C2) = one_one(B) )
         => algebr8660921524188924756oprime(B,A2,C2) ) ) ).

% invertible_coprime
tff(fact_6604_gcd__coprime__exists,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) != zero_zero(B) )
         => ? [A10: B,B6: B] :
              ( ( A2 = aa(B,B,aa(B,fun(B,B),times_times(B),A10),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)) )
              & ( B2 = aa(B,B,aa(B,fun(B,B),times_times(B),B6),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)) )
              & algebr8660921524188924756oprime(B,A10,B6) ) ) ) ).

% gcd_coprime_exists
tff(fact_6605_gcd__coprime,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,A5: B,B5: B] :
          ( ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2) != zero_zero(B) )
         => ( ( A2 = aa(B,B,aa(B,fun(B,B),times_times(B),A5),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)) )
           => ( ( B2 = aa(B,B,aa(B,fun(B,B),times_times(B),B5),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)) )
             => algebr8660921524188924756oprime(B,A5,B5) ) ) ) ) ).

% gcd_coprime
tff(fact_6606_div__gcd__coprime,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( ( A2 != zero_zero(B) )
            | ( B2 != zero_zero(B) ) )
         => algebr8660921524188924756oprime(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2))) ) ) ).

% div_gcd_coprime
tff(fact_6607_Rat__cases,axiom,
    ! [Q2: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q2 = aa(int,rat,aa(int,fun(int,rat),fract,A3),B3) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
           => ~ algebr8660921524188924756oprime(int,A3,B3) ) ) ).

% Rat_cases
tff(fact_6608_Rat__induct,axiom,
    ! [Pa: fun(rat,$o),Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => ( algebr8660921524188924756oprime(int,A3,B3)
           => aa(rat,$o,Pa,aa(int,rat,aa(int,fun(int,rat),fract,A3),B3)) ) )
     => aa(rat,$o,Pa,Q2) ) ).

% Rat_induct
tff(fact_6609_coprime__common__divisor__int,axiom,
    ! [A2: int,B2: int,X: int] :
      ( algebr8660921524188924756oprime(int,A2,B2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),A2)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),X),B2)
         => ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_6610_inj__graph,axiom,
    ! [C: $tType,B: $tType] : inj_on(fun(B,C),set(product_prod(B,C)),aTP_Lamp_afj(fun(B,C),set(product_prod(B,C))),top_top(set(fun(B,C)))) ).

% inj_graph
tff(fact_6611_inj__on__UNION__chain,axiom,
    ! [D: $tType,C: $tType,B: $tType,I5: set(B),A4: fun(B,set(C)),F: fun(C,D)] :
      ( ! [I2: B,J2: B] :
          ( aa(set(B),$o,member(B,I2),I5)
         => ( aa(set(B),$o,member(B,J2),I5)
           => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),A4,I2)),aa(B,set(C),A4,J2))
              | aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),A4,J2)),aa(B,set(C),A4,I2)) ) ) )
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),I5)
           => inj_on(C,D,F,aa(B,set(C),A4,I2)) )
       => inj_on(C,D,F,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_6612_inj__on__filter__key__eq,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),Y: B,Xs: list(B)] :
      ( inj_on(B,C,F,aa(set(B),set(B),insert(B,Y),aa(list(B),set(B),set2(B),Xs)))
     => ( filter2(B,aa(B,fun(B,$o),aTP_Lamp_afk(fun(B,C),fun(B,fun(B,$o)),F),Y),Xs) = filter2(B,aa(B,fun(B,$o),fequal(B),Y),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_6613_inj__on__INTER,axiom,
    ! [D: $tType,C: $tType,B: $tType,I5: set(B),F: fun(C,D),A4: fun(B,set(C))] :
      ( ( I5 != bot_bot(set(B)) )
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),I5)
           => inj_on(C,D,F,aa(B,set(C),A4,I2)) )
       => inj_on(C,D,F,aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) ) ) ).

% inj_on_INTER
tff(fact_6614_finite__imp__nat__seg__image__inj__on,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ? [N2: nat,F4: fun(nat,B)] :
          ( ( A4 = aa(set(nat),set(B),image2(nat,B,F4),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N2))) )
          & inj_on(nat,B,F4,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6615_finite__imp__inj__to__nat__seg_H,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ~ ! [F4: fun(B,nat)] :
            ( ? [N2: nat] : aa(set(B),set(nat),image2(B,nat,F4),A4) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N2))
           => ~ inj_on(B,nat,F4,A4) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_6616_finite__imp__inj__to__nat__seg,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ? [F4: fun(B,nat),N2: nat] :
          ( ( aa(set(B),set(nat),image2(B,nat,F4),A4) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),N2)) )
          & inj_on(B,nat,F4,A4) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6617_set__to__map__inverse,axiom,
    ! [C: $tType,B: $tType,S: set(product_prod(B,C))] :
      ( inj_on(product_prod(B,C),B,product_fst(B,C),S)
     => ( map_to_set(B,C,set_to_map(B,C,S)) = S ) ) ).

% set_to_map_inverse
tff(fact_6618_inj__on__nth,axiom,
    ! [B: $tType,Xs: list(B),I5: set(nat)] :
      ( distinct(B,Xs)
     => ( ! [X2: nat] :
            ( aa(set(nat),$o,member(nat,X2),I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(list(B),nat,size_size(list(B)),Xs)) )
       => inj_on(nat,B,nth(B,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_6619_infinite__countable__subset,axiom,
    ! [B: $tType,S: set(B)] :
      ( ~ aa(set(B),$o,finite_finite2(B),S)
     => ? [F4: fun(nat,B)] :
          ( inj_on(nat,B,F4,top_top(set(nat)))
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(nat),set(B),image2(nat,B,F4),top_top(set(nat)))),S) ) ) ).

% infinite_countable_subset
tff(fact_6620_infinite__iff__countable__subset,axiom,
    ! [B: $tType,S: set(B)] :
      ( ~ aa(set(B),$o,finite_finite2(B),S)
    <=> ? [F5: fun(nat,B)] :
          ( inj_on(nat,B,F5,top_top(set(nat)))
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(nat),set(B),image2(nat,B,F5),top_top(set(nat)))),S) ) ) ).

% infinite_iff_countable_subset
tff(fact_6621_Rat__cases__nonzero,axiom,
    ! [Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q2 = aa(int,rat,aa(int,fun(int,rat),fract,A3),B3) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
           => ( ( A3 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A3,B3) ) ) )
     => ( Q2 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_6622_inj__on__disjoint__Un,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),A4: set(B),G: fun(B,C),B4: set(B)] :
      ( inj_on(B,C,F,A4)
     => ( inj_on(B,C,G,B4)
       => ( ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(B),set(C),image2(B,C,F),A4)),aa(set(B),set(C),image2(B,C,G),B4)) = bot_bot(set(C)) )
         => inj_on(B,C,aa(fun(B,C),fun(B,C),aa(set(B),fun(fun(B,C),fun(B,C)),aTP_Lamp_afl(fun(B,C),fun(set(B),fun(fun(B,C),fun(B,C))),F),A4),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_6623_set__to__map__simp,axiom,
    ! [C: $tType,B: $tType,S: set(product_prod(B,C)),K: B,V2: C] :
      ( inj_on(product_prod(B,C),B,product_fst(B,C),S)
     => ( ( aa(B,option(C),set_to_map(B,C,S),K) = aa(C,option(C),some(C),V2) )
      <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),K),V2)),S) ) ) ).

% set_to_map_simp
tff(fact_6624_image__INT,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(B,C),C6: set(B),A4: set(D),B4: fun(D,set(B)),J: D] :
      ( inj_on(B,C,F,C6)
     => ( ! [X2: D] :
            ( aa(set(D),$o,member(D,X2),A4)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(D,set(B),B4,X2)),C6) )
       => ( aa(set(D),$o,member(D,J),A4)
         => ( aa(set(B),set(C),image2(B,C,F),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),B4),A4))) = 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(B)),fun(D,set(C)),aTP_Lamp_afm(fun(B,C),fun(fun(D,set(B)),fun(D,set(C))),F),B4)),A4)) ) ) ) ) ).

% image_INT
tff(fact_6625_inj__on__funpow__least,axiom,
    ! [B: $tType,N: nat,F: fun(B,B),S2: B] :
      ( ( aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),S2) = S2 )
     => ( ! [M2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
             => ( aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),M2),F),S2) != S2 ) ) )
       => inj_on(nat,B,aa(B,fun(nat,B),aTP_Lamp_afn(fun(B,B),fun(B,fun(nat,B)),F),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_6626_card__le__inj,axiom,
    ! [C: $tType,B: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(C),nat,finite_card(C),B4))
         => ? [F4: fun(B,C)] :
              ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F4),A4)),B4)
              & inj_on(B,C,F4,A4) ) ) ) ) ).

% card_le_inj
tff(fact_6627_card__inj__on__le,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),A4: set(B),B4: set(C)] :
      ( inj_on(B,C,F,A4)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F),A4)),B4)
       => ( aa(set(C),$o,finite_finite2(C),B4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(C),nat,finite_card(C),B4)) ) ) ) ).

% card_inj_on_le
tff(fact_6628_inj__on__iff__card__le,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => ( ? [F5: fun(B,C)] :
              ( inj_on(B,C,F5,A4)
              & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F5),A4)),B4) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(C),nat,finite_card(C),B4)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_6629_card__vimage__inj__on__le,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),D3: set(B),A4: set(C)] :
      ( inj_on(B,C,F,D3)
     => ( aa(set(C),$o,finite_finite2(C),A4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),A4)),D3))),aa(set(C),nat,finite_card(C),A4)) ) ) ).

% card_vimage_inj_on_le
tff(fact_6630_quotient__of__unique,axiom,
    ! [R: rat] :
    ? [X2: product_prod(int,int)] :
      ( ( R = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X2)),aa(product_prod(int,int),int,product_snd(int,int),X2)) )
      & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X2))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X2),aa(product_prod(int,int),int,product_snd(int,int),X2))
      & ! [Y3: product_prod(int,int)] :
          ( ( ( R = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y3)),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y3))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y3),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
         => ( Y3 = X2 ) ) ) ).

% quotient_of_unique
tff(fact_6631_Ex__inj__on__UNION__Sigma,axiom,
    ! [B: $tType,C: $tType,A4: fun(C,set(B)),I5: set(C)] :
    ? [F4: fun(B,product_prod(C,B))] :
      ( inj_on(B,product_prod(C,B),F4,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5)))
      & aa(set(product_prod(C,B)),$o,aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),$o),ord_less_eq(set(product_prod(C,B))),aa(set(B),set(product_prod(C,B)),image2(B,product_prod(C,B),F4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),A4),I5)))),product_Sigma(C,B,I5,A4)) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_6632_map__sorted__distinct__set__unique,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),Xs: list(B),Ys: list(B)] :
          ( inj_on(B,C,F,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),Ys)))
         => ( sorted_wrt(C,ord_less_eq(C),aa(list(B),list(C),map(B,C,F),Xs))
           => ( distinct(C,aa(list(B),list(C),map(B,C,F),Xs))
             => ( sorted_wrt(C,ord_less_eq(C),aa(list(B),list(C),map(B,C,F),Ys))
               => ( distinct(C,aa(list(B),list(C),map(B,C,F),Ys))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_6633_sort__key__inj__key__eq,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Xs: list(B),Ys: list(B),F: fun(B,C)] :
          ( ( mset(B,Xs) = mset(B,Ys) )
         => ( inj_on(B,C,F,aa(list(B),set(B),set2(B),Xs))
           => ( sorted_wrt(C,ord_less_eq(C),aa(list(B),list(C),map(B,C,F),Ys))
             => ( aa(list(B),list(B),linorder_sort_key(B,C,F),Xs) = Ys ) ) ) ) ) ).

% sort_key_inj_key_eq
tff(fact_6634_sum__mult__sum__if__inj,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( semiring_0(D)
     => ! [F: fun(B,D),G: fun(C,D),A4: set(B),B4: set(C)] :
          ( inj_on(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),aa(fun(C,D),fun(B,fun(C,D)),aTP_Lamp_afo(fun(B,D),fun(fun(C,D),fun(B,fun(C,D))),F),G)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
         => ( aa(D,D,aa(D,fun(D,D),times_times(D),aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),F),A4)),aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),G),B4)) = aa(set(D),D,aa(fun(D,D),fun(set(D),D),groups7311177749621191930dd_sum(D,D),id(D)),aa(fun(D,$o),set(D),collect(D),aa(set(C),fun(D,$o),aa(set(B),fun(set(C),fun(D,$o)),aa(fun(C,D),fun(set(B),fun(set(C),fun(D,$o))),aTP_Lamp_afp(fun(B,D),fun(fun(C,D),fun(set(B),fun(set(C),fun(D,$o)))),F),G),A4),B4))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_6635_funpow__inj__finite,axiom,
    ! [B: $tType,P2: fun(B,B),X: B] :
      ( inj_on(B,B,P2,top_top(set(B)))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_afq(fun(B,B),fun(B,fun(B,$o)),P2),X)))
       => ~ ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N2),P2),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_6636_inj__vimage__singleton,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),A2: C] :
      ( inj_on(B,C,F,top_top(set(B)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(set(C),set(C),insert(C,A2),bot_bot(set(C))))),aa(set(B),set(B),insert(B,the(B,aa(C,fun(B,$o),aTP_Lamp_afr(fun(B,C),fun(C,fun(B,$o)),F),A2))),bot_bot(set(B)))) ) ).

% inj_vimage_singleton
tff(fact_6637_inj__on__vimage__singleton,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),A4: set(B),A2: C] :
      ( inj_on(B,C,F,A4)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(set(C),set(C),insert(C,A2),bot_bot(set(C))))),A4)),aa(set(B),set(B),insert(B,the(B,aa(C,fun(B,$o),aa(set(B),fun(C,fun(B,$o)),aTP_Lamp_afs(fun(B,C),fun(set(B),fun(C,fun(B,$o))),F),A4),A2))),bot_bot(set(B)))) ) ).

% inj_on_vimage_singleton
tff(fact_6638_inj__on__map__inv__f,axiom,
    ! [C: $tType,B: $tType,L: list(B),A4: set(B),F: fun(B,C)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),L)),A4)
     => ( inj_on(B,C,F,A4)
       => ( aa(list(C),list(B),map(C,B,inv_on(B,C,F,A4)),aa(list(B),list(C),map(B,C,F),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_6639_Rats__cases_H,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [X: B] :
          ( aa(set(B),$o,member(B,X),field_char_0_Rats(B))
         => ~ ! [A3: int,B3: int] :
                ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
               => ( algebr8660921524188924756oprime(int,A3,B3)
                 => ( X != aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(int,B,ring_1_of_int(B),A3)),aa(int,B,ring_1_of_int(B),B3)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_6640_coprime__int__iff,axiom,
    ! [Ma: nat,N: nat] :
      ( algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),N))
    <=> algebr8660921524188924756oprime(nat,Ma,N) ) ).

% coprime_int_iff
tff(fact_6641_inv__on__f__f,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),A4: set(B),X: B] :
      ( inj_on(B,C,F,A4)
     => ( aa(set(B),$o,member(B,X),A4)
       => ( aa(C,B,inv_on(B,C,F,A4),aa(B,C,F,X)) = X ) ) ) ).

% inv_on_f_f
tff(fact_6642_coprime__nat__abs__left__iff,axiom,
    ! [K: int,N: nat] :
      ( algebr8660921524188924756oprime(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),N)
    <=> algebr8660921524188924756oprime(int,K,aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% coprime_nat_abs_left_iff
tff(fact_6643_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K: int] :
      ( algebr8660921524188924756oprime(nat,N,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),N),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_6644_coprime__common__divisor__nat,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( algebr8660921524188924756oprime(nat,A2,B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),A2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),B2)
         => ( X = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_6645_coprime__Suc__right__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,N)) ).

% coprime_Suc_right_nat
tff(fact_6646_coprime__Suc__left__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,N),N) ).

% coprime_Suc_left_nat
tff(fact_6647_coprime__crossproduct__nat,axiom,
    ! [A2: nat,D2: nat,B2: nat,C2: nat] :
      ( algebr8660921524188924756oprime(nat,A2,D2)
     => ( algebr8660921524188924756oprime(nat,B2,C2)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D2) )
        <=> ( ( A2 = B2 )
            & ( C2 = D2 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_6648_coprime__Suc__0__left,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).

% coprime_Suc_0_left
tff(fact_6649_coprime__Suc__0__right,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_6650_Rats__0,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => aa(set(B),$o,member(B,zero_zero(B)),field_char_0_Rats(B)) ) ).

% Rats_0
tff(fact_6651_Rats__1,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => aa(set(B),$o,member(B,one_one(B)),field_char_0_Rats(B)) ) ).

% Rats_1
tff(fact_6652_Rats__add,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [A2: B,B2: B] :
          ( aa(set(B),$o,member(B,A2),field_char_0_Rats(B))
         => ( aa(set(B),$o,member(B,B2),field_char_0_Rats(B))
           => aa(set(B),$o,member(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),field_char_0_Rats(B)) ) ) ) ).

% Rats_add
tff(fact_6653_Rats__of__nat,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [N: nat] : aa(set(B),$o,member(B,aa(nat,B,semiring_1_of_nat(B),N)),field_char_0_Rats(B)) ) ).

% Rats_of_nat
tff(fact_6654_inv__on__f__range,axiom,
    ! [B: $tType,C: $tType,Y: B,F: fun(C,B),A4: set(C)] :
      ( aa(set(B),$o,member(B,Y),aa(set(C),set(B),image2(C,B,F),A4))
     => aa(set(C),$o,member(C,aa(B,C,inv_on(C,B,F,A4),Y)),A4) ) ).

% inv_on_f_range
tff(fact_6655_f__inv__on__f,axiom,
    ! [C: $tType,B: $tType,Y: B,F: fun(C,B),A4: set(C)] :
      ( aa(set(B),$o,member(B,Y),aa(set(C),set(B),image2(C,B,F),A4))
     => ( aa(C,B,F,aa(B,C,inv_on(C,B,F,A4),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_6656_mult__inj__if__coprime__nat,axiom,
    ! [B: $tType,C: $tType,F: fun(B,nat),A4: set(B),G: fun(C,nat),B4: set(C)] :
      ( inj_on(B,nat,F,A4)
     => ( inj_on(C,nat,G,B4)
       => ( ! [A3: B,B3: C] :
              ( aa(set(B),$o,member(B,A3),A4)
             => ( aa(set(C),$o,member(C,B3),B4)
               => algebr8660921524188924756oprime(nat,aa(B,nat,F,A3),aa(C,nat,G,B3)) ) )
         => inj_on(product_prod(B,C),nat,aa(fun(B,fun(C,nat)),fun(product_prod(B,C),nat),product_case_prod(B,C,nat),aa(fun(C,nat),fun(B,fun(C,nat)),aTP_Lamp_aft(fun(B,nat),fun(fun(C,nat),fun(B,fun(C,nat))),F),G)),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_6657_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_6658_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_6659_inv__on__def,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),A4: set(B),X: C] : aa(C,B,inv_on(B,C,F,A4),X) = fChoice(B,aa(C,fun(B,$o),aa(set(B),fun(C,fun(B,$o)),aTP_Lamp_afs(fun(B,C),fun(set(B),fun(C,fun(B,$o))),F),A4),X)) ).

% inv_on_def
tff(fact_6660_inj__on__apfst,axiom,
    ! [C: $tType,D: $tType,B: $tType,F: fun(B,D),A4: set(B)] :
      ( inj_on(product_prod(B,C),product_prod(D,C),product_apfst(B,D,C,F),product_Sigma(B,C,A4,aTP_Lamp_vv(B,set(C))))
    <=> inj_on(B,D,F,A4) ) ).

% inj_on_apfst
tff(fact_6661_If__the__inv__into__f__f,axiom,
    ! [C: $tType,B: $tType,I: B,C6: set(B),G: fun(B,C),X: B] :
      ( aa(set(B),$o,member(B,I),C6)
     => ( inj_on(B,C,G,C6)
       => ( aa(B,B,aa(fun(B,C),fun(B,B),aa(fun(C,B),fun(fun(B,C),fun(B,B)),comp(C,B,B),aa(B,fun(C,B),aa(fun(B,C),fun(B,fun(C,B)),aTP_Lamp_afu(set(B),fun(fun(B,C),fun(B,fun(C,B))),C6),G),X)),G),I) = aa(B,B,id(B),I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_6662_the__inv__into__def,axiom,
    ! [C: $tType,B: $tType,A4: set(C),F: fun(C,B),X4: B] : the_inv_into(C,B,A4,F,X4) = the(C,aa(B,fun(C,$o),aa(fun(C,B),fun(B,fun(C,$o)),aTP_Lamp_afv(set(C),fun(fun(C,B),fun(B,fun(C,$o))),A4),F),X4)) ).

% the_inv_into_def
tff(fact_6663_If__the__inv__into__in__Func,axiom,
    ! [C: $tType,B: $tType,G: fun(B,C),C6: set(B),B4: set(B),X: B] :
      ( inj_on(B,C,G,C6)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))
       => aa(set(fun(C,B)),$o,member(fun(C,B),aa(B,fun(C,B),aa(set(B),fun(B,fun(C,B)),aTP_Lamp_afw(fun(B,C),fun(set(B),fun(B,fun(C,B))),G),C6),X)),bNF_Wellorder_Func(C,B,top_top(set(C)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6664_card__quotient__disjoint,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( inj_on(B,set(set(B)),aTP_Lamp_afx(set(product_prod(B,B)),fun(B,set(set(B))),R),A4)
       => ( aa(set(set(B)),nat,finite_card(set(B)),equiv_quotient(B,A4,R)) = aa(set(B),nat,finite_card(B),A4) ) ) ) ).

% card_quotient_disjoint
tff(fact_6665_quotient__diff1,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),A4: set(B),A2: B] :
      ( inj_on(B,set(set(B)),aTP_Lamp_afx(set(product_prod(B,B)),fun(B,set(set(B))),R),A4)
     => ( aa(set(B),$o,member(B,A2),A4)
       => ( equiv_quotient(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))),R) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),minus_minus(set(set(B))),equiv_quotient(B,A4,R)),equiv_quotient(B,aa(set(B),set(B),insert(B,A2),bot_bot(set(B))),R)) ) ) ) ).

% quotient_diff1
tff(fact_6666_Func__empty,axiom,
    ! [C: $tType,B: $tType,B4: set(C)] : bNF_Wellorder_Func(B,C,bot_bot(set(B)),B4) = aa(set(fun(B,C)),set(fun(B,C)),insert(fun(B,C),aTP_Lamp_afy(B,C)),bot_bot(set(fun(B,C)))) ).

% Func_empty
tff(fact_6667_finite__equiv__class,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B)),X5: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4)))
       => ( aa(set(set(B)),$o,member(set(B),X5),equiv_quotient(B,A4,R))
         => aa(set(B),$o,finite_finite2(B),X5) ) ) ) ).

% finite_equiv_class
tff(fact_6668_finite__quotient,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4)))
       => aa(set(set(B)),$o,finite_finite2(set(B)),equiv_quotient(B,A4,R)) ) ) ).

% finite_quotient
tff(fact_6669_quotient__def,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B))] : equiv_quotient(B,A4,R) = aa(set(set(set(B))),set(set(B)),complete_Sup_Sup(set(set(B))),aa(set(B),set(set(set(B))),image2(B,set(set(B)),aTP_Lamp_afz(set(product_prod(B,B)),fun(B,set(set(B))),R)),A4)) ).

% quotient_def
tff(fact_6670_positive_Orsp,axiom,
    aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,ratrel,fequal($o)),aTP_Lamp_tr(product_prod(int,int),$o)),aTP_Lamp_tr(product_prod(int,int),$o)) ).

% positive.rsp
tff(fact_6671_pair__vimage__is__Image,axiom,
    ! [B: $tType,C: $tType,U: C,E6: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(B),aa(fun(B,product_prod(C,B)),fun(set(product_prod(C,B)),set(B)),vimage(B,product_prod(C,B)),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),U)),E6) = aa(set(C),set(B),image(C,B,E6),aa(set(C),set(C),insert(C,U),bot_bot(set(C)))) ).

% pair_vimage_is_Image
tff(fact_6672_Image__UN,axiom,
    ! [C: $tType,B: $tType,D: $tType,R: set(product_prod(C,B)),B4: fun(D,set(C)),A4: set(D)] : aa(set(C),set(B),image(C,B,R),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_aga(set(product_prod(C,B)),fun(fun(D,set(C)),fun(D,set(B))),R),B4)),A4)) ).

% Image_UN
tff(fact_6673_id__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] : aa(fun(C,C),$o,aa(fun(B,B),fun(fun(C,C),$o),bNF_rel_fun(B,C,B,C,A4,A4),id(B)),id(C)) ).

% id_transfer
tff(fact_6674_finite__Image__subset,axiom,
    ! [B: $tType,C: $tType,A4: set(product_prod(C,B)),B4: set(C),C6: set(product_prod(C,B))] :
      ( aa(set(B),$o,finite_finite2(B),aa(set(C),set(B),image(C,B,A4),B4))
     => ( aa(set(product_prod(C,B)),$o,aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),$o),ord_less_eq(set(product_prod(C,B))),C6),A4)
       => aa(set(B),$o,finite_finite2(B),aa(set(C),set(B),image(C,B,C6),B4)) ) ) ).

% finite_Image_subset
tff(fact_6675_transfer__rule__of__int,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ring_1(C)
        & ring_1(B) )
     => ! [R2: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,zero_zero(B)),zero_zero(C))
         => ( aa(C,$o,aa(B,fun(C,$o),R2,one_one(B)),one_one(C))
           => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),R2,bNF_rel_fun(B,C,B,C,R2,R2)),plus_plus(B)),plus_plus(C))
             => ( aa(fun(C,C),$o,aa(fun(B,B),fun(fun(C,C),$o),bNF_rel_fun(B,C,B,C,R2,R2),uminus_uminus(B)),uminus_uminus(C))
               => aa(fun(int,C),$o,aa(fun(int,B),fun(fun(int,C),$o),bNF_rel_fun(int,int,B,C,fequal(int),R2),ring_1_of_int(B)),ring_1_of_int(C)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_6676_comp__transfer,axiom,
    ! [B: $tType,C: $tType,F6: $tType,G4: $tType,E: $tType,D: $tType,B4: fun(B,fun(D,$o)),C6: fun(C,fun(E,$o)),A4: fun(F6,fun(G4,$o))] : aa(fun(fun(D,E),fun(fun(G4,D),fun(G4,E))),$o,aa(fun(fun(B,C),fun(fun(F6,B),fun(F6,C))),fun(fun(fun(D,E),fun(fun(G4,D),fun(G4,E))),$o),bNF_rel_fun(fun(B,C),fun(D,E),fun(fun(F6,B),fun(F6,C)),fun(fun(G4,D),fun(G4,E)),bNF_rel_fun(B,D,C,E,B4,C6),bNF_rel_fun(fun(F6,B),fun(G4,D),fun(F6,C),fun(G4,E),bNF_rel_fun(F6,G4,B,D,A4,B4),bNF_rel_fun(F6,G4,C,E,A4,C6))),comp(B,C,F6)),comp(D,E,G4)) ).

% comp_transfer
tff(fact_6677_If__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] : aa(fun($o,fun(C,fun(C,C))),$o,aa(fun($o,fun(B,fun(B,B))),fun(fun($o,fun(C,fun(C,C))),$o),bNF_rel_fun($o,$o,fun(B,fun(B,B)),fun(C,fun(C,C)),fequal($o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),A4,bNF_rel_fun(B,C,B,C,A4,A4))),if(B)),if(C)) ).

% If_transfer
tff(fact_6678_rel__fun__def__butlast,axiom,
    ! [B: $tType,E: $tType,D: $tType,F6: $tType,G4: $tType,C: $tType,R2: fun(B,fun(C,$o)),S: fun(D,fun(F6,$o)),T4: fun(E,fun(G4,$o)),F: fun(B,fun(D,E)),G: fun(C,fun(F6,G4))] :
      ( aa(fun(C,fun(F6,G4)),$o,aa(fun(B,fun(D,E)),fun(fun(C,fun(F6,G4)),$o),bNF_rel_fun(B,C,fun(D,E),fun(F6,G4),R2,bNF_rel_fun(D,F6,E,G4,S,T4)),F),G)
    <=> ! [X3: B,Y5: C] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,X3),Y5)
         => aa(fun(F6,G4),$o,aa(fun(D,E),fun(fun(F6,G4),$o),bNF_rel_fun(D,F6,E,G4,S,T4),aa(B,fun(D,E),F,X3)),aa(C,fun(F6,G4),G,Y5)) ) ) ).

% rel_fun_def_butlast
tff(fact_6679_power__transfer,axiom,
    ! [B: $tType,C: $tType] :
      ( ( power(C)
        & power(B) )
     => ! [R2: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,one_one(B)),one_one(C))
         => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),R2,bNF_rel_fun(B,C,B,C,R2,R2)),times_times(B)),times_times(C))
           => aa(fun(C,fun(nat,C)),$o,aa(fun(B,fun(nat,B)),fun(fun(C,fun(nat,C)),$o),bNF_rel_fun(B,C,fun(nat,B),fun(nat,C),R2,bNF_rel_fun(nat,nat,B,C,fequal(nat),R2)),power_power(B)),power_power(C)) ) ) ) ).

% power_transfer
tff(fact_6680_transfer__rule__numeral,axiom,
    ! [B: $tType,C: $tType] :
      ( ( monoid_add(C)
        & semiring_numeral(C)
        & monoid_add(B)
        & semiring_numeral(B) )
     => ! [R2: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,zero_zero(B)),zero_zero(C))
         => ( aa(C,$o,aa(B,fun(C,$o),R2,one_one(B)),one_one(C))
           => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),R2,bNF_rel_fun(B,C,B,C,R2,R2)),plus_plus(B)),plus_plus(C))
             => aa(fun(num,C),$o,aa(fun(num,B),fun(fun(num,C),$o),bNF_rel_fun(num,num,B,C,fequal(num),R2),numeral_numeral(B)),numeral_numeral(C)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_6681_rtrancl__image__unfold__right,axiom,
    ! [B: $tType,E6: set(product_prod(B,B)),V: set(B)] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,E6),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),V))),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),V)) ).

% rtrancl_image_unfold_right
tff(fact_6682_rtrancl__reachable__induct,axiom,
    ! [B: $tType,I5: set(B),INV: set(B),E6: set(product_prod(B,B))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),I5),INV)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,E6),INV)),INV)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),I5)),INV) ) ) ).

% rtrancl_reachable_induct
tff(fact_6683_rtrancl__image__advance__rtrancl,axiom,
    ! [B: $tType,Q2: B,R2: set(product_prod(B,B)),Q0: set(B),X: B] :
      ( aa(set(B),$o,member(B,Q2),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),Q0))
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),X)),transitive_rtrancl(B,R2))
       => aa(set(B),$o,member(B,X),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),Q0)) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_6684_rtrancl__image__advance,axiom,
    ! [B: $tType,Q2: B,R2: set(product_prod(B,B)),Q0: set(B),X: B] :
      ( aa(set(B),$o,member(B,Q2),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),Q0))
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Q2),X)),R2)
       => aa(set(B),$o,member(B,X),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),Q0)) ) ) ).

% rtrancl_image_advance
tff(fact_6685_trancl__Image__unfold__right,axiom,
    ! [B: $tType,E6: set(product_prod(B,B)),S: set(B)] : aa(set(B),set(B),image(B,B,transitive_trancl(B,E6)),S) = aa(set(B),set(B),image(B,B,E6),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),S)) ).

% trancl_Image_unfold_right
tff(fact_6686_trancl__Image__unfold__left,axiom,
    ! [B: $tType,E6: set(product_prod(B,B)),S: set(B)] : aa(set(B),set(B),image(B,B,transitive_trancl(B,E6)),S) = aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),aa(set(B),set(B),image(B,B,E6),S)) ).

% trancl_Image_unfold_left
tff(fact_6687_Image__empty__rtrancl__Image__id,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),V2: B] :
      ( ( aa(set(B),set(B),image(B,B,R2),aa(set(B),set(B),insert(B,V2),bot_bot(set(B)))) = bot_bot(set(B)) )
     => ( aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),aa(set(B),set(B),insert(B,V2),bot_bot(set(B)))) = aa(set(B),set(B),insert(B,V2),bot_bot(set(B))) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_6688_Image__empty__trancl__Image__empty,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),V2: B] :
      ( ( aa(set(B),set(B),image(B,B,R2),aa(set(B),set(B),insert(B,V2),bot_bot(set(B)))) = bot_bot(set(B)) )
     => ( aa(set(B),set(B),image(B,B,transitive_trancl(B,R2)),aa(set(B),set(B),insert(B,V2),bot_bot(set(B)))) = bot_bot(set(B)) ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_6689_reachable__mono,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),R7: set(product_prod(B,B)),X5: set(B),X10: set(B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),R7)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),X10)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),X5)),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R7)),X10)) ) ) ).

% reachable_mono
tff(fact_6690_Image__subset__snd__image,axiom,
    ! [B: $tType,C: $tType,A4: set(product_prod(C,B)),B4: set(C)] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(C),set(B),image(C,B,A4),B4)),aa(set(product_prod(C,B)),set(B),image2(product_prod(C,B),B,product_snd(C,B)),A4)) ).

% Image_subset_snd_image
tff(fact_6691_trancl__image__by__rtrancl,axiom,
    ! [B: $tType,E6: set(product_prod(B,B)),Vi: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),image(B,B,transitive_trancl(B,E6)),Vi)),Vi) = aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),Vi) ).

% trancl_image_by_rtrancl
tff(fact_6692_Image__singleton,axiom,
    ! [B: $tType,C: $tType,R: set(product_prod(C,B)),A2: C] : aa(set(C),set(B),image(C,B,R),aa(set(C),set(C),insert(C,A2),bot_bot(set(C)))) = aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aTP_Lamp_mh(set(product_prod(C,B)),fun(C,fun(B,$o)),R),A2)) ).

% Image_singleton
tff(fact_6693_Image__subset,axiom,
    ! [B: $tType,C: $tType,R: set(product_prod(B,C)),A4: set(B),B4: set(C),C6: set(B)] :
      ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),R),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)))
     => aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,R),C6)),B4) ) ).

% Image_subset
tff(fact_6694_Image__INT__subset,axiom,
    ! [C: $tType,B: $tType,D: $tType,R: set(product_prod(C,B)),B4: fun(D,set(C)),A4: set(D)] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(C),set(B),image(C,B,R),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4)))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_aga(set(product_prod(C,B)),fun(fun(D,set(C)),fun(D,set(B))),R),B4)),A4))) ).

% Image_INT_subset
tff(fact_6695_rtrancl__apply__insert,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),X: B,S: set(B)] : aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),aa(set(B),set(B),insert(B,X),S)) = aa(set(B),set(B),insert(B,X),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,R2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),S),aa(set(B),set(B),image(B,B,R2),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))))) ).

% rtrancl_apply_insert
tff(fact_6696_E__closed__restr__reach__cases,axiom,
    ! [B: $tType,U: B,V2: B,E6: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_rtrancl(B,E6))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,E6),R2)),R2)
       => ( ~ aa(set(B),$o,member(B,V2),R2)
         => ~ ( ~ aa(set(B),$o,member(B,U),R2)
             => ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_rtrancl(B,rel_restrict(B,E6,R2))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_6697_rel__restrict__tranclI,axiom,
    ! [B: $tType,X: B,Y: B,E6: set(product_prod(B,B)),R2: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),transitive_trancl(B,E6))
     => ( ~ aa(set(B),$o,member(B,X),R2)
       => ( ~ aa(set(B),$o,member(B,Y),R2)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,E6),R2)),R2)
           => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X),Y)),transitive_trancl(B,rel_restrict(B,E6,R2))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_6698_Image__eq__UN,axiom,
    ! [B: $tType,C: $tType,R: set(product_prod(C,B)),B4: set(C)] : aa(set(C),set(B),image(C,B,R),B4) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_agb(set(product_prod(C,B)),fun(C,set(B)),R)),B4)) ).

% Image_eq_UN
tff(fact_6699_Sigma__Image,axiom,
    ! [B: $tType,C: $tType,A4: set(C),B4: fun(C,set(B)),X5: set(C)] : aa(set(C),set(B),image(C,B,product_Sigma(C,B,A4,B4)),X5) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),X5),A4))) ).

% Sigma_Image
tff(fact_6700_UN__Image,axiom,
    ! [C: $tType,B: $tType,D: $tType,X5: fun(D,set(product_prod(C,B))),I5: set(D),S: set(C)] : aa(set(C),set(B),image(C,B,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)),X5),I5))),S) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(set(C),fun(D,set(B)),aTP_Lamp_agc(fun(D,set(product_prod(C,B))),fun(set(C),fun(D,set(B))),X5),S)),I5)) ).

% UN_Image
tff(fact_6701_finite__reachable__advance,axiom,
    ! [B: $tType,E6: set(product_prod(B,B)),V0: B,V2: B] :
      ( aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),aa(set(B),set(B),insert(B,V0),bot_bot(set(B)))))
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),V0),V2)),transitive_rtrancl(B,E6))
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),aa(set(B),set(B),insert(B,V2),bot_bot(set(B))))) ) ) ).

% finite_reachable_advance
tff(fact_6702_rtrancl__Image__advance__ss,axiom,
    ! [B: $tType,U: B,V2: B,E6: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),E6)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),aa(set(B),set(B),insert(B,V2),bot_bot(set(B))))),aa(set(B),set(B),image(B,B,transitive_rtrancl(B,E6)),aa(set(B),set(B),insert(B,U),bot_bot(set(B))))) ) ).

% rtrancl_Image_advance_ss
tff(fact_6703_trancl__Image__advance__ss,axiom,
    ! [B: $tType,U: B,V2: B,E6: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),E6)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,transitive_trancl(B,E6)),aa(set(B),set(B),insert(B,V2),bot_bot(set(B))))),aa(set(B),set(B),image(B,B,transitive_trancl(B,E6)),aa(set(B),set(B),insert(B,U),bot_bot(set(B))))) ) ).

% trancl_Image_advance_ss
tff(fact_6704_trancl__restrict__reachable,axiom,
    ! [B: $tType,U: B,V2: B,E6: set(product_prod(B,B)),S: set(B)] :
      ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_trancl(B,E6))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),image(B,B,E6),S)),S)
       => ( aa(set(B),$o,member(B,U),S)
         => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),U),V2)),transitive_trancl(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),E6),product_Sigma(B,B,S,aTP_Lamp_zv(set(B),fun(B,set(B)),S))))) ) ) ) ).

% trancl_restrict_reachable
tff(fact_6705_of__rat_Orsp,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => aa(fun(product_prod(int,int),B),$o,aa(fun(product_prod(int,int),B),fun(fun(product_prod(int,int),B),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),B,B,ratrel,fequal(B)),aTP_Lamp_tx(product_prod(int,int),B)),aTP_Lamp_tx(product_prod(int,int),B)) ) ).

% of_rat.rsp
tff(fact_6706_Image__fold,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(B)] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),R2)
     => ( aa(set(B),set(C),image(B,C,R2),S) = finite_fold(product_prod(B,C),set(C),aa(fun(B,fun(C,fun(set(C),set(C)))),fun(product_prod(B,C),fun(set(C),set(C))),product_case_prod(B,C,fun(set(C),set(C))),aTP_Lamp_acl(set(B),fun(B,fun(C,fun(set(C),set(C)))),S)),bot_bot(set(C)),R2) ) ) ).

% Image_fold
tff(fact_6707_fun_Oin__rel,axiom,
    ! [C: $tType,D: $tType,B: $tType,R2: fun(C,fun(D,$o)),A2: fun(B,C),B2: fun(B,D)] :
      ( aa(fun(B,D),$o,aa(fun(B,C),fun(fun(B,D),$o),bNF_rel_fun(B,B,C,D,fequal(B),R2),A2),B2)
    <=> ? [Z5: fun(B,product_prod(C,D))] :
          ( aa(set(fun(B,product_prod(C,D))),$o,member(fun(B,product_prod(C,D)),Z5),aa(fun(fun(B,product_prod(C,D)),$o),set(fun(B,product_prod(C,D))),collect(fun(B,product_prod(C,D))),aTP_Lamp_agd(fun(C,fun(D,$o)),fun(fun(B,product_prod(C,D)),$o),R2)))
          & ( aa(fun(B,product_prod(C,D)),fun(B,C),aa(fun(product_prod(C,D),C),fun(fun(B,product_prod(C,D)),fun(B,C)),comp(product_prod(C,D),C,B),product_fst(C,D)),Z5) = A2 )
          & ( aa(fun(B,product_prod(C,D)),fun(B,D),aa(fun(product_prod(C,D),D),fun(fun(B,product_prod(C,D)),fun(B,D)),comp(product_prod(C,D),D,B),product_snd(C,D)),Z5) = B2 ) ) ) ).

% fun.in_rel
tff(fact_6708_transfer__rule__of__nat,axiom,
    ! [B: $tType,C: $tType] :
      ( ( semiring_1(C)
        & semiring_1(B) )
     => ! [R2: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,zero_zero(B)),zero_zero(C))
         => ( aa(C,$o,aa(B,fun(C,$o),R2,one_one(B)),one_one(C))
           => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),R2,bNF_rel_fun(B,C,B,C,R2,R2)),plus_plus(B)),plus_plus(C))
             => aa(fun(nat,C),$o,aa(fun(nat,B),fun(fun(nat,C),$o),bNF_rel_fun(nat,nat,B,C,fequal(nat),R2),semiring_1_of_nat(B)),semiring_1_of_nat(C)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_6709_Fract_Orsp,axiom,
    aa(fun(int,fun(int,product_prod(int,int))),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),$o),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_tu(int,fun(int,product_prod(int,int)))),aTP_Lamp_tu(int,fun(int,product_prod(int,int)))) ).

% Fract.rsp
tff(fact_6710_less__eq__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less_eq(int)),ord_less_eq(int)) ).

% less_eq_integer.rsp
tff(fact_6711_less__eq__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less_eq(nat)),ord_less_eq(nat)) ).

% less_eq_natural.rsp
tff(fact_6712_uminus__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),uminus_uminus(int)),uminus_uminus(int)) ).

% uminus_integer.rsp
tff(fact_6713_minus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),minus_minus(nat)),minus_minus(nat)) ).

% minus_natural.rsp
tff(fact_6714_minus__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),minus_minus(int)),minus_minus(int)) ).

% minus_integer.rsp
tff(fact_6715_sub_Orsp,axiom,
    aa(fun(num,fun(num,int)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),$o),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_uc(num,fun(num,int))),aTP_Lamp_uc(num,fun(num,int))) ).

% sub.rsp
tff(fact_6716_gcd__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),gcd_gcd(int)),gcd_gcd(int)) ).

% gcd_integer.rsp
tff(fact_6717_integer__of__natural_Orsp,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),semiring_1_of_nat(int)),semiring_1_of_nat(int)) ).

% integer_of_natural.rsp
tff(fact_6718_dup_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_ty(int,int)),aTP_Lamp_ty(int,int)) ).

% dup.rsp
tff(fact_6719_sgn__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),sgn_sgn(int)),sgn_sgn(int)) ).

% sgn_integer.rsp
tff(fact_6720_plus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),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_6721_plus__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),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_6722_num_Ocase__transfer,axiom,
    ! [B: $tType,C: $tType,S: fun(B,fun(C,$o))] : aa(fun(C,fun(fun(num,C),fun(fun(num,C),fun(num,C)))),$o,aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),fun(fun(C,fun(fun(num,C),fun(fun(num,C),fun(num,C)))),$o),bNF_rel_fun(B,C,fun(fun(num,B),fun(fun(num,B),fun(num,B))),fun(fun(num,C),fun(fun(num,C),fun(num,C))),S,bNF_rel_fun(fun(num,B),fun(num,C),fun(fun(num,B),fun(num,B)),fun(fun(num,C),fun(num,C)),bNF_rel_fun(num,num,B,C,fequal(num),S),bNF_rel_fun(fun(num,B),fun(num,C),fun(num,B),fun(num,C),bNF_rel_fun(num,num,B,C,fequal(num),S),bNF_rel_fun(num,num,B,C,fequal(num),S)))),case_num(B)),case_num(C)) ).

% num.case_transfer
tff(fact_6723_bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),fequal(fun(nat,$o))),bit_se5641148757651400278ts_bit(nat)),bit_se5641148757651400278ts_bit(nat)) ).

% bit_natural.rsp
tff(fact_6724_Suc_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),suc),suc) ).

% Suc.rsp
tff(fact_6725_natural__of__integer_Orsp,axiom,
    aa(fun(int,nat),$o,aa(fun(int,nat),fun(fun(int,nat),$o),bNF_rel_fun(int,int,nat,nat,fequal(int),fequal(nat)),nat2),nat2) ).

% natural_of_integer.rsp
tff(fact_6726_not__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),bit_ri4277139882892585799ns_not(int)),bit_ri4277139882892585799ns_not(int)) ).

% not_integer.rsp
tff(fact_6727_and__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5824344872417868541ns_and(nat)),bit_se5824344872417868541ns_and(nat)) ).

% and_natural.rsp
tff(fact_6728_and__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5824344872417868541ns_and(int)),bit_se5824344872417868541ns_and(int)) ).

% and_integer.rsp
tff(fact_6729_take__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se2584673776208193580ke_bit(nat)),bit_se2584673776208193580ke_bit(nat)) ).

% take_bit_natural.rsp
tff(fact_6730_take__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2584673776208193580ke_bit(int)),bit_se2584673776208193580ke_bit(int)) ).

% take_bit_integer.rsp
tff(fact_6731_modulo__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),modulo_modulo(nat)),modulo_modulo(nat)) ).

% modulo_natural.rsp
tff(fact_6732_modulo__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),modulo_modulo(int)),modulo_modulo(int)) ).

% modulo_integer.rsp
tff(fact_6733_unset__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se2638667681897837118et_bit(nat)),bit_se2638667681897837118et_bit(nat)) ).

% unset_bit_natural.rsp
tff(fact_6734_unset__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(int)) ).

% unset_bit_integer.rsp
tff(fact_6735_set__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5668285175392031749et_bit(nat)),bit_se5668285175392031749et_bit(nat)) ).

% set_bit_natural.rsp
tff(fact_6736_set__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(int)) ).

% set_bit_integer.rsp
tff(fact_6737_flip__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(int)) ).

% flip_bit_integer.rsp
tff(fact_6738_flip__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se8732182000553998342ip_bit(nat)),bit_se8732182000553998342ip_bit(nat)) ).

% flip_bit_natural.rsp
tff(fact_6739_push__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se4730199178511100633sh_bit(nat)),bit_se4730199178511100633sh_bit(nat)) ).

% push_bit_natural.rsp
tff(fact_6740_push__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se4730199178511100633sh_bit(int)),bit_se4730199178511100633sh_bit(int)) ).

% push_bit_integer.rsp
tff(fact_6741_xor__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5824344971392196577ns_xor(nat)),bit_se5824344971392196577ns_xor(nat)) ).

% xor_natural.rsp
tff(fact_6742_xor__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5824344971392196577ns_xor(int)),bit_se5824344971392196577ns_xor(int)) ).

% xor_integer.rsp
tff(fact_6743_bit__integer_Orsp,axiom,
    aa(fun(int,fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(int,fun(nat,$o)),$o),bNF_rel_fun(int,int,fun(nat,$o),fun(nat,$o),fequal(int),fequal(fun(nat,$o))),bit_se5641148757651400278ts_bit(int)),bit_se5641148757651400278ts_bit(int)) ).

% bit_integer.rsp
tff(fact_6744_or__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se1065995026697491101ons_or(int)),bit_se1065995026697491101ons_or(int)) ).

% or_integer.rsp
tff(fact_6745_or__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se1065995026697491101ons_or(nat)),bit_se1065995026697491101ons_or(nat)) ).

% or_natural.rsp
tff(fact_6746_drop__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se4197421643247451524op_bit(int)),bit_se4197421643247451524op_bit(int)) ).

% drop_bit_integer.rsp
tff(fact_6747_drop__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se4197421643247451524op_bit(nat)),bit_se4197421643247451524op_bit(nat)) ).

% drop_bit_natural.rsp
tff(fact_6748_mask__integer_Orsp,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),bit_se2239418461657761734s_mask(int)),bit_se2239418461657761734s_mask(int)) ).

% mask_integer.rsp
tff(fact_6749_mask__natural_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),bit_se2239418461657761734s_mask(nat)),bit_se2239418461657761734s_mask(nat)) ).

% mask_natural.rsp
tff(fact_6750_abs__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),abs_abs(int)),abs_abs(int)) ).

% abs_integer.rsp
tff(fact_6751_divide__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),divide_divide(int)),divide_divide(int)) ).

% divide_integer.rsp
tff(fact_6752_divide__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),divide_divide(nat)),divide_divide(nat)) ).

% divide_natural.rsp
tff(fact_6753_times__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),times_times(nat)),times_times(nat)) ).

% times_natural.rsp
tff(fact_6754_times__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),times_times(int)),times_times(int)) ).

% times_integer.rsp
tff(fact_6755_less__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less(int)),ord_less(int)) ).

% less_integer.rsp
tff(fact_6756_less__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less(nat)),ord_less(nat)) ).

% less_natural.rsp
tff(fact_6757_transfer__rule__of__bool,axiom,
    ! [B: $tType,C: $tType] :
      ( ( zero_neq_one(C)
        & zero_neq_one(B) )
     => ! [R2: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,zero_zero(B)),zero_zero(C))
         => ( aa(C,$o,aa(B,fun(C,$o),R2,one_one(B)),one_one(C))
           => aa(fun($o,C),$o,aa(fun($o,B),fun(fun($o,C),$o),bNF_rel_fun($o,$o,B,C,fequal($o),R2),zero_neq_one_of_bool(B)),zero_neq_one_of_bool(C)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_6758_times__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),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_tw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_tw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat.rsp
tff(fact_6759_plus__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),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_ts(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_ts(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat.rsp
tff(fact_6760_num__of__integer_Orsp,axiom,
    aa(fun(int,num),$o,aa(fun(int,num),fun(fun(int,num),$o),bNF_rel_fun(int,int,num,num,fequal(int),fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)) ).

% num_of_integer.rsp
tff(fact_6761_fun_Omap__ident,axiom,
    ! [C: $tType,B: $tType,Ta: fun(B,C)] : aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aTP_Lamp_rv(C,C)),Ta) = Ta ).

% fun.map_ident
tff(fact_6762_predicate2__transferD,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,R12: fun(B,fun(C,$o)),R23: fun(D,fun(E,$o)),Pa: fun(B,fun(D,$o)),Q: fun(C,fun(E,$o)),A2: product_prod(B,C),A4: set(product_prod(B,C)),B2: product_prod(D,E),B4: set(product_prod(D,E))] :
      ( aa(fun(C,fun(E,$o)),$o,aa(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),$o),bNF_rel_fun(B,C,fun(D,$o),fun(E,$o),R12,bNF_rel_fun(D,E,$o,$o,R23,fequal($o))),Pa),Q)
     => ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),A2),A4)
       => ( aa(set(product_prod(D,E)),$o,member(product_prod(D,E),B2),B4)
         => ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),A4),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R12)))
           => ( aa(set(product_prod(D,E)),$o,aa(set(product_prod(D,E)),fun(set(product_prod(D,E)),$o),ord_less_eq(set(product_prod(D,E))),B4),aa(fun(product_prod(D,E),$o),set(product_prod(D,E)),collect(product_prod(D,E)),aa(fun(D,fun(E,$o)),fun(product_prod(D,E),$o),product_case_prod(D,E,$o),R23)))
             => ( aa(D,$o,aa(B,fun(D,$o),Pa,aa(product_prod(B,C),B,product_fst(B,C),A2)),aa(product_prod(D,E),D,product_fst(D,E),B2))
              <=> aa(E,$o,aa(C,fun(E,$o),Q,aa(product_prod(B,C),C,product_snd(B,C),A2)),aa(product_prod(D,E),E,product_snd(D,E),B2)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_6763_prod_Omap__ident,axiom,
    ! [C: $tType,B: $tType,Ta: product_prod(B,C)] : aa(product_prod(B,C),product_prod(B,C),product_map_prod(B,B,C,C,aTP_Lamp_il(B,B),aTP_Lamp_rv(C,C)),Ta) = Ta ).

% prod.map_ident
tff(fact_6764_uminus__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_tv(product_prod(int,int),product_prod(int,int))),aTP_Lamp_tv(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat.rsp
tff(fact_6765_inverse__rat_Orsp,axiom,
    aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_tt(product_prod(int,int),product_prod(int,int))),aTP_Lamp_tt(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat.rsp
tff(fact_6766_rec__nat__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] : aa(fun(C,fun(fun(nat,fun(C,C)),fun(nat,C))),$o,aa(fun(B,fun(fun(nat,fun(B,B)),fun(nat,B))),fun(fun(C,fun(fun(nat,fun(C,C)),fun(nat,C))),$o),bNF_rel_fun(B,C,fun(fun(nat,fun(B,B)),fun(nat,B)),fun(fun(nat,fun(C,C)),fun(nat,C)),A4,bNF_rel_fun(fun(nat,fun(B,B)),fun(nat,fun(C,C)),fun(nat,B),fun(nat,C),bNF_rel_fun(nat,nat,fun(B,B),fun(C,C),fequal(nat),bNF_rel_fun(B,C,B,C,A4,A4)),bNF_rel_fun(nat,nat,B,C,fequal(nat),A4))),rec_nat(B)),rec_nat(C)) ).

% rec_nat_transfer
tff(fact_6767_fun_Orel__map_I1_J,axiom,
    ! [E: $tType,C: $tType,D: $tType,B: $tType,Sb: fun(C,fun(D,$o)),I: fun(E,C),X: fun(B,E),Y: fun(B,D)] :
      ( aa(fun(B,D),$o,aa(fun(B,C),fun(fun(B,D),$o),bNF_rel_fun(B,B,C,D,fequal(B),Sb),aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),I),X)),Y)
    <=> aa(fun(B,D),$o,aa(fun(B,E),fun(fun(B,D),$o),bNF_rel_fun(B,B,E,D,fequal(B),aa(fun(E,C),fun(E,fun(D,$o)),aTP_Lamp_age(fun(C,fun(D,$o)),fun(fun(E,C),fun(E,fun(D,$o))),Sb),I)),X),Y) ) ).

% fun.rel_map(1)
tff(fact_6768_fun_Orel__map_I2_J,axiom,
    ! [C: $tType,D: $tType,E: $tType,B: $tType,Sa: fun(C,fun(D,$o)),X: fun(B,C),G: fun(E,D),Y: fun(B,E)] :
      ( aa(fun(B,D),$o,aa(fun(B,C),fun(fun(B,D),$o),bNF_rel_fun(B,B,C,D,fequal(B),Sa),X),aa(fun(B,E),fun(B,D),aa(fun(E,D),fun(fun(B,E),fun(B,D)),comp(E,D,B),G),Y))
    <=> aa(fun(B,E),$o,aa(fun(B,C),fun(fun(B,E),$o),bNF_rel_fun(B,B,C,E,fequal(B),aa(fun(E,D),fun(C,fun(E,$o)),aTP_Lamp_agf(fun(C,fun(D,$o)),fun(fun(E,D),fun(C,fun(E,$o))),Sa),G)),X),Y) ) ).

% fun.rel_map(2)
tff(fact_6769_plus__rat_Otransfer,axiom,
    aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),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_ts(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat)) ).

% plus_rat.transfer
tff(fact_6770_inverse__rat_Otransfer,axiom,
    aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_tt(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat)) ).

% inverse_rat.transfer
tff(fact_6771_one__rat_Otransfer,axiom,
    aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),one_one(rat)) ).

% one_rat.transfer
tff(fact_6772_zero__rat_Otransfer,axiom,
    aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),zero_zero(rat)) ).

% zero_rat.transfer
tff(fact_6773_Fract_Otransfer,axiom,
    aa(fun(int,fun(int,rat)),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),$o),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_tu(int,fun(int,product_prod(int,int)))),fract) ).

% Fract.transfer
tff(fact_6774_of__rat_Otransfer,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => aa(fun(rat,B),$o,aa(fun(product_prod(int,int),B),fun(fun(rat,B),$o),bNF_rel_fun(product_prod(int,int),rat,B,B,pcr_rat,fequal(B)),aTP_Lamp_tx(product_prod(int,int),B)),field_char_0_of_rat(B)) ) ).

% of_rat.transfer
tff(fact_6775_uminus__rat_Otransfer,axiom,
    aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_tv(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat)) ).

% uminus_rat.transfer
tff(fact_6776_times__rat_Otransfer,axiom,
    aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),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_tw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat)) ).

% times_rat.transfer
tff(fact_6777_positive_Otransfer,axiom,
    aa(fun(rat,$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(rat,$o),$o),bNF_rel_fun(product_prod(int,int),rat,$o,$o,pcr_rat,fequal($o)),aTP_Lamp_tr(product_prod(int,int),$o)),positive) ).

% positive.transfer
tff(fact_6778_times__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),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_np(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).

% times_int.transfer
tff(fact_6779_num_Orec__transfer,axiom,
    ! [B: $tType,C: $tType,S: fun(B,fun(C,$o))] : aa(fun(C,fun(fun(num,fun(C,C)),fun(fun(num,fun(C,C)),fun(num,C)))),$o,aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),fun(fun(C,fun(fun(num,fun(C,C)),fun(fun(num,fun(C,C)),fun(num,C)))),$o),bNF_rel_fun(B,C,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),fun(fun(num,fun(C,C)),fun(fun(num,fun(C,C)),fun(num,C))),S,bNF_rel_fun(fun(num,fun(B,B)),fun(num,fun(C,C)),fun(fun(num,fun(B,B)),fun(num,B)),fun(fun(num,fun(C,C)),fun(num,C)),bNF_rel_fun(num,num,fun(B,B),fun(C,C),fequal(num),bNF_rel_fun(B,C,B,C,S,S)),bNF_rel_fun(fun(num,fun(B,B)),fun(num,fun(C,C)),fun(num,B),fun(num,C),bNF_rel_fun(num,num,fun(B,B),fun(C,C),fequal(num),bNF_rel_fun(B,C,B,C,S,S)),bNF_rel_fun(num,num,B,C,fequal(num),S)))),rec_num(B)),rec_num(C)) ).

% num.rec_transfer
tff(fact_6780_verit__eq__simplify_I21_J,axiom,
    ! [B: $tType,F1: B,F22: fun(num,fun(B,B)),F32: fun(num,fun(B,B)),X32: num] : aa(num,B,aa(fun(num,fun(B,B)),fun(num,B),aa(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)),aa(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),rec_num(B),F1),F22),F32),aa(num,num,bit1,X32)) = aa(B,B,aa(num,fun(B,B),F32,X32),aa(num,B,aa(fun(num,fun(B,B)),fun(num,B),aa(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)),aa(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),rec_num(B),F1),F22),F32),X32)) ).

% verit_eq_simplify(21)
tff(fact_6781_verit__eq__simplify_I19_J,axiom,
    ! [B: $tType,F1: B,F22: fun(num,fun(B,B)),F32: fun(num,fun(B,B))] : aa(num,B,aa(fun(num,fun(B,B)),fun(num,B),aa(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)),aa(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),rec_num(B),F1),F22),F32),one2) = F1 ).

% verit_eq_simplify(19)
tff(fact_6782_verit__eq__simplify_I20_J,axiom,
    ! [B: $tType,F1: B,F22: fun(num,fun(B,B)),F32: fun(num,fun(B,B)),X22: num] : aa(num,B,aa(fun(num,fun(B,B)),fun(num,B),aa(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)),aa(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),rec_num(B),F1),F22),F32),aa(num,num,bit0,X22)) = aa(B,B,aa(num,fun(B,B),F22,X22),aa(num,B,aa(fun(num,fun(B,B)),fun(num,B),aa(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)),aa(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),rec_num(B),F1),F22),F32),X22)) ).

% verit_eq_simplify(20)
tff(fact_6783_zero__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),zero_zero(int)) ).

% zero_int.transfer
tff(fact_6784_int__transfer,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_agg(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).

% int_transfer
tff(fact_6785_uminus__int_Otransfer,axiom,
    aa(fun(int,int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),$o),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_nq(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int)) ).

% uminus_int.transfer
tff(fact_6786_nat_Otransfer,axiom,
    aa(fun(int,nat),$o,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),$o),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_6787_one__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),one_one(int)) ).

% one_int.transfer
tff(fact_6788_of__int_Otransfer,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => aa(fun(int,B),$o,aa(fun(product_prod(nat,nat),B),fun(fun(int,B),$o),bNF_rel_fun(product_prod(nat,nat),int,B,B,pcr_int,fequal(B)),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),aTP_Lamp_nr(nat,fun(nat,B)))),ring_1_of_int(B)) ) ).

% of_int.transfer
tff(fact_6789_less__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less(int)) ).

% less_int.transfer
tff(fact_6790_less__eq__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less_eq(int)) ).

% less_eq_int.transfer
tff(fact_6791_plus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),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_nx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int)) ).

% plus_int.transfer
tff(fact_6792_minus__int_Otransfer,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),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_nz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int)) ).

% minus_int.transfer
tff(fact_6793_times__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),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_np(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_np(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int.rsp
tff(fact_6794_antimono__funpow,axiom,
    ! [B: $tType] :
      ( ( lattice(B)
        & order_top(B) )
     => ! [Q: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),Q)
         => order_antimono(nat,B,aTP_Lamp_agh(fun(B,B),fun(nat,B),Q)) ) ) ).

% antimono_funpow
tff(fact_6795_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),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_6796_int_Orel__eq__transfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),intrel),fequal(int)) ).

% int.rel_eq_transfer
tff(fact_6797_zero__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).

% zero_int.rsp
tff(fact_6798_antimonoD,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C),X: B,Y: B] :
          ( order_antimono(B,C,F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y)),aa(B,C,F,X)) ) ) ) ).

% antimonoD
tff(fact_6799_antimonoE,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C),X: B,Y: B] :
          ( order_antimono(B,C,F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y)),aa(B,C,F,X)) ) ) ) ).

% antimonoE
tff(fact_6800_antimonoI,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C)] :
          ( ! [X2: B,Y2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y2)),aa(B,C,F,X2)) )
         => order_antimono(B,C,F) ) ) ).

% antimonoI
tff(fact_6801_antimono__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C)] :
          ( order_antimono(B,C,F)
        <=> ! [X3: B,Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y5)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,Y5)),aa(B,C,F,X3)) ) ) ) ).

% antimono_def
tff(fact_6802_int_Oabs__eq__iff,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),int,abs_Integ,X) = aa(product_prod(nat,nat),int,abs_Integ,Y) )
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,X),Y) ) ).

% int.abs_eq_iff
tff(fact_6803_uminus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),$o),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_nq(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_nq(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int.rsp
tff(fact_6804_nat_Orsp,axiom,
    aa(fun(product_prod(nat,nat),nat),$o,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),$o),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_6805_one__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).

% one_int.rsp
tff(fact_6806_antimono__iff__le__Suc,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [F: fun(nat,B)] :
          ( order_antimono(nat,B,F)
        <=> ! [N4: nat] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(nat,B,F,aa(nat,nat,suc,N4))),aa(nat,B,F,N4)) ) ) ).

% antimono_iff_le_Suc
tff(fact_6807_of__int_Orsp,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => aa(fun(product_prod(nat,nat),B),$o,aa(fun(product_prod(nat,nat),B),fun(fun(product_prod(nat,nat),B),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),B,B,intrel,fequal(B)),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),aTP_Lamp_nr(nat,fun(nat,B)))),aa(fun(nat,fun(nat,B)),fun(product_prod(nat,nat),B),product_case_prod(nat,nat,B),aTP_Lamp_nr(nat,fun(nat,B)))) ) ).

% of_int.rsp
tff(fact_6808_intrel__def,axiom,
    intrel = aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_agj(nat,fun(nat,fun(product_prod(nat,nat),$o)))) ).

% intrel_def
tff(fact_6809_less__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int.rsp
tff(fact_6810_less__eq__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int.rsp
tff(fact_6811_minus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),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_nz(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_nz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int.rsp
tff(fact_6812_plus__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),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_nx(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_nx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int.rsp
tff(fact_6813_sum__mset__constant,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Y: B,A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aTP_Lamp_go(B,fun(C,B),Y)),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(multiset(C),nat,size_size(multiset(C)),A4))),Y) ) ).

% sum_mset_constant
tff(fact_6814_finite__refines__card__le,axiom,
    ! [B: $tType,A4: set(B),R2: set(product_prod(B,B)),S: set(product_prod(B,B))] :
      ( aa(set(set(B)),$o,finite_finite2(set(B)),equiv_quotient(B,A4,R2))
     => ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),S)
       => ( equiv_equiv(B,A4,R2)
         => ( equiv_equiv(B,A4,S)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(set(B)),nat,finite_card(set(B)),equiv_quotient(B,A4,S))),aa(set(set(B)),nat,finite_card(set(B)),equiv_quotient(B,A4,R2))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_6815_of__nat__sum__mset,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [A4: multiset(nat)] : aa(nat,B,semiring_1_of_nat(B),comm_m7189776963980413722m_mset(nat,A4)) = comm_m7189776963980413722m_mset(B,aa(multiset(nat),multiset(B),image_mset(nat,B,semiring_1_of_nat(B)),A4)) ) ).

% of_nat_sum_mset
tff(fact_6816_Union__image__single__mset,axiom,
    ! [B: $tType,Ma: multiset(B)] : comm_m7189776963980413722m_mset(multiset(B),aa(multiset(B),multiset(multiset(B)),image_mset(B,multiset(B),aTP_Lamp_agk(B,multiset(B))),Ma)) = Ma ).

% Union_image_single_mset
tff(fact_6817_sum__mset_Oempty,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ( comm_m7189776963980413722m_mset(B,zero_zero(multiset(B))) = zero_zero(B) ) ) ).

% sum_mset.empty
tff(fact_6818_sum__mset_Oadd__mset,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [X: B,N5: multiset(B)] : comm_m7189776963980413722m_mset(B,aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),N5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),X),comm_m7189776963980413722m_mset(B,N5)) ) ).

% sum_mset.add_mset
tff(fact_6819_sum__mset_Ounion,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [M: multiset(B),N5: multiset(B)] : comm_m7189776963980413722m_mset(B,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),comm_m7189776963980413722m_mset(B,M)),comm_m7189776963980413722m_mset(B,N5)) ) ).

% sum_mset.union
tff(fact_6820_sum__mset_Oneutral__const,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aTP_Lamp_bj(C,B)),A4)) = zero_zero(B) ) ).

% sum_mset.neutral_const
tff(fact_6821_sum__mset_Oinsert,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B),X: C,A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,G),aa(multiset(C),multiset(C),aa(C,fun(multiset(C),multiset(C)),add_mset(C),X),A4))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(C,B,G,X)),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,G),A4))) ) ).

% sum_mset.insert
tff(fact_6822_equiv__type,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B))] :
      ( equiv_equiv(B,A4,R)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4))) ) ).

% equiv_type
tff(fact_6823_sum__mset_Oswap,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,fun(D,B)),B4: multiset(D),A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(multiset(D),fun(C,B),aTP_Lamp_agl(fun(C,fun(D,B)),fun(multiset(D),fun(C,B)),G),B4)),A4)) = comm_m7189776963980413722m_mset(B,aa(multiset(D),multiset(B),image_mset(D,B,aa(multiset(C),fun(D,B),aTP_Lamp_agm(fun(C,fun(D,B)),fun(multiset(C),fun(D,B)),G),A4)),B4)) ) ).

% sum_mset.swap
tff(fact_6824_sum__mset_Odistrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(C,B),H: fun(C,B),A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bo(fun(C,B),fun(fun(C,B),fun(C,B)),G),H)),A4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,G),A4))),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,H),A4))) ) ).

% sum_mset.distrib
tff(fact_6825_sum__mset__distrib__left,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [C2: B,F: fun(C,B),M: multiset(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),C2),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,F),M))) = comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bp(B,fun(fun(C,B),fun(C,B)),C2),F)),M)) ) ).

% sum_mset_distrib_left
tff(fact_6826_sum__mset__distrib__right,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F: fun(C,B),M: multiset(C),C2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,F),M))),C2) = comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(B,fun(C,B),aTP_Lamp_bq(fun(C,B),fun(B,fun(C,B)),F),C2)),M)) ) ).

% sum_mset_distrib_right
tff(fact_6827_sum__mset__product,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(C)
        & times(C)
        & semiring_0(B) )
     => ! [F: fun(C,B),A4: multiset(C),G: fun(D,B),B4: multiset(D)] : aa(B,B,aa(B,fun(B,B),times_times(B),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,F),A4))),comm_m7189776963980413722m_mset(B,aa(multiset(D),multiset(B),image_mset(D,B,G),B4))) = comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(multiset(D),fun(C,B),aa(fun(D,B),fun(multiset(D),fun(C,B)),aTP_Lamp_ago(fun(C,B),fun(fun(D,B),fun(multiset(D),fun(C,B))),F),G),B4)),A4)) ) ).

% sum_mset_product
tff(fact_6828_equiv__Eps__preserves,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B)),X5: set(B)] :
      ( equiv_equiv(B,A4,R)
     => ( aa(set(set(B)),$o,member(set(B),X5),equiv_quotient(B,A4,R))
       => aa(set(B),$o,member(B,fChoice(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),X5))),A4) ) ) ).

% equiv_Eps_preserves
tff(fact_6829_equiv__Eps__in,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B)),X5: set(B)] :
      ( equiv_equiv(B,A4,R)
     => ( aa(set(set(B)),$o,member(set(B),X5),equiv_quotient(B,A4,R))
       => aa(set(B),$o,member(B,fChoice(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),X5))),X5) ) ) ).

% equiv_Eps_in
tff(fact_6830_size__eq__sum__mset,axiom,
    ! [B: $tType,M: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),M) = comm_m7189776963980413722m_mset(nat,aa(multiset(B),multiset(nat),image_mset(B,nat,aTP_Lamp_gr(B,nat)),M)) ).

% size_eq_sum_mset
tff(fact_6831_refines__equiv__image__eq,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),S: set(product_prod(B,B)),A4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),S)
     => ( equiv_equiv(B,A4,R2)
       => ( equiv_equiv(B,A4,S)
         => ( aa(set(set(B)),set(set(B)),image2(set(B),set(B),image(B,B,S)),equiv_quotient(B,A4,R2)) = equiv_quotient(B,A4,S) ) ) ) ) ).

% refines_equiv_image_eq
tff(fact_6832_UN__equiv__class2,axiom,
    ! [B: $tType,D: $tType,C: $tType,A14: set(B),R1: set(product_prod(B,B)),A25: set(C),R22: set(product_prod(C,C)),F: fun(B,fun(C,set(D))),A12: B,A23: C] :
      ( equiv_equiv(B,A14,R1)
     => ( equiv_equiv(C,A25,R22)
       => ( equiv_congruent2(B,C,set(D),R1,R22,F)
         => ( aa(set(B),$o,member(B,A12),A14)
           => ( aa(set(C),$o,member(C,A23),A25)
             => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),aa(C,fun(B,set(D)),aa(fun(B,fun(C,set(D))),fun(C,fun(B,set(D))),aTP_Lamp_agp(set(product_prod(C,C)),fun(fun(B,fun(C,set(D))),fun(C,fun(B,set(D)))),R22),F),A23)),aa(set(B),set(B),image(B,B,R1),aa(set(B),set(B),insert(B,A12),bot_bot(set(B)))))) = aa(C,set(D),aa(B,fun(C,set(D)),F,A12),A23) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_6833_UN__equiv__class,axiom,
    ! [C: $tType,B: $tType,A4: set(B),R: set(product_prod(B,B)),F: fun(B,set(C)),A2: B] :
      ( equiv_equiv(B,A4,R)
     => ( equiv_congruent(B,set(C),R,F)
       => ( aa(set(B),$o,member(B,A2),A4)
         => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),aa(set(B),set(B),image(B,B,R),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) = aa(B,set(C),F,A2) ) ) ) ) ).

% UN_equiv_class
tff(fact_6834_congruent2__implies__congruent__UN,axiom,
    ! [B: $tType,D: $tType,C: $tType,A14: set(B),R1: set(product_prod(B,B)),A25: set(C),R22: set(product_prod(C,C)),F: fun(B,fun(C,set(D))),A2: C] :
      ( equiv_equiv(B,A14,R1)
     => ( equiv_equiv(C,A25,R22)
       => ( equiv_congruent2(B,C,set(D),R1,R22,F)
         => ( aa(set(C),$o,member(C,A2),A25)
           => equiv_congruent(B,set(D),R1,aa(C,fun(B,set(D)),aa(fun(B,fun(C,set(D))),fun(C,fun(B,set(D))),aTP_Lamp_agp(set(product_prod(C,C)),fun(fun(B,fun(C,set(D))),fun(C,fun(B,set(D)))),R22),F),A2)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_6835_UN__equiv__class__type,axiom,
    ! [B: $tType,C: $tType,A4: set(B),R: set(product_prod(B,B)),F: fun(B,set(C)),X5: set(B),B4: set(set(C))] :
      ( equiv_equiv(B,A4,R)
     => ( equiv_congruent(B,set(C),R,F)
       => ( aa(set(set(B)),$o,member(set(B),X5),equiv_quotient(B,A4,R))
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(set(set(C)),$o,member(set(C),aa(B,set(C),F,X2)),B4) )
           => aa(set(set(C)),$o,member(set(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),X5))),B4) ) ) ) ) ).

% UN_equiv_class_type
tff(fact_6836_UN__equiv__class__type2,axiom,
    ! [B: $tType,C: $tType,D: $tType,A14: set(B),R1: set(product_prod(B,B)),A25: set(C),R22: set(product_prod(C,C)),F: fun(B,fun(C,set(D))),X13: set(B),X24: set(C),B4: set(set(D))] :
      ( equiv_equiv(B,A14,R1)
     => ( equiv_equiv(C,A25,R22)
       => ( equiv_congruent2(B,C,set(D),R1,R22,F)
         => ( aa(set(set(B)),$o,member(set(B),X13),equiv_quotient(B,A14,R1))
           => ( aa(set(set(C)),$o,member(set(C),X24),equiv_quotient(C,A25,R22))
             => ( ! [X1: B,X23: C] :
                    ( aa(set(B),$o,member(B,X1),A14)
                   => ( aa(set(C),$o,member(C,X23),A25)
                     => aa(set(set(D)),$o,member(set(D),aa(C,set(D),aa(B,fun(C,set(D)),F,X1),X23)),B4) ) )
               => aa(set(set(D)),$o,member(set(D),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),aa(set(C),fun(B,set(D)),aTP_Lamp_agq(fun(B,fun(C,set(D))),fun(set(C),fun(B,set(D))),F),X24)),X13))),B4) ) ) ) ) ) ) ).

% UN_equiv_class_type2
tff(fact_6837_univ__preserves,axiom,
    ! [B: $tType,C: $tType,A4: set(B),R: set(product_prod(B,B)),F: fun(B,C),B4: set(C)] :
      ( equiv_equiv(B,A4,R)
     => ( equiv_congruent(B,C,R,F)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(set(C),$o,member(C,aa(B,C,F,X2)),B4) )
         => ! [X4: set(B)] :
              ( aa(set(set(B)),$o,member(set(B),X4),equiv_quotient(B,A4,R))
             => aa(set(C),$o,member(C,bNF_Greatest_univ(B,C,F,X4)),B4) ) ) ) ) ).

% univ_preserves
tff(fact_6838_UN__equiv__class__inject,axiom,
    ! [C: $tType,B: $tType,A4: set(B),R: set(product_prod(B,B)),F: fun(B,set(C)),X5: set(B),Y4: set(B)] :
      ( equiv_equiv(B,A4,R)
     => ( equiv_congruent(B,set(C),R,F)
       => ( ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),X5)) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),F),Y4)) )
         => ( aa(set(set(B)),$o,member(set(B),X5),equiv_quotient(B,A4,R))
           => ( aa(set(set(B)),$o,member(set(B),Y4),equiv_quotient(B,A4,R))
             => ( ! [X2: B,Y2: B] :
                    ( aa(set(B),$o,member(B,X2),A4)
                   => ( aa(set(B),$o,member(B,Y2),A4)
                     => ( ( aa(B,set(C),F,X2) = aa(B,set(C),F,Y2) )
                       => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X2),Y2)),R) ) ) )
               => ( X5 = Y4 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_6839_univ__commute,axiom,
    ! [C: $tType,B: $tType,A4: set(B),R: set(product_prod(B,B)),F: fun(B,C),X: B] :
      ( equiv_equiv(B,A4,R)
     => ( equiv_congruent(B,C,R,F)
       => ( aa(set(B),$o,member(B,X),A4)
         => ( bNF_Greatest_univ(B,C,F,aa(B,set(B),equiv_proj(B,B,R),X)) = aa(B,C,F,X) ) ) ) ) ).

% univ_commute
tff(fact_6840_proj__Eps,axiom,
    ! [B: $tType,A4: set(B),R: set(product_prod(B,B)),X5: set(B)] :
      ( equiv_equiv(B,A4,R)
     => ( aa(set(set(B)),$o,member(set(B),X5),equiv_quotient(B,A4,R))
       => ( aa(B,set(B),equiv_proj(B,B,R),fChoice(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),X5))) = X5 ) ) ) ).

% proj_Eps
tff(fact_6841_equiv__proj,axiom,
    ! [B: $tType,A4: set(B),R2: set(product_prod(B,B)),Z: product_prod(B,B)] :
      ( equiv_equiv(B,A4,R2)
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),Z),R2)
       => ( aa(product_prod(B,B),set(B),aa(fun(product_prod(B,B),B),fun(product_prod(B,B),set(B)),aa(fun(B,set(B)),fun(fun(product_prod(B,B),B),fun(product_prod(B,B),set(B))),comp(B,set(B),product_prod(B,B)),equiv_proj(B,B,R2)),product_fst(B,B)),Z) = aa(product_prod(B,B),set(B),aa(fun(product_prod(B,B),B),fun(product_prod(B,B),set(B)),aa(fun(B,set(B)),fun(fun(product_prod(B,B),B),fun(product_prod(B,B),set(B))),comp(B,set(B),product_prod(B,B)),equiv_proj(B,B,R2)),product_snd(B,B)),Z) ) ) ) ).

% equiv_proj
tff(fact_6842_relImage__proj,axiom,
    ! [B: $tType,A4: set(B),R2: set(product_prod(B,B))] :
      ( equiv_equiv(B,A4,R2)
     => aa(set(product_prod(set(B),set(B))),$o,aa(set(product_prod(set(B),set(B))),fun(set(product_prod(set(B),set(B))),$o),ord_less_eq(set(product_prod(set(B),set(B)))),bNF_Gr4221423524335903396lImage(B,set(B),R2,equiv_proj(B,B,R2))),id_on(set(B),equiv_quotient(B,A4,R2))) ) ).

% relImage_proj
tff(fact_6843_list_Oin__rel,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),A2: list(B),B2: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,R2),A2),B2)
    <=> ? [Z5: list(product_prod(B,C))] :
          ( aa(set(list(product_prod(B,C))),$o,member(list(product_prod(B,C)),Z5),aa(fun(list(product_prod(B,C)),$o),set(list(product_prod(B,C))),collect(list(product_prod(B,C))),aTP_Lamp_agr(fun(B,fun(C,$o)),fun(list(product_prod(B,C)),$o),R2)))
          & ( aa(list(product_prod(B,C)),list(B),map(product_prod(B,C),B,product_fst(B,C)),Z5) = A2 )
          & ( aa(list(product_prod(B,C)),list(C),map(product_prod(B,C),C,product_snd(B,C)),Z5) = B2 ) ) ) ).

% list.in_rel
tff(fact_6844_list_Odisc__transfer_I2_J,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o))] : aa(fun(list(C),$o),$o,aa(fun(list(B),$o),fun(fun(list(C),$o),$o),bNF_rel_fun(list(B),list(C),$o,$o,list_all2(B,C,R2),fequal($o)),aTP_Lamp_pt(list(B),$o)),aTP_Lamp_pp(list(C),$o)) ).

% list.disc_transfer(2)
tff(fact_6845_list_Odisc__transfer_I1_J,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o))] : aa(fun(list(C),$o),$o,aa(fun(list(B),$o),fun(fun(list(C),$o),$o),bNF_rel_fun(list(B),list(C),$o,$o,list_all2(B,C,R2),fequal($o)),aTP_Lamp_ags(list(B),$o)),aTP_Lamp_agt(list(C),$o)) ).

% list.disc_transfer(1)
tff(fact_6846_length__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] : aa(fun(list(C),nat),$o,aa(fun(list(B),nat),fun(fun(list(C),nat),$o),bNF_rel_fun(list(B),list(C),nat,nat,list_all2(B,C,A4),fequal(nat)),size_size(list(B))),size_size(list(C))) ).

% length_transfer
tff(fact_6847_list_Orel__map_I2_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,Sa: fun(B,fun(C,$o)),X: list(B),G: fun(D,C),Y: list(D)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Sa),X),aa(list(D),list(C),map(D,C,G),Y))
    <=> aa(list(D),$o,aa(list(B),fun(list(D),$o),list_all2(B,D,aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_agu(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Sa),G)),X),Y) ) ).

% list.rel_map(2)
tff(fact_6848_list_Orel__map_I1_J,axiom,
    ! [D: $tType,B: $tType,C: $tType,Sb: fun(B,fun(C,$o)),I: fun(D,B),X: list(D),Y: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Sb),aa(list(D),list(B),map(D,B,I),X)),Y)
    <=> aa(list(C),$o,aa(list(D),fun(list(C),$o),list_all2(D,C,aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_agv(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Sb),I)),X),Y) ) ).

% list.rel_map(1)
tff(fact_6849_list__all2__map1,axiom,
    ! [D: $tType,B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),F: fun(D,B),As: list(D),Bs: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),aa(list(D),list(B),map(D,B,F),As)),Bs)
    <=> aa(list(C),$o,aa(list(D),fun(list(C),$o),list_all2(D,C,aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_agv(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Pa),F)),As),Bs) ) ).

% list_all2_map1
tff(fact_6850_list__all2__map2,axiom,
    ! [B: $tType,C: $tType,D: $tType,Pa: fun(B,fun(C,$o)),As: list(B),F: fun(D,C),Bs: list(D)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),As),aa(list(D),list(C),map(D,C,F),Bs))
    <=> aa(list(D),$o,aa(list(B),fun(list(D),$o),list_all2(B,D,aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_agu(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Pa),F)),As),Bs) ) ).

% list_all2_map2
tff(fact_6851_Misc_Olist__all2__induct,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),L: list(B),L3: list(C),Q: fun(list(B),fun(list(C),$o))] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),L),L3)
     => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Q,nil(B)),nil(C))
       => ( ! [X2: B,X8: C,Ls2: list(B),Ls3: list(C)] :
              ( aa(C,$o,aa(B,fun(C,$o),Pa,X2),X8)
             => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Ls2),Ls3)
               => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),Q,Ls2),Ls3)
                 => aa(list(C),$o,aa(list(B),fun(list(C),$o),Q,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Ls2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),X8),Ls3)) ) ) )
         => aa(list(C),$o,aa(list(B),fun(list(C),$o),Q,L),L3) ) ) ) ).

% Misc.list_all2_induct
tff(fact_6852_list__all2__append2,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C),Zs: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys),Zs))
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
          & ( aa(list(B),nat,size_size(list(B)),Us2) = aa(list(C),nat,size_size(list(C)),Ys) )
          & ( aa(list(B),nat,size_size(list(B)),Vs2) = aa(list(C),nat,size_size(list(C)),Zs) )
          & aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Us2),Ys)
          & aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Vs2),Zs) ) ) ).

% list_all2_append2
tff(fact_6853_list__all2__append1,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(B),Zs: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),Zs)
    <=> ? [Us2: list(C),Vs2: list(C)] :
          ( ( Zs = aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Us2),Vs2) )
          & ( aa(list(C),nat,size_size(list(C)),Us2) = aa(list(B),nat,size_size(list(B)),Xs) )
          & ( aa(list(C),nat,size_size(list(C)),Vs2) = aa(list(B),nat,size_size(list(B)),Ys) )
          & aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Us2)
          & aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Ys),Vs2) ) ) ).

% list_all2_append1
tff(fact_6854_list__all2__append,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),Pa: fun(B,fun(C,$o)),Us: list(B),Vs: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Us)),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),Ys),Vs))
      <=> ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Ys)
          & aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Us),Vs) ) ) ) ).

% list_all2_append
tff(fact_6855_list__all2__lengthD,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Ys)
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% list_all2_lengthD
tff(fact_6856_relImage__mono,axiom,
    ! [C: $tType,B: $tType,R12: set(product_prod(B,B)),R23: set(product_prod(B,B)),F: fun(B,C)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R12),R23)
     => aa(set(product_prod(C,C)),$o,aa(set(product_prod(C,C)),fun(set(product_prod(C,C)),$o),ord_less_eq(set(product_prod(C,C))),bNF_Gr4221423524335903396lImage(B,C,R12,F)),bNF_Gr4221423524335903396lImage(B,C,R23,F)) ) ).

% relImage_mono
tff(fact_6857_list__all2__conv__all__nth,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Ys)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Xs))
           => aa(C,$o,aa(B,fun(C,$o),Pa,aa(nat,B,nth(B,Xs),I3)),aa(nat,C,nth(C,Ys),I3)) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_6858_list__all2__all__nthI,axiom,
    ! [B: $tType,C: $tType,A2: list(B),B2: list(C),Pa: fun(B,fun(C,$o))] :
      ( ( aa(list(B),nat,size_size(list(B)),A2) = aa(list(C),nat,size_size(list(C)),B2) )
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(B),nat,size_size(list(B)),A2))
           => aa(C,$o,aa(B,fun(C,$o),Pa,aa(nat,B,nth(B,A2),N2)),aa(nat,C,nth(C,B2),N2)) )
       => aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),A2),B2) ) ) ).

% list_all2_all_nthI
tff(fact_6859_list__all2__nthD2,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C),P2: nat] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(list(C),nat,size_size(list(C)),Ys))
       => aa(C,$o,aa(B,fun(C,$o),Pa,aa(nat,B,nth(B,Xs),P2)),aa(nat,C,nth(C,Ys),P2)) ) ) ).

% list_all2_nthD2
tff(fact_6860_list__all2__nthD,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C),P2: nat] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(list(B),nat,size_size(list(B)),Xs))
       => aa(C,$o,aa(B,fun(C,$o),Pa,aa(nat,B,nth(B,Xs),P2)),aa(nat,C,nth(C,Ys),P2)) ) ) ).

% list_all2_nthD
tff(fact_6861_product__lists__set,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),set(list(B)),set2(list(B)),product_lists(B,Xss)) = aa(fun(list(B),$o),set(list(B)),collect(list(B)),aTP_Lamp_agx(list(list(B)),fun(list(B),$o),Xss)) ).

% product_lists_set
tff(fact_6862_list__all2I,axiom,
    ! [B: $tType,C: $tType,A2: list(B),B2: list(C),Pa: fun(B,fun(C,$o))] :
      ( ! [X2: product_prod(B,C)] :
          ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X2),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,A2,B2)))
         => aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Pa),X2) )
     => ( ( aa(list(B),nat,size_size(list(B)),A2) = aa(list(C),nat,size_size(list(C)),B2) )
       => aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),A2),B2) ) ) ).

% list_all2I
tff(fact_6863_sum__list__transfer,axiom,
    ! [B: $tType,C: $tType] :
      ( ( monoid_add(C)
        & monoid_add(B) )
     => ! [A4: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),A4,zero_zero(B)),zero_zero(C))
         => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),A4,bNF_rel_fun(B,C,B,C,A4,A4)),plus_plus(B)),plus_plus(C))
           => aa(fun(list(C),C),$o,aa(fun(list(B),B),fun(fun(list(C),C),$o),bNF_rel_fun(list(B),list(C),B,C,list_all2(B,C,A4),A4),groups8242544230860333062m_list(B)),groups8242544230860333062m_list(C)) ) ) ) ).

% sum_list_transfer
tff(fact_6864_relImage__def,axiom,
    ! [B: $tType,C: $tType,R2: set(product_prod(C,C)),F: fun(C,B)] : bNF_Gr4221423524335903396lImage(C,B,R2,F) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(C,B),fun(product_prod(B,B),$o),aTP_Lamp_agy(set(product_prod(C,C)),fun(fun(C,B),fun(product_prod(B,B),$o)),R2),F)) ).

% relImage_def
tff(fact_6865_horner__sum__transfer,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType] :
      ( ( comm_semiring_0(C)
        & comm_semiring_0(B) )
     => ! [A4: fun(B,fun(C,$o)),B4: fun(D,fun(E,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),A4,zero_zero(B)),zero_zero(C))
         => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),A4,bNF_rel_fun(B,C,B,C,A4,A4)),plus_plus(B)),plus_plus(C))
           => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),A4,bNF_rel_fun(B,C,B,C,A4,A4)),times_times(B)),times_times(C))
             => aa(fun(fun(E,C),fun(C,fun(list(E),C))),$o,aa(fun(fun(D,B),fun(B,fun(list(D),B))),fun(fun(fun(E,C),fun(C,fun(list(E),C))),$o),bNF_rel_fun(fun(D,B),fun(E,C),fun(B,fun(list(D),B)),fun(C,fun(list(E),C)),bNF_rel_fun(D,E,B,C,B4,A4),bNF_rel_fun(B,C,fun(list(D),B),fun(list(E),C),A4,bNF_rel_fun(list(D),list(E),B,C,list_all2(D,E,B4),A4))),groups4207007520872428315er_sum(D,B)),groups4207007520872428315er_sum(E,C)) ) ) ) ) ).

% horner_sum_transfer
tff(fact_6866_relImage__relInvImage,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,B)),F: fun(C,B),A4: set(C)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),product_Sigma(B,B,aa(set(C),set(B),image2(C,B,F),A4),aa(set(C),fun(B,set(B)),aTP_Lamp_wa(fun(C,B),fun(set(C),fun(B,set(B))),F),A4)))
     => ( bNF_Gr4221423524335903396lImage(C,B,bNF_Gr7122648621184425601vImage(C,B,A4,R2,F),F) = R2 ) ) ).

% relImage_relInvImage
tff(fact_6867_prod__list__transfer,axiom,
    ! [B: $tType,C: $tType] :
      ( ( monoid_mult(C)
        & monoid_mult(B) )
     => ! [A4: fun(B,fun(C,$o))] :
          ( aa(C,$o,aa(B,fun(C,$o),A4,one_one(B)),one_one(C))
         => ( aa(fun(C,fun(C,C)),$o,aa(fun(B,fun(B,B)),fun(fun(C,fun(C,C)),$o),bNF_rel_fun(B,C,fun(B,B),fun(C,C),A4,bNF_rel_fun(B,C,B,C,A4,A4)),times_times(B)),times_times(C))
           => aa(fun(list(C),C),$o,aa(fun(list(B),B),fun(fun(list(C),C),$o),bNF_rel_fun(list(B),list(C),B,C,list_all2(B,C,A4),A4),groups5270119922927024881d_list(B)),groups5270119922927024881d_list(C)) ) ) ) ).

% prod_list_transfer
tff(fact_6868_prod__list_ONil,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ( aa(list(B),B,groups5270119922927024881d_list(B),nil(B)) = one_one(B) ) ) ).

% prod_list.Nil
tff(fact_6869_relInvImage__mono,axiom,
    ! [B: $tType,C: $tType,R12: set(product_prod(B,B)),R23: set(product_prod(B,B)),A4: set(C),F: fun(C,B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R12),R23)
     => aa(set(product_prod(C,C)),$o,aa(set(product_prod(C,C)),fun(set(product_prod(C,C)),$o),ord_less_eq(set(product_prod(C,C))),bNF_Gr7122648621184425601vImage(C,B,A4,R12,F)),bNF_Gr7122648621184425601vImage(C,B,A4,R23,F)) ) ).

% relInvImage_mono
tff(fact_6870_prod__list__zero__iff,axiom,
    ! [B: $tType] :
      ( ( semiring_1(B)
        & semiri3467727345109120633visors(B) )
     => ! [Xs: list(B)] :
          ( ( aa(list(B),B,groups5270119922927024881d_list(B),Xs) = zero_zero(B) )
        <=> aa(set(B),$o,member(B,zero_zero(B)),aa(list(B),set(B),set2(B),Xs)) ) ) ).

% prod_list_zero_iff
tff(fact_6871_prod__list__coprime__left,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Xs: list(B),A2: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => algebr8660921524188924756oprime(B,X2,A2) )
         => algebr8660921524188924756oprime(B,aa(list(B),B,groups5270119922927024881d_list(B),Xs),A2) ) ) ).

% prod_list_coprime_left
tff(fact_6872_prod__list__coprime__right,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Xs: list(B),A2: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(list(B),set(B),set2(B),Xs))
             => algebr8660921524188924756oprime(B,A2,X2) )
         => algebr8660921524188924756oprime(B,A2,aa(list(B),B,groups5270119922927024881d_list(B),Xs)) ) ) ).

% prod_list_coprime_right
tff(fact_6873_prod__list_Oeq__foldr,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups5270119922927024881d_list(B),Xs) = aa(B,B,foldr(B,B,times_times(B),Xs),one_one(B)) ) ).

% prod_list.eq_foldr
tff(fact_6874_relInvImage__def,axiom,
    ! [C: $tType,B: $tType,A4: set(B),R2: set(product_prod(C,C)),F: fun(B,C)] : bNF_Gr7122648621184425601vImage(B,C,A4,R2,F) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,C),fun(product_prod(B,B),$o),aa(set(product_prod(C,C)),fun(fun(B,C),fun(product_prod(B,B),$o)),aTP_Lamp_agz(set(B),fun(set(product_prod(C,C)),fun(fun(B,C),fun(product_prod(B,B),$o))),A4),R2),F)) ).

% relInvImage_def
tff(fact_6875_relInvImage__UNIV__relImage,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,B)),F: fun(B,C)] : aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),bNF_Gr7122648621184425601vImage(B,C,top_top(set(B)),bNF_Gr4221423524335903396lImage(B,C,R2,F),F)) ).

% relInvImage_UNIV_relImage
tff(fact_6876_Some__image__these__eq,axiom,
    ! [B: $tType,A4: set(option(B))] : aa(set(B),set(option(B)),image2(B,option(B),some(B)),these(B,A4)) = aa(fun(option(B),$o),set(option(B)),collect(option(B)),aTP_Lamp_aha(set(option(B)),fun(option(B),$o),A4)) ).

% Some_image_these_eq
tff(fact_6877_possible__bit__def,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Tyrep: itself(B),N: nat] :
          ( bit_se6407376104438227557le_bit(B,Tyrep,N)
        <=> ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) != zero_zero(B) ) ) ) ).

% possible_bit_def
tff(fact_6878_possible__bit__0,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ty: itself(B)] : bit_se6407376104438227557le_bit(B,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_6879_possible__bit__less__imp,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Tyrep: itself(B),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(B,Tyrep,I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(B,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_6880_Option_Othese__def,axiom,
    ! [B: $tType,A4: set(option(B))] : these(B,A4) = aa(set(option(B)),set(B),image2(option(B),B,the2(B)),aa(fun(option(B),$o),set(option(B)),collect(option(B)),aTP_Lamp_aha(set(option(B)),fun(option(B),$o),A4))) ).

% Option.these_def
tff(fact_6881_drop__bit__exp__eq,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,N: nat] :
          aa(B,B,aa(nat,fun(B,B),bit_se4197421643247451524op_bit(B),Ma),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)) = aa(B,B,
            aa(B,fun(B,B),times_times(B),
              aa($o,B,zero_neq_one_of_bool(B),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
                & bit_se6407376104438227557le_bit(B,type2(B),N) ))),
            aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma))) ) ).

% drop_bit_exp_eq
tff(fact_6882_bit__minus__2__iff,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),N)
        <=> ( bit_se6407376104438227557le_bit(B,type2(B),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% bit_minus_2_iff
tff(fact_6883_CHAR__eq__0,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ( semiri4206861660011772517g_char(B,type2(B)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_6884_of__nat__CHAR,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( aa(nat,B,semiring_1_of_nat(B),semiri4206861660011772517g_char(B,type2(B))) = zero_zero(B) ) ) ).

% of_nat_CHAR
tff(fact_6885_bit__minus__1__iff,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,uminus_uminus(B),one_one(B))),N)
        <=> bit_se6407376104438227557le_bit(B,type2(B),N) ) ) ).

% bit_minus_1_iff
tff(fact_6886_CHAR__eqI,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [C2: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),C2) = zero_zero(B) )
         => ( ! [X2: nat] :
                ( ( aa(nat,B,semiring_1_of_nat(B),X2) = zero_zero(B) )
               => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),C2),X2) )
           => ( semiri4206861660011772517g_char(B,type2(B)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_6887_of__nat__eq__0__iff__char__dvd,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat] :
          ( ( aa(nat,B,semiring_1_of_nat(B),N) = zero_zero(B) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),semiri4206861660011772517g_char(B,type2(B))),N) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_6888_bit__of__nat__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(nat,B,semiring_1_of_nat(B),Ma)),N)
        <=> ( bit_se6407376104438227557le_bit(B,type2(B),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Ma),N) ) ) ) ).

% bit_of_nat_iff
tff(fact_6889_bit__minus__iff,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,uminus_uminus(B),A2)),N)
        <=> ( bit_se6407376104438227557le_bit(B,type2(B),N)
            & ~ aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),one_one(B))),N) ) ) ) ).

% bit_minus_iff
tff(fact_6890_bit__push__bit__iff,axiom,
    ! [B: $tType] :
      ( bit_se359711467146920520ations(B)
     => ! [Ma: nat,A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Ma),A2)),N)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N)
            & bit_se6407376104438227557le_bit(B,type2(B),N)
            & aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Ma)) ) ) ) ).

% bit_push_bit_iff
tff(fact_6891_CHAR__pos__iff,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(B,type2(B)))
      <=> ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
            & ( aa(nat,B,semiring_1_of_nat(B),N4) = zero_zero(B) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_6892_CHAR__eq__posI,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
         => ( ( aa(nat,B,semiring_1_of_nat(B),C2) = zero_zero(B) )
           => ( ! [X2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),C2)
                   => ( aa(nat,B,semiring_1_of_nat(B),X2) != zero_zero(B) ) ) )
             => ( semiri4206861660011772517g_char(B,type2(B)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_6893_CHAR__eq0__iff,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ( ( semiri4206861660011772517g_char(B,type2(B)) = zero_zero(nat) )
      <=> ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
           => ( aa(nat,B,semiring_1_of_nat(B),N4) != zero_zero(B) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_6894_fold__possible__bit,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [N: nat] :
          ( ( aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N) = zero_zero(B) )
        <=> ~ bit_se6407376104438227557le_bit(B,type2(B),N) ) ) ).

% fold_possible_bit
tff(fact_6895_bit__2__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N)
        <=> ( bit_se6407376104438227557le_bit(B,type2(B),one_one(nat))
            & ( N = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_6896_bit__minus__exp__iff,axiom,
    ! [B: $tType] :
      ( bit_ri3973907225187159222ations(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,uminus_uminus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma))),N)
        <=> ( bit_se6407376104438227557le_bit(B,type2(B),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6897_bit__mask__sub__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [Ma: nat,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Ma)),one_one(B))),N)
        <=> ( bit_se6407376104438227557le_bit(B,type2(B),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),Ma) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6898_bit__double__iff,axiom,
    ! [B: $tType] :
      ( bit_semiring_bits(B)
     => ! [A2: B,N: nat] :
          ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),A2)),N)
        <=> ( aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
            & ( N != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(B,type2(B),N) ) ) ) ).

% bit_double_iff
tff(fact_6899_Gcd__fin_Oeq__fold,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B)] :
          aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) = $ite(aa(set(B),$o,finite_finite2(B),A4),finite_fold(B,B,gcd_gcd(B),zero_zero(B),A4),one_one(B)) ) ).

% Gcd_fin.eq_fold
tff(fact_6900_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [B: $tType,F: fun(nat,fun(B,B)),A2: nat,B2: nat,Acc: B] :
      ( accp(product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),set_fo1817059534552279752at_rel(B),aa(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),aa(fun(nat,fun(B,B)),fun(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))),product_Pair(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),F),aa(product_prod(nat,B),product_prod(nat,product_prod(nat,B)),aa(nat,fun(product_prod(nat,B),product_prod(nat,product_prod(nat,B))),product_Pair(nat,product_prod(nat,B)),A2),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),B2),Acc))))
     => ( set_fo6178422350223883121st_nat(B,F,A2,B2,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc,set_fo6178422350223883121st_nat(B,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(B,B,aa(nat,fun(B,B),F,A2),Acc))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_6901_Gcd__fin_Oempty,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ( aa(set(B),B,semiring_gcd_Gcd_fin(B),bot_bot(set(B))) = zero_zero(B) ) ) ).

% Gcd_fin.empty
tff(fact_6902_Gcd__fin_Oinfinite,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B)] :
          ( ~ aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) = one_one(B) ) ) ) ).

% Gcd_fin.infinite
tff(fact_6903_is__unit__Gcd__fin__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)),one_one(B))
        <=> ( aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) = one_one(B) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_6904_Gcd__fin_Oinsert,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,A4: set(B)] : aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(set(B),set(B),insert(B,A2),A4)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)) ) ).

% Gcd_fin.insert
tff(fact_6905_Gcd__fin__eq__Gcd,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [A4: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) = gcd_Gcd(B,A4) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_6906_Gcd__fin_Ounion,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B),B4: set(B)] : aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)),aa(set(B),B,semiring_gcd_Gcd_fin(B),B4)) ) ).

% Gcd_fin.union
tff(fact_6907_Gcd__fin_Oin__idem,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,A4: set(B)] :
          ( aa(set(B),$o,member(B,A2),A4)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)) = aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) ) ) ) ).

% Gcd_fin.in_idem
tff(fact_6908_Gcd__fin__dvd,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,A4: set(B)] :
          ( aa(set(B),$o,member(B,A2),A4)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)),A2) ) ) ).

% Gcd_fin_dvd
tff(fact_6909_gcd__list__greatest,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Bs: list(B),A2: B] :
          ( ! [B3: B] :
              ( aa(set(B),$o,member(B,B3),aa(list(B),set(B),set2(B),Bs))
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B3) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(list(B),set(B),set2(B),Bs))) ) ) ).

% gcd_list_greatest
tff(fact_6910_dvd__gcd__list__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,Xs: list(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(list(B),set(B),set2(B),Xs)))
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Xs))
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),X3) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_6911_Gcd__fin__greatest,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B),A2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ! [B3: B] :
                ( aa(set(B),$o,member(B,B3),A4)
               => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B3) )
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)) ) ) ) ).

% Gcd_fin_greatest
tff(fact_6912_dvd__Gcd__fin__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B),B2: B] :
          ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4))
          <=> ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),A4)
               => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),X3) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_6913_Gcd__fin_Osubset,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B4: set(B),A4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
         => ( aa(B,B,aa(B,fun(B,B),gcd_gcd(B),aa(set(B),B,semiring_gcd_Gcd_fin(B),B4)),aa(set(B),B,semiring_gcd_Gcd_fin(B),A4)) = aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) ) ) ) ).

% Gcd_fin.subset
tff(fact_6914_Gcd__fin_Oremove,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,A4: set(B)] :
          ( aa(set(B),$o,member(B,A2),A4)
         => ( aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_6915_Gcd__fin_Oinsert__remove,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,A4: set(B)] : aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(set(B),set(B),insert(B,A2),A4)) = aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_6916_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [B: $tType,A0: fun(nat,fun(B,B)),A12: nat,A23: nat,A32: B,Pa: fun(fun(nat,fun(B,B)),fun(nat,fun(nat,fun(B,$o))))] :
      ( accp(product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),set_fo1817059534552279752at_rel(B),aa(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),aa(fun(nat,fun(B,B)),fun(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))),product_Pair(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),A0),aa(product_prod(nat,B),product_prod(nat,product_prod(nat,B)),aa(nat,fun(product_prod(nat,B),product_prod(nat,product_prod(nat,B))),product_Pair(nat,product_prod(nat,B)),A12),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),A23),A32))))
     => ( ! [F4: fun(nat,fun(B,B)),A3: nat,B3: nat,Acc3: B] :
            ( accp(product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),set_fo1817059534552279752at_rel(B),aa(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),aa(fun(nat,fun(B,B)),fun(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))),product_Pair(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),F4),aa(product_prod(nat,B),product_prod(nat,product_prod(nat,B)),aa(nat,fun(product_prod(nat,B),product_prod(nat,product_prod(nat,B))),product_Pair(nat,product_prod(nat,B)),A3),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),B3),Acc3))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A3)
               => aa(B,$o,aa(nat,fun(B,$o),aa(nat,fun(nat,fun(B,$o)),aa(fun(nat,fun(B,B)),fun(nat,fun(nat,fun(B,$o))),Pa,F4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat))),B3),aa(B,B,aa(nat,fun(B,B),F4,A3),Acc3)) )
             => aa(B,$o,aa(nat,fun(B,$o),aa(nat,fun(nat,fun(B,$o)),aa(fun(nat,fun(B,B)),fun(nat,fun(nat,fun(B,$o))),Pa,F4),A3),B3),Acc3) ) )
       => aa(B,$o,aa(nat,fun(B,$o),aa(nat,fun(nat,fun(B,$o)),aa(fun(nat,fun(B,B)),fun(nat,fun(nat,fun(B,$o))),Pa,A0),A12),A23),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_6917_Gcd__fin__0__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A4: set(B)] :
          ( ( aa(set(B),B,semiring_gcd_Gcd_fin(B),A4) = zero_zero(B) )
        <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(B),set(B),insert(B,zero_zero(B)),bot_bot(set(B))))
            & aa(set(B),$o,finite_finite2(B),A4) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_6918_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [B: $tType,X: fun(nat,fun(B,B)),Xa2: nat,Xb: nat,Xc: B,Y: B] :
      ( ( set_fo6178422350223883121st_nat(B,X,Xa2,Xb,Xc) = Y )
     => ( accp(product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),set_fo1817059534552279752at_rel(B),aa(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),aa(fun(nat,fun(B,B)),fun(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))),product_Pair(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),X),aa(product_prod(nat,B),product_prod(nat,product_prod(nat,B)),aa(nat,fun(product_prod(nat,B),product_prod(nat,product_prod(nat,B))),product_Pair(nat,product_prod(nat,B)),Xa2),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),Xb),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa2),Xc,set_fo6178422350223883121st_nat(B,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(B,B,aa(nat,fun(B,B),X,Xa2),Xc))) )
           => ~ accp(product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),set_fo1817059534552279752at_rel(B),aa(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),aa(fun(nat,fun(B,B)),fun(product_prod(nat,product_prod(nat,B)),product_prod(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B)))),product_Pair(fun(nat,fun(B,B)),product_prod(nat,product_prod(nat,B))),X),aa(product_prod(nat,B),product_prod(nat,product_prod(nat,B)),aa(nat,fun(product_prod(nat,B),product_prod(nat,product_prod(nat,B))),product_Pair(nat,product_prod(nat,B)),Xa2),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_6919_pairself_Opelims,axiom,
    ! [B: $tType,C: $tType,X: fun(C,B),Xa2: product_prod(C,C),Y: product_prod(B,B)] :
      ( ( aa(product_prod(C,C),product_prod(B,B),pairself(C,B,X),Xa2) = Y )
     => ( accp(product_prod(fun(C,B),product_prod(C,C)),pairself_rel(C,B),aa(product_prod(C,C),product_prod(fun(C,B),product_prod(C,C)),aa(fun(C,B),fun(product_prod(C,C),product_prod(fun(C,B),product_prod(C,C))),product_Pair(fun(C,B),product_prod(C,C)),X),Xa2))
       => ~ ! [A3: C,B3: C] :
              ( ( Xa2 = aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),A3),B3) )
             => ( ( Y = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(C,B,X,A3)),aa(C,B,X,B3)) )
               => ~ accp(product_prod(fun(C,B),product_prod(C,C)),pairself_rel(C,B),aa(product_prod(C,C),product_prod(fun(C,B),product_prod(C,C)),aa(fun(C,B),fun(product_prod(C,C),product_prod(fun(C,B),product_prod(C,C))),product_Pair(fun(C,B),product_prod(C,C)),X),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),A3),B3))) ) ) ) ) ).

% pairself.pelims
tff(fact_6920_finite__enum__subset,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [X5: set(B),Y4: set(B)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(B),nat,finite_card(B),X5))
             => ( infini527867602293511546merate(B,X5,I2) = infini527867602293511546merate(B,Y4,I2) ) )
         => ( aa(set(B),$o,finite_finite2(B),X5)
           => ( aa(set(B),$o,finite_finite2(B),Y4)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),X5)),aa(set(B),nat,finite_card(B),Y4))
               => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),Y4) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_6921_le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N)) ) ).

% le_enumerate
tff(fact_6922_finite__le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(nat),nat,finite_card(nat),S))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N)) ) ) ).

% finite_le_enumerate
tff(fact_6923_enumerate__Suc_H,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [S: set(B),N: nat] : infini527867602293511546merate(B,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,infini527867602293511546merate(B,S,zero_zero(nat))),bot_bot(set(B)))),N) ) ).

% enumerate_Suc'
tff(fact_6924_partition__rev_Opelims,axiom,
    ! [B: $tType,X: fun(B,$o),Xa2: product_prod(list(B),list(B)),Xb: list(B),Y: product_prod(list(B),list(B))] :
      ( ( partition_rev(B,X,Xa2,Xb) = Y )
     => ( accp(product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),partition_rev_rel(B),aa(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),aa(fun(B,$o),fun(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B)))),product_Pair(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),X),aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),Xa2),Xb)))
       => ( ! [Yes: list(B),No: list(B)] :
              ( ( Xa2 = aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Yes),No) )
             => ( ( Xb = nil(B) )
               => ( ( Y = aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Yes),No) )
                 => ~ accp(product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),partition_rev_rel(B),aa(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),aa(fun(B,$o),fun(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B)))),product_Pair(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),X),aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),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)),Yes),No)),nil(B)))) ) ) )
         => ~ ! [Yes: list(B),No: list(B)] :
                ( ( Xa2 = 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)),Yes),No) )
               => ! [X2: B,Xs2: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
                   => ( ( Y = partition_rev(B,X,
                            $ite(aa(B,$o,X,X2),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),X2),Yes)),No),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)),Yes),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),No))),
                            Xs2) )
                     => ~ accp(product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),partition_rev_rel(B),aa(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),aa(fun(B,$o),fun(product_prod(product_prod(list(B),list(B)),list(B)),product_prod(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B)))),product_Pair(fun(B,$o),product_prod(product_prod(list(B),list(B)),list(B))),X),aa(list(B),product_prod(product_prod(list(B),list(B)),list(B)),aa(product_prod(list(B),list(B)),fun(list(B),product_prod(product_prod(list(B),list(B)),list(B))),product_Pair(product_prod(list(B),list(B)),list(B)),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)),Yes),No)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)))) ) ) ) ) ) ) ).

% partition_rev.pelims
tff(fact_6925_quicksort__by__rel_Opsimps_I2_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B),X: B,Xs: list(B)] :
      ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R2),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Sl2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))))
     => ( aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(product_prod(list(B),list(B)),list(B),aa(fun(list(B),fun(list(B),list(B))),fun(product_prod(list(B),list(B)),list(B)),product_case_prod(list(B),list(B),list(B)),aa(B,fun(list(B),fun(list(B),list(B))),aa(list(B),fun(B,fun(list(B),fun(list(B),list(B)))),aTP_Lamp_rp(fun(B,fun(B,$o)),fun(list(B),fun(B,fun(list(B),fun(list(B),list(B))))),R2),Sl2),X)),partition_rev(B,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),R2),X),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)),nil(B)),nil(B)),Xs)) ) ) ).

% quicksort_by_rel.psimps(2)
tff(fact_6926_quicksort__by__rel_Opsimps_I1_J,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),Sl2: list(B)] :
      ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R2),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)),Sl2),nil(B))))
     => ( aa(list(B),list(B),quicksort_by_rel(B,R2,Sl2),nil(B)) = Sl2 ) ) ).

% quicksort_by_rel.psimps(1)
tff(fact_6927_quicksort__by__rel_Opinduct,axiom,
    ! [B: $tType,A0: fun(B,fun(B,$o)),A12: list(B),A23: list(B),Pa: fun(fun(B,fun(B,$o)),fun(list(B),fun(list(B),$o)))] :
      ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),A0),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),A12),A23)))
     => ( ! [R4: fun(B,fun(B,$o)),Sl: list(B)] :
            ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Sl),nil(B))))
           => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(B,fun(B,$o)),fun(list(B),fun(list(B),$o)),Pa,R4),Sl),nil(B)) )
       => ( ! [R4: fun(B,fun(B,$o)),Sl: list(B),X2: B,Xs2: list(B)] :
              ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),R4),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Sl),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2))))
             => ( ! [Xa: product_prod(list(B),list(B)),Xb2: list(B),Y3: list(B)] :
                    ( ( Xa = partition_rev(B,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),R4),X2),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)),nil(B)),nil(B)),Xs2) )
                   => ( ( 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)),Xb2),Y3) = Xa )
                     => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(B,fun(B,$o)),fun(list(B),fun(list(B),$o)),Pa,R4),Sl),Y3) ) )
               => ( ! [Xa: product_prod(list(B),list(B)),Xb2: list(B),Y3: list(B)] :
                      ( ( Xa = partition_rev(B,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),R4),X2),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B)),Xs2) )
                     => ( ( aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Xb2),Y3) = Xa )
                       => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(B,fun(B,$o)),fun(list(B),fun(list(B),$o)),Pa,R4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),aa(list(B),list(B),quicksort_by_rel(B,R4,Sl),Y3))),Xb2) ) )
                 => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(B,fun(B,$o)),fun(list(B),fun(list(B),$o)),Pa,R4),Sl),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)) ) ) )
         => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(B,fun(B,$o)),fun(list(B),fun(list(B),$o)),Pa,A0),A12),A23) ) ) ) ).

% quicksort_by_rel.pinduct
tff(fact_6928_quicksort__by__rel_Opelims,axiom,
    ! [B: $tType,X: fun(B,fun(B,$o)),Xa2: list(B),Xb: list(B),Y: list(B)] :
      ( ( aa(list(B),list(B),quicksort_by_rel(B,X,Xa2),Xb) = Y )
     => ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Xa2),Xb)))
       => ( ( ( Xb = nil(B) )
           => ( ( Y = Xa2 )
             => ~ accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),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)),Xa2),nil(B)))) ) )
         => ~ ! [X2: B,Xs2: list(B)] :
                ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
               => ( ( Y = aa(product_prod(list(B),list(B)),list(B),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(B,fun(list(B),fun(list(B),list(B))),aa(list(B),fun(B,fun(list(B),fun(list(B),list(B)))),aTP_Lamp_rp(fun(B,fun(B,$o)),fun(list(B),fun(B,fun(list(B),fun(list(B),list(B))))),X),Xa2),X2)),partition_rev(B,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),X),X2),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)),nil(B)),nil(B)),Xs2)) )
                 => ~ accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),quicksort_by_rel_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),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)),Xa2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2)))) ) ) ) ) ) ).

% quicksort_by_rel.pelims
tff(fact_6929_mergesort__by__rel__merge_Opelims,axiom,
    ! [B: $tType,X: fun(B,fun(B,$o)),Xa2: list(B),Xb: list(B),Y: list(B)] :
      ( ( merges9089515139780605204_merge(B,X,Xa2,Xb) = Y )
     => ( accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),merges2244889521215249637ge_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),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)),Xa2),Xb)))
       => ( ! [X2: B,Xs2: list(B)] :
              ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2) )
             => ! [Y2: B,Ys3: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
                 => ( ( Y = $ite(aa(B,$o,aa(B,fun(B,$o),X,X2),Y2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),merges9089515139780605204_merge(B,X,Xs2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),merges9089515139780605204_merge(B,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs2),Ys3))) )
                   => ~ accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),merges2244889521215249637ge_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),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),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3)))) ) ) )
         => ( ( ( Xb = nil(B) )
             => ( ( Y = Xa2 )
               => ~ accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),merges2244889521215249637ge_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Xa2),nil(B)))) ) )
           => ~ ( ( Xa2 = nil(B) )
               => ! [V4: B,Va: list(B)] :
                    ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
                   => ( ( Y = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
                     => ~ accp(product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),merges2244889521215249637ge_rel(B),aa(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B))),aa(fun(B,fun(B,$o)),fun(product_prod(list(B),list(B)),product_prod(fun(B,fun(B,$o)),product_prod(list(B),list(B)))),product_Pair(fun(B,fun(B,$o)),product_prod(list(B),list(B))),X),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)),nil(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_6930_finite__enumerate__Suc_H_H,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [S: set(B),N: nat] :
          ( aa(set(B),$o,finite_finite2(B),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),aa(set(B),nat,finite_card(B),S))
           => ( infini527867602293511546merate(B,S,aa(nat,nat,suc,N)) = ord_Least(B,aa(nat,fun(B,$o),aTP_Lamp_ahb(set(B),fun(nat,fun(B,$o)),S),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_6931_Least__eq__0,axiom,
    ! [Pa: fun(nat,$o)] :
      ( aa(nat,$o,Pa,zero_zero(nat))
     => ( ord_Least(nat,Pa) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_6932_Least__le,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),K: B] :
          ( aa(B,$o,Pa,K)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),ord_Least(B,Pa)),K) ) ) ).

% Least_le
tff(fact_6933_Least1I,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o)] :
          ( ? [X4: B] :
              ( aa(B,$o,Pa,X4)
              & ! [Y2: B] :
                  ( aa(B,$o,Pa,Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y2) )
              & ! [Y2: B] :
                  ( ( aa(B,$o,Pa,Y2)
                    & ! [Ya2: B] :
                        ( aa(B,$o,Pa,Ya2)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),Ya2) ) )
                 => ( Y2 = X4 ) ) )
         => aa(B,$o,Pa,ord_Least(B,Pa)) ) ) ).

% Least1I
tff(fact_6934_Least1__le,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o),Z: B] :
          ( ? [X4: B] :
              ( aa(B,$o,Pa,X4)
              & ! [Y2: B] :
                  ( aa(B,$o,Pa,Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y2) )
              & ! [Y2: B] :
                  ( ( aa(B,$o,Pa,Y2)
                    & ! [Ya2: B] :
                        ( aa(B,$o,Pa,Ya2)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y2),Ya2) ) )
                 => ( Y2 = X4 ) ) )
         => ( aa(B,$o,Pa,Z)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),ord_Least(B,Pa)),Z) ) ) ) ).

% Least1_le
tff(fact_6935_LeastI2__order,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o),X: B,Q: fun(B,$o)] :
          ( aa(B,$o,Pa,X)
         => ( ! [Y2: B] :
                ( aa(B,$o,Pa,Y2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2) )
           => ( ! [X2: B] :
                  ( aa(B,$o,Pa,X2)
                 => ( ! [Y3: B] :
                        ( aa(B,$o,Pa,Y3)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y3) )
                   => aa(B,$o,Q,X2) ) )
             => aa(B,$o,Q,ord_Least(B,Pa)) ) ) ) ) ).

% LeastI2_order
tff(fact_6936_Least__equality,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o),X: B] :
          ( aa(B,$o,Pa,X)
         => ( ! [Y2: B] :
                ( aa(B,$o,Pa,Y2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y2) )
           => ( ord_Least(B,Pa) = X ) ) ) ) ).

% Least_equality
tff(fact_6937_LeastI2__wellorder,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),A2: B,Q: fun(B,$o)] :
          ( aa(B,$o,Pa,A2)
         => ( ! [A3: B] :
                ( aa(B,$o,Pa,A3)
               => ( ! [B7: B] :
                      ( aa(B,$o,Pa,B7)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B7) )
                 => aa(B,$o,Q,A3) ) )
           => aa(B,$o,Q,ord_Least(B,Pa)) ) ) ) ).

% LeastI2_wellorder
tff(fact_6938_LeastI2__wellorder__ex,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),Q: fun(B,$o)] :
          ( ? [X_1: B] : aa(B,$o,Pa,X_1)
         => ( ! [A3: B] :
                ( aa(B,$o,Pa,A3)
               => ( ! [B7: B] :
                      ( aa(B,$o,Pa,B7)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B7) )
                 => aa(B,$o,Q,A3) ) )
           => aa(B,$o,Q,ord_Least(B,Pa)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_6939_not__less__Least,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [K: B,Pa: fun(B,$o)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),K),ord_Least(B,Pa))
         => ~ aa(B,$o,Pa,K) ) ) ).

% not_less_Least
tff(fact_6940_LeastI2__ex,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),Q: fun(B,$o)] :
          ( ? [X_1: B] : aa(B,$o,Pa,X_1)
         => ( ! [X2: B] :
                ( aa(B,$o,Pa,X2)
               => aa(B,$o,Q,X2) )
           => aa(B,$o,Q,ord_Least(B,Pa)) ) ) ) ).

% LeastI2_ex
tff(fact_6941_LeastI__ex,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o)] :
          ( ? [X_1: B] : aa(B,$o,Pa,X_1)
         => aa(B,$o,Pa,ord_Least(B,Pa)) ) ) ).

% LeastI_ex
tff(fact_6942_LeastI2,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),A2: B,Q: fun(B,$o)] :
          ( aa(B,$o,Pa,A2)
         => ( ! [X2: B] :
                ( aa(B,$o,Pa,X2)
               => aa(B,$o,Q,X2) )
           => aa(B,$o,Q,ord_Least(B,Pa)) ) ) ) ).

% LeastI2
tff(fact_6943_LeastI,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Pa: fun(B,$o),K: B] :
          ( aa(B,$o,Pa,K)
         => aa(B,$o,Pa,ord_Least(B,Pa)) ) ) ).

% LeastI
tff(fact_6944_Least__Suc2,axiom,
    ! [Pa: fun(nat,$o),N: nat,Q: fun(nat,$o),Ma: nat] :
      ( aa(nat,$o,Pa,N)
     => ( aa(nat,$o,Q,Ma)
       => ( ~ aa(nat,$o,Pa,zero_zero(nat))
         => ( ! [K4: nat] :
                ( aa(nat,$o,Pa,aa(nat,nat,suc,K4))
              <=> aa(nat,$o,Q,K4) )
           => ( ord_Least(nat,Pa) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_6945_Least__Suc,axiom,
    ! [Pa: fun(nat,$o),N: nat] :
      ( aa(nat,$o,Pa,N)
     => ( ~ aa(nat,$o,Pa,zero_zero(nat))
       => ( ord_Least(nat,Pa) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ahc(fun(nat,$o),fun(nat,$o),Pa))) ) ) ) ).

% Least_Suc
tff(fact_6946_Least__Min,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o)] :
          ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Pa))
         => ( ? [X_1: B] : aa(B,$o,Pa,X_1)
           => ( ord_Least(B,Pa) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(fun(B,$o),set(B),collect(B),Pa)) ) ) ) ) ).

% Least_Min
tff(fact_6947_abort__Bleast__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [S: set(B),Pa: fun(B,$o)] : abort_Bleast(B,S,Pa) = ord_Least(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ahd(set(B),fun(fun(B,$o),fun(B,$o)),S),Pa)) ) ).

% abort_Bleast_def
tff(fact_6948_Bleast__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [S: set(B),Pa: fun(B,$o)] : bleast(B,S,Pa) = ord_Least(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ahd(set(B),fun(fun(B,$o),fun(B,$o)),S),Pa)) ) ).

% Bleast_def
tff(fact_6949_enumerate__0,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [S: set(B)] : infini527867602293511546merate(B,S,zero_zero(nat)) = ord_Least(B,aTP_Lamp_ahe(set(B),fun(B,$o),S)) ) ).

% enumerate_0
tff(fact_6950_Least__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [F: fun(B,C),S: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( ? [X4: B] :
                ( aa(set(B),$o,member(B,X4),S)
                & ! [Xa3: B] :
                    ( aa(set(B),$o,member(B,Xa3),S)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Xa3) ) )
           => ( ord_Least(C,aa(set(B),fun(C,$o),aTP_Lamp_ahf(fun(B,C),fun(set(B),fun(C,$o)),F),S)) = aa(B,C,F,ord_Least(B,aTP_Lamp_ahg(set(B),fun(B,$o),S))) ) ) ) ) ).

% Least_mono
tff(fact_6951_enumerate__Suc_H_H,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [S: set(B),N: nat] :
          ( ~ aa(set(B),$o,finite_finite2(B),S)
         => ( infini527867602293511546merate(B,S,aa(nat,nat,suc,N)) = ord_Least(B,aa(nat,fun(B,$o),aTP_Lamp_ahb(set(B),fun(nat,fun(B,$o)),S),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_6952_enumerate__Suc,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [S: set(B),N: nat] : infini527867602293511546merate(B,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert(B,ord_Least(B,aTP_Lamp_ahe(set(B),fun(B,$o),S))),bot_bot(set(B)))),N) ) ).

% enumerate_Suc
tff(fact_6953_size__Diff__singleton,axiom,
    ! [B: $tType,X: B,M: multiset(B)] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M))
     => ( aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(B),nat,size_size(multiset(B)),M)),one_one(nat)) ) ) ).

% size_Diff_singleton
tff(fact_6954_size__Diff__singleton__if,axiom,
    ! [B: $tType,A4: multiset(B),X: B] :
      aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B))))) = $ite(aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),A4)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(B),nat,size_size(multiset(B)),A4)),one_one(nat)),aa(multiset(B),nat,size_size(multiset(B)),A4)) ).

% size_Diff_singleton_if
tff(fact_6955_set__mset__union,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] : aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(multiset(B),set(B),set_mset(B),M)),aa(multiset(B),set(B),set_mset(B),N5)) ).

% set_mset_union
tff(fact_6956_sum__mset__0__iff,axiom,
    ! [B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [M: multiset(B)] :
          ( ( comm_m7189776963980413722m_mset(B,M) = zero_zero(B) )
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),aa(multiset(B),set(B),set_mset(B),M))
             => ( X3 = zero_zero(B) ) ) ) ) ).

% sum_mset_0_iff
tff(fact_6957_mset__diff__cancel1elem,axiom,
    ! [B: $tType,A2: B,B4: multiset(B)] :
      ( ~ aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4))
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))),B4) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))) ) ) ).

% mset_diff_cancel1elem
tff(fact_6958_set__mset__Union__mset,axiom,
    ! [B: $tType,MM: multiset(multiset(B))] : aa(multiset(B),set(B),set_mset(B),comm_m7189776963980413722m_mset(multiset(B),MM)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(multiset(B)),set(set(B)),image2(multiset(B),set(B),set_mset(B)),aa(multiset(multiset(B)),set(multiset(B)),set_mset(multiset(B)),MM))) ).

% set_mset_Union_mset
tff(fact_6959_Inf__nat__def,axiom,
    ! [X5: set(nat)] : aa(set(nat),nat,complete_Inf_Inf(nat),X5) = ord_Least(nat,aTP_Lamp_ahh(set(nat),fun(nat,$o),X5)) ).

% Inf_nat_def
tff(fact_6960_in__image__mset,axiom,
    ! [B: $tType,C: $tType,Y: B,F: fun(C,B),M: multiset(C)] :
      ( aa(set(B),$o,member(B,Y),aa(multiset(B),set(B),set_mset(B),aa(multiset(C),multiset(B),image_mset(C,B,F),M)))
    <=> aa(set(B),$o,member(B,Y),aa(set(C),set(B),image2(C,B,F),aa(multiset(C),set(C),set_mset(C),M))) ) ).

% in_image_mset
tff(fact_6961_set__mset__Inf,axiom,
    ! [B: $tType,A4: set(multiset(B))] :
      ( ( A4 != bot_bot(set(multiset(B))) )
     => ( aa(multiset(B),set(B),set_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),A4)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(multiset(B)),set(set(B)),image2(multiset(B),set(B),set_mset(B)),A4)) ) ) ).

% set_mset_Inf
tff(fact_6962_union__iff,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B4: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)))
    <=> ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4))
        | aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4)) ) ) ).

% union_iff
tff(fact_6963_mset__un__cases,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B4: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)))
     => ( ~ aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4))
       => aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4)) ) ) ).

% mset_un_cases
tff(fact_6964_mset__left__cancel__union,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B4: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)))
     => ( ~ aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4))
       => aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4)) ) ) ).

% mset_left_cancel_union
tff(fact_6965_mset__right__cancel__union,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B4: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)))
     => ( ~ aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4))
       => aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4)) ) ) ).

% mset_right_cancel_union
tff(fact_6966_ex__Melem__conv,axiom,
    ! [B: $tType,A4: multiset(B)] :
      ( ? [X3: B] : aa(set(B),$o,member(B,X3),aa(multiset(B),set(B),set_mset(B),A4))
    <=> ( A4 != zero_zero(multiset(B)) ) ) ).

% ex_Melem_conv
tff(fact_6967_size__eq__Suc__imp__elem,axiom,
    ! [B: $tType,M: multiset(B),N: nat] :
      ( ( aa(multiset(B),nat,size_size(multiset(B)),M) = aa(nat,nat,suc,N) )
     => ? [A3: B] : aa(set(B),$o,member(B,A3),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% size_eq_Suc_imp_elem
tff(fact_6968_image__mset__cong__pair,axiom,
    ! [D: $tType,C: $tType,B: $tType,M: multiset(product_prod(B,C)),F: fun(B,fun(C,D)),G: fun(B,fun(C,D))] :
      ( ! [X2: B,Y2: C] :
          ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2)),aa(multiset(product_prod(B,C)),set(product_prod(B,C)),set_mset(product_prod(B,C)),M))
         => ( aa(C,D,aa(B,fun(C,D),F,X2),Y2) = aa(C,D,aa(B,fun(C,D),G,X2),Y2) ) )
     => ( aa(multiset(product_prod(B,C)),multiset(D),image_mset(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),F)),M) = aa(multiset(product_prod(B,C)),multiset(D),image_mset(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G)),M) ) ) ).

% image_mset_cong_pair
tff(fact_6969_image__mset__cong,axiom,
    ! [C: $tType,B: $tType,M: multiset(B),F: fun(B,C),G: fun(B,C)] :
      ( ! [X2: B] :
          ( aa(set(B),$o,member(B,X2),aa(multiset(B),set(B),set_mset(B),M))
         => ( aa(B,C,F,X2) = aa(B,C,G,X2) ) )
     => ( aa(multiset(B),multiset(C),image_mset(B,C,F),M) = aa(multiset(B),multiset(C),image_mset(B,C,G),M) ) ) ).

% image_mset_cong
tff(fact_6970_sum__mset_Oneutral,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: multiset(B)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(multiset(B),set(B),set_mset(B),A4))
             => ( X2 = zero_zero(B) ) )
         => ( comm_m7189776963980413722m_mset(B,A4) = zero_zero(B) ) ) ) ).

% sum_mset.neutral
tff(fact_6971_multiset__induct__min,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(multiset(B),$o),M: multiset(B)] :
          ( aa(multiset(B),$o,Pa,zero_zero(multiset(B)))
         => ( ! [X2: B,M8: multiset(B)] :
                ( aa(multiset(B),$o,Pa,M8)
               => ( ! [Xa: B] :
                      ( aa(set(B),$o,member(B,Xa),aa(multiset(B),set(B),set_mset(B),M8))
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Xa) )
                 => aa(multiset(B),$o,Pa,aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X2),M8)) ) )
           => aa(multiset(B),$o,Pa,M) ) ) ) ).

% multiset_induct_min
tff(fact_6972_multiset__induct__max,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(multiset(B),$o),M: multiset(B)] :
          ( aa(multiset(B),$o,Pa,zero_zero(multiset(B)))
         => ( ! [X2: B,M8: multiset(B)] :
                ( aa(multiset(B),$o,Pa,M8)
               => ( ! [Xa: B] :
                      ( aa(set(B),$o,member(B,Xa),aa(multiset(B),set(B),set_mset(B),M8))
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),X2) )
                 => aa(multiset(B),$o,Pa,aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X2),M8)) ) )
           => aa(multiset(B),$o,Pa,M) ) ) ) ).

% multiset_induct_max
tff(fact_6973_mset__right__cancel__elem,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B2: B] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B))))))
     => ( ( A2 != B2 )
       => aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4)) ) ) ).

% mset_right_cancel_elem
tff(fact_6974_multi__member__this,axiom,
    ! [B: $tType,X: B,XS: multiset(B)] : aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),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),X),zero_zero(multiset(B)))),XS))) ).

% multi_member_this
tff(fact_6975_multi__member__skip,axiom,
    ! [B: $tType,X: B,XS: multiset(B),Y: B] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),XS))
     => aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),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),Y),zero_zero(multiset(B)))),XS))) ) ).

% multi_member_skip
tff(fact_6976_mset__left__cancel__elem,axiom,
    ! [B: $tType,A2: B,B2: B,A4: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),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),B2),zero_zero(multiset(B)))),A4)))
     => ( ( A2 != B2 )
       => aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4)) ) ) ).

% mset_left_cancel_elem
tff(fact_6977_sum__mset__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [K3: multiset(B),F: fun(B,C),G: fun(B,C)] :
          ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),aa(multiset(B),set(B),set_mset(B),K3))
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,I2)),aa(B,C,G,I2)) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),comm_m7189776963980413722m_mset(C,aa(multiset(B),multiset(C),image_mset(B,C,F),K3))),comm_m7189776963980413722m_mset(C,aa(multiset(B),multiset(C),image_mset(B,C,G),K3))) ) ) ).

% sum_mset_mono
tff(fact_6978_mset__2dist2__cases,axiom,
    ! [B: $tType,A2: B,B2: B,A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B))))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4))
     => ( ~ aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B))))),A4)
       => ( ~ aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B))))),B4)
         => ( ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4))
             => ~ aa(set(B),$o,member(B,B2),aa(multiset(B),set(B),set_mset(B),B4)) )
           => ~ ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4))
               => ~ aa(set(B),$o,member(B,B2),aa(multiset(B),set(B),set_mset(B),A4)) ) ) ) ) ) ).

% mset_2dist2_cases
tff(fact_6979_mset__union__subset__s,axiom,
    ! [B: $tType,A2: B,B4: multiset(B),C6: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))),B4)),C6)
     => ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),C6))
        & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),C6) ) ) ).

% mset_union_subset_s
tff(fact_6980_mset__le__mono__add__single,axiom,
    ! [B: $tType,A2: B,Ys: multiset(B),B2: B,Ws: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),Ys))
     => ( aa(set(B),$o,member(B,B2),aa(multiset(B),set(B),set_mset(B),Ws))
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),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),A2),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B))))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Ys),Ws)) ) ) ).

% mset_le_mono_add_single
tff(fact_6981_nth__mem__mset,axiom,
    ! [B: $tType,I: nat,Ls: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ls))
     => aa(set(B),$o,member(B,aa(nat,B,nth(B,Ls),I)),aa(multiset(B),set(B),set_mset(B),mset(B,Ls))) ) ).

% nth_mem_mset
tff(fact_6982_diff__union__single__conv,axiom,
    ! [B: $tType,A2: B,J4: multiset(B),I5: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),J4))
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I5),J4)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I5),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),J4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))))) ) ) ).

% diff_union_single_conv
tff(fact_6983_mset__un__single__un__cases,axiom,
    ! [B: $tType,A2: B,A4: multiset(B),B4: multiset(B),C6: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),A4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6) )
     => ( ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),B4))
         => ( A4 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))))),C6) ) )
       => ~ ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),C6))
           => ( A4 != aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))))) ) ) ) ) ).

% mset_un_single_un_cases
tff(fact_6984_diff__union__single__conv2,axiom,
    ! [B: $tType,A2: B,J4: multiset(B),I5: multiset(B)] :
      ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),J4))
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),J4),I5)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B)))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),J4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))))),I5) ) ) ).

% diff_union_single_conv2
tff(fact_6985_mset__union__diff__comm,axiom,
    ! [B: $tType,Ta: B,S: multiset(B),T4: multiset(B)] :
      ( aa(set(B),$o,member(B,Ta),aa(multiset(B),set(B),set_mset(B),S))
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),T4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),S),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Ta),zero_zero(multiset(B))))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),T4),S)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Ta),zero_zero(multiset(B)))) ) ) ).

% mset_union_diff_comm
tff(fact_6986_insert__DiffM2,axiom,
    ! [B: $tType,X: B,M: multiset(B)] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M))
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B))))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))) = M ) ) ).

% insert_DiffM2
tff(fact_6987_mset__contains__eq,axiom,
    ! [B: $tType,Ma: B,M: multiset(B)] :
      ( aa(set(B),$o,member(B,Ma),aa(multiset(B),set(B),set_mset(B),M))
    <=> ( 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),Ma),zero_zero(multiset(B)))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Ma),zero_zero(multiset(B))))) = M ) ) ).

% mset_contains_eq
tff(fact_6988_mset__size1elem,axiom,
    ! [B: $tType,Pa: multiset(B),Q2: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(B),nat,size_size(multiset(B)),Pa)),one_one(nat))
     => ( aa(set(B),$o,member(B,Q2),aa(multiset(B),set(B),set_mset(B),Pa))
       => ( Pa = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Q2),zero_zero(multiset(B))) ) ) ) ).

% mset_size1elem
tff(fact_6989_size__Suc__Diff1,axiom,
    ! [B: $tType,X: B,M: multiset(B)] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M))
     => ( aa(nat,nat,suc,aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))))) = aa(multiset(B),nat,size_size(multiset(B)),M) ) ) ).

% size_Suc_Diff1
tff(fact_6990_size__Diff1__less,axiom,
    ! [B: $tType,X: B,M: multiset(B)] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))))),aa(multiset(B),nat,size_size(multiset(B)),M)) ) ).

% size_Diff1_less
tff(fact_6991_size__Diff2__less,axiom,
    ! [B: $tType,X: B,M: multiset(B),Y: B] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M))
     => ( aa(set(B),$o,member(B,Y),aa(multiset(B),set(B),set_mset(B),M))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B))))),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Y),zero_zero(multiset(B)))))),aa(multiset(B),nat,size_size(multiset(B)),M)) ) ) ).

% size_Diff2_less
tff(fact_6992_sum__mset_Oremove,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [X: B,A4: multiset(B)] :
          ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),A4))
         => ( comm_m7189776963980413722m_mset(B,A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),X),comm_m7189776963980413722m_mset(B,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),zero_zero(multiset(B)))))) ) ) ) ).

% sum_mset.remove
tff(fact_6993_size__diff__se,axiom,
    ! [B: $tType,Ta: B,S: multiset(B)] :
      ( aa(set(B),$o,member(B,Ta),aa(multiset(B),set(B),set_mset(B),S))
     => ( aa(multiset(B),nat,size_size(multiset(B)),S) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),S),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Ta),zero_zero(multiset(B)))))),one_one(nat)) ) ) ).

% size_diff_se
tff(fact_6994_subset__mset_Osum__mset__mono,axiom,
    ! [C: $tType,B: $tType,K3: multiset(B),F: fun(B,multiset(C)),G: fun(B,multiset(C))] :
      ( ! [I2: B] :
          ( aa(set(B),$o,member(B,I2),aa(multiset(B),set(B),set_mset(B),K3))
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),aa(B,multiset(C),F,I2)),aa(B,multiset(C),G,I2)) )
     => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),comm_monoid_sum_mset(multiset(C),plus_plus(multiset(C)),zero_zero(multiset(C)),aa(multiset(B),multiset(multiset(C)),image_mset(B,multiset(C),F),K3))),comm_monoid_sum_mset(multiset(C),plus_plus(multiset(C)),zero_zero(multiset(C)),aa(multiset(B),multiset(multiset(C)),image_mset(B,multiset(C),G),K3))) ) ).

% subset_mset.sum_mset_mono
tff(fact_6995_subset__mset_Osum__mset__0__iff,axiom,
    ! [B: $tType,M: multiset(multiset(B))] :
      ( ( comm_monoid_sum_mset(multiset(B),plus_plus(multiset(B)),zero_zero(multiset(B)),M) = zero_zero(multiset(B)) )
    <=> ! [X3: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X3),aa(multiset(multiset(B)),set(multiset(B)),set_mset(multiset(B)),M))
         => ( X3 = zero_zero(multiset(B)) ) ) ) ).

% subset_mset.sum_mset_0_iff
tff(fact_6996_multiset_Oin__rel,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),A2: multiset(B),B2: multiset(C)] :
      ( aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),rel_mset(B,C,R2),A2),B2)
    <=> ? [Z5: multiset(product_prod(B,C))] :
          ( aa(set(multiset(product_prod(B,C))),$o,member(multiset(product_prod(B,C)),Z5),aa(fun(multiset(product_prod(B,C)),$o),set(multiset(product_prod(B,C))),collect(multiset(product_prod(B,C))),aTP_Lamp_ahi(fun(B,fun(C,$o)),fun(multiset(product_prod(B,C)),$o),R2)))
          & ( aa(multiset(product_prod(B,C)),multiset(B),image_mset(product_prod(B,C),B,product_fst(B,C)),Z5) = A2 )
          & ( aa(multiset(product_prod(B,C)),multiset(C),image_mset(product_prod(B,C),C,product_snd(B,C)),Z5) = B2 ) ) ) ).

% multiset.in_rel
tff(fact_6997_count__image__mset,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A4: multiset(C),X: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(C),multiset(B),image_mset(C,B,F),A4)),X) = aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7311177749621191930dd_sum(C,nat),aa(multiset(C),fun(C,nat),count(C),A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(B),set(C),aa(fun(C,B),fun(set(B),set(C)),vimage(C,B),F),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))),aa(multiset(C),set(C),set_mset(C),A4))) ).

% count_image_mset
tff(fact_6998_mset__empty__count,axiom,
    ! [B: $tType,M: multiset(B)] :
      ( ! [P6: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),P6) = zero_zero(nat)
    <=> ( M = zero_zero(multiset(B)) ) ) ).

% mset_empty_count
tff(fact_6999_count__empty,axiom,
    ! [B: $tType,A2: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),zero_zero(multiset(B))),A2) = zero_zero(nat) ).

% count_empty
tff(fact_7000_count__union,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B),A2: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)),A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),A2)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),N5),A2)) ).

% count_union
tff(fact_7001_count__greater__zero__iff,axiom,
    ! [B: $tType,M: multiset(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X))
    <=> aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% count_greater_zero_iff
tff(fact_7002_count__mset__0__iff,axiom,
    ! [B: $tType,Xs: list(B),X: B] :
      ( ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset(B,Xs)),X) = zero_zero(nat) )
    <=> ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs)) ) ).

% count_mset_0_iff
tff(fact_7003_count__greater__eq__one__iff,axiom,
    ! [B: $tType,M: multiset(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X))
    <=> aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% count_greater_eq_one_iff
tff(fact_7004_count__greater__eq__Suc__zero__iff,axiom,
    ! [B: $tType,M: multiset(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X))
    <=> aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% count_greater_eq_Suc_zero_iff
tff(fact_7005_count__eq__zero__iff,axiom,
    ! [B: $tType,M: multiset(B),X: B] :
      ( ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X) = zero_zero(nat) )
    <=> ~ aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% count_eq_zero_iff
tff(fact_7006_count__inI,axiom,
    ! [B: $tType,M: multiset(B),X: B] :
      ( ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X) != zero_zero(nat) )
     => aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% count_inI
tff(fact_7007_multiset_Orel__map_I2_J,axiom,
    ! [B: $tType,C: $tType,D: $tType,Sa: fun(B,fun(C,$o)),X: multiset(B),G: fun(D,C),Y: multiset(D)] :
      ( aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),rel_mset(B,C,Sa),X),aa(multiset(D),multiset(C),image_mset(D,C,G),Y))
    <=> aa(multiset(D),$o,aa(multiset(B),fun(multiset(D),$o),rel_mset(B,D,aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_agu(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Sa),G)),X),Y) ) ).

% multiset.rel_map(2)
tff(fact_7008_multiset_Orel__map_I1_J,axiom,
    ! [D: $tType,B: $tType,C: $tType,Sb: fun(B,fun(C,$o)),I: fun(D,B),X: multiset(D),Y: multiset(C)] :
      ( aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),rel_mset(B,C,Sb),aa(multiset(D),multiset(B),image_mset(D,B,I),X)),Y)
    <=> aa(multiset(C),$o,aa(multiset(D),fun(multiset(C),$o),rel_mset(D,C,aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_agv(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Sb),I)),X),Y) ) ).

% multiset.rel_map(1)
tff(fact_7009_zero__multiset_Orep__eq,axiom,
    ! [B: $tType,X4: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),zero_zero(multiset(B))),X4) = zero_zero(nat) ).

% zero_multiset.rep_eq
tff(fact_7010_plus__multiset_Orep__eq,axiom,
    ! [B: $tType,X: multiset(B),Xa2: multiset(B),X4: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X),Xa2)),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),X),X4)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Xa2),X4)) ).

% plus_multiset.rep_eq
tff(fact_7011_count__sum,axiom,
    ! [B: $tType,C: $tType,F: fun(C,multiset(B)),A4: set(C),X: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(set(C),multiset(B),aa(fun(C,multiset(B)),fun(set(C),multiset(B)),groups7311177749621191930dd_sum(C,multiset(B)),F),A4)),X) = aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7311177749621191930dd_sum(C,nat),aa(B,fun(C,nat),aTP_Lamp_ahj(fun(C,multiset(B)),fun(B,fun(C,nat)),F),X)),A4) ).

% count_sum
tff(fact_7012_rel__mset__size,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),M: multiset(B),N5: multiset(C)] :
      ( aa(multiset(C),$o,aa(multiset(B),fun(multiset(C),$o),rel_mset(B,C,R2),M),N5)
     => ( aa(multiset(B),nat,size_size(multiset(B)),M) = aa(multiset(C),nat,size_size(multiset(C)),N5) ) ) ).

% rel_mset_size
tff(fact_7013_mset__subset__eq__count,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),A2: B] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),A2)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),B4),A2)) ) ).

% mset_subset_eq_count
tff(fact_7014_subseteq__mset__def,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4)
    <=> ! [A6: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),A6)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),B4),A6)) ) ).

% subseteq_mset_def
tff(fact_7015_mset__subset__eqI,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] :
      ( ! [A3: B] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),A3)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),B4),A3))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),B4) ) ).

% mset_subset_eqI
tff(fact_7016_count__ne__remove,axiom,
    ! [B: $tType,X: B,Ta: B,S: multiset(B)] :
      ( ( X != Ta )
     => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),S),X) = aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),S),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Ta),zero_zero(multiset(B))))),X) ) ) ).

% count_ne_remove
tff(fact_7017_count__in__diffI,axiom,
    ! [B: $tType,N5: multiset(B),X: B,M: multiset(B)] :
      ( ! [N2: nat] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),N5),X) != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X))
     => aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),N5))) ) ).

% count_in_diffI
tff(fact_7018_set__mset__def,axiom,
    ! [B: $tType,M: multiset(B)] : aa(multiset(B),set(B),set_mset(B),M) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ahk(multiset(B),fun(B,$o),M)) ).

% set_mset_def
tff(fact_7019_set__mset__diff,axiom,
    ! [B: $tType,M: multiset(B),N5: multiset(B)] : aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),N5)) = aa(fun(B,$o),set(B),collect(B),aa(multiset(B),fun(B,$o),aTP_Lamp_ahl(multiset(B),fun(multiset(B),fun(B,$o)),M),N5)) ).

% set_mset_diff
tff(fact_7020_count__mset,axiom,
    ! [B: $tType,Xs: list(B),X: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset(B,Xs)),X) = aa(list(B),nat,size_size(list(B)),filter2(B,aa(B,fun(B,$o),fequal(B),X),Xs)) ).

% count_mset
tff(fact_7021_count__image__mset_H,axiom,
    ! [C: $tType,B: $tType,F: fun(C,B),X5: multiset(C),Y: B] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(C),multiset(B),image_mset(C,B,F),X5)),Y) = aa(set(C),nat,aa(fun(C,nat),fun(set(C),nat),groups7311177749621191930dd_sum(C,nat),aa(multiset(C),fun(C,nat),count(C),X5)),aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),aa(multiset(C),fun(B,fun(C,$o)),aTP_Lamp_ahm(fun(C,B),fun(multiset(C),fun(B,fun(C,$o))),F),X5),Y))) ).

% count_image_mset'
tff(fact_7022_size__multiset__overloaded__eq,axiom,
    ! [B: $tType,X: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),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_7023_count,axiom,
    ! [B: $tType,X: multiset(B)] : aa(set(fun(B,nat)),$o,member(fun(B,nat),aa(multiset(B),fun(B,nat),count(B),X)),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o))) ).

% count
tff(fact_7024_count__cases,axiom,
    ! [B: $tType,Y: fun(B,nat)] :
      ( aa(set(fun(B,nat)),$o,member(fun(B,nat),Y),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o)))
     => ~ ! [X2: multiset(B)] : Y != aa(multiset(B),fun(B,nat),count(B),X2) ) ).

% count_cases
tff(fact_7025_count__induct,axiom,
    ! [B: $tType,Y: fun(B,nat),Pa: fun(fun(B,nat),$o)] :
      ( aa(set(fun(B,nat)),$o,member(fun(B,nat),Y),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o)))
     => ( ! [X2: multiset(B)] : aa(fun(B,nat),$o,Pa,aa(multiset(B),fun(B,nat),count(B),X2))
       => aa(fun(B,nat),$o,Pa,Y) ) ) ).

% count_induct
tff(fact_7026_Inf__multiset_Orep__eq,axiom,
    ! [B: $tType,X: set(multiset(B)),X4: B] :
      aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),X)),X4) = $ite(aa(set(multiset(B)),set(fun(B,nat)),image2(multiset(B),fun(B,nat),count(B)),X) = bot_bot(set(fun(B,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(B,nat)),set(nat),image2(fun(B,nat),nat,aTP_Lamp_aho(B,fun(fun(B,nat),nat),X4)),aa(set(multiset(B)),set(fun(B,nat)),image2(multiset(B),fun(B,nat),count(B)),X)))) ).

% Inf_multiset.rep_eq
tff(fact_7027_count__Inf__multiset__nonempty,axiom,
    ! [B: $tType,A4: set(multiset(B)),X: B] :
      ( ( A4 != bot_bot(set(multiset(B))) )
     => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),A4)),X) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(multiset(B)),set(nat),image2(multiset(B),nat,aTP_Lamp_ahp(B,fun(multiset(B),nat),X)),A4)) ) ) ).

% count_Inf_multiset_nonempty
tff(fact_7028_count__single,axiom,
    ! [B: $tType,B2: B,A2: B] :
      aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),B2),zero_zero(multiset(B)))),A2) = $ite(B2 = A2,one_one(nat),zero_zero(nat)) ).

% count_single
tff(fact_7029_count__mset__gt__0,axiom,
    ! [B: $tType,X: B,Xs: list(B)] :
      ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset(B,Xs)),X)) ) ).

% count_mset_gt_0
tff(fact_7030_in__diff__countE,axiom,
    ! [B: $tType,X: B,M: multiset(B),N5: multiset(B)] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),N5)))
     => ~ ! [N2: nat] : aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X) != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,N2)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),N5),X)) ) ).

% in_diff_countE
tff(fact_7031_distinct__count__atmost__1,axiom,
    ! [B: $tType,X: list(B)] :
      ( distinct(B,X)
    <=> ! [A6: B] :
          aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset(B,X)),A6) = $ite(aa(set(B),$o,member(B,A6),aa(list(B),set(B),set2(B),X)),one_one(nat),zero_zero(nat)) ) ).

% distinct_count_atmost_1
tff(fact_7032_sum__mset__delta_H,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_1(B)
     => ! [Y: C,C2: B,A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(B,fun(C,B),aTP_Lamp_ahq(C,fun(B,fun(C,B)),Y),C2)),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),aa(nat,B,semiring_1_of_nat(B),aa(C,nat,aa(multiset(C),fun(C,nat),count(C),A4),Y))) ) ).

% sum_mset_delta'
tff(fact_7033_sum__mset__delta,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_1(B)
     => ! [Y: C,C2: B,A4: multiset(C)] : comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(B,fun(C,B),aTP_Lamp_ahr(C,fun(B,fun(C,B)),Y),C2)),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),C2),aa(nat,B,semiring_1_of_nat(B),aa(C,nat,aa(multiset(C),fun(C,nat),count(C),A4),Y))) ) ).

% sum_mset_delta
tff(fact_7034_size__multiset__eq,axiom,
    ! [B: $tType,F: fun(B,nat),M: multiset(B)] : aa(multiset(B),nat,size_multiset(B,F),M) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),aTP_Lamp_ahs(fun(B,nat),fun(multiset(B),fun(B,nat)),F),M)),aa(multiset(B),set(B),set_mset(B),M)) ).

% size_multiset_eq
tff(fact_7035_bdd__above__multiset__imp__finite__support,axiom,
    ! [B: $tType,A4: set(multiset(B))] :
      ( ( A4 != bot_bot(set(multiset(B))) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4)
       => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(multiset(B)),set(set(B)),image2(multiset(B),set(B),aTP_Lamp_aht(multiset(B),set(B))),A4))) ) ) ).

% bdd_above_multiset_imp_finite_support
tff(fact_7036_Sup__multiset__in__multiset,axiom,
    ! [B: $tType,A4: set(multiset(B))] :
      ( ( A4 != bot_bot(set(multiset(B))) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4)
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ahu(set(multiset(B)),fun(B,$o),A4))) ) ) ).

% Sup_multiset_in_multiset
tff(fact_7037_subset__mset_Obdd__above__UN,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(multiset(C)))] :
      ( aa(set(B),$o,finite_finite2(B),I5)
     => ( condit8047198070973881523_above(multiset(C),subseteq_mset(C),aa(set(set(multiset(C))),set(multiset(C)),complete_Sup_Sup(set(multiset(C))),aa(set(B),set(set(multiset(C))),image2(B,set(multiset(C)),A4),I5)))
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),I5)
           => condit8047198070973881523_above(multiset(C),subseteq_mset(C),aa(B,set(multiset(C)),A4,X3)) ) ) ) ).

% subset_mset.bdd_above_UN
tff(fact_7038_set__mset__Sup,axiom,
    ! [B: $tType,A4: set(multiset(B))] :
      ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4)
     => ( aa(multiset(B),set(B),set_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),A4)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(multiset(B)),set(set(B)),image2(multiset(B),set(B),set_mset(B)),A4)) ) ) ).

% set_mset_Sup
tff(fact_7039_count__Sup__multiset__nonempty,axiom,
    ! [B: $tType,A4: set(multiset(B)),X: B] :
      ( ( A4 != bot_bot(set(multiset(B))) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4)
       => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),A4)),X) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(B)),set(nat),image2(multiset(B),nat,aTP_Lamp_ahp(B,fun(multiset(B),nat),X)),A4)) ) ) ) ).

% count_Sup_multiset_nonempty
tff(fact_7040_subset__mset_Omono__cSUP,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [F: fun(multiset(B),C),A4: fun(D,multiset(B)),I5: set(D)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(D),set(multiset(B)),image2(D,multiset(B),A4),I5))
           => ( ( I5 != bot_bot(set(D)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(D),set(C),image2(D,C,aa(fun(D,multiset(B)),fun(D,C),aTP_Lamp_ahv(fun(multiset(B),C),fun(fun(D,multiset(B)),fun(D,C)),F),A4)),I5))),aa(multiset(B),C,F,aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(D),set(multiset(B)),image2(D,multiset(B),A4),I5)))) ) ) ) ) ).

% subset_mset.mono_cSUP
tff(fact_7041_subset__mset_Omono__cSup,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [F: fun(multiset(B),C),A4: set(multiset(B))] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4)
           => ( ( A4 != bot_bot(set(multiset(B))) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(multiset(B)),set(C),image2(multiset(B),C,F),A4))),aa(multiset(B),C,F,aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),A4))) ) ) ) ) ).

% subset_mset.mono_cSup
tff(fact_7042_order_Omono_Ocong,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [Less_eq: fun(B,fun(B,$o))] : mono(B,C,Less_eq) = mono(B,C,Less_eq) ) ).

% order.mono.cong
tff(fact_7043_subset__mset_Omono__def,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [F: fun(multiset(B),C)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
        <=> ! [X3: multiset(B),Y5: multiset(B)] :
              ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X3),Y5)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,X3)),aa(multiset(B),C,F,Y5)) ) ) ) ).

% subset_mset.mono_def
tff(fact_7044_subset__mset_OmonoI,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [F: fun(multiset(B),C)] :
          ( ! [X2: multiset(B),Y2: multiset(B)] :
              ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X2),Y2)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,X2)),aa(multiset(B),C,F,Y2)) )
         => aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F) ) ) ).

% subset_mset.monoI
tff(fact_7045_subset__mset_OmonoE,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [F: fun(multiset(B),C),X: multiset(B),Y: multiset(B)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X),Y)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,X)),aa(multiset(B),C,F,Y)) ) ) ) ).

% subset_mset.monoE
tff(fact_7046_subset__mset_OmonoD,axiom,
    ! [C: $tType,B: $tType] :
      ( order(C)
     => ! [F: fun(multiset(B),C),X: multiset(B),Y: multiset(B)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X),Y)
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,X)),aa(multiset(B),C,F,Y)) ) ) ) ).

% subset_mset.monoD
tff(fact_7047_Inf__multiset__def,axiom,
    ! [B: $tType] : complete_Inf_Inf(multiset(B)) = aa(fun(set(fun(B,nat)),fun(B,nat)),fun(set(multiset(B)),multiset(B)),map_fun(set(multiset(B)),set(fun(B,nat)),fun(B,nat),multiset(B),image2(multiset(B),fun(B,nat),count(B)),abs_multiset(B)),aTP_Lamp_ahw(set(fun(B,nat)),fun(B,nat))) ).

% Inf_multiset_def
tff(fact_7048_Sup__multiset__def,axiom,
    ! [B: $tType,A4: set(multiset(B))] :
      aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),A4) = $ite(
        ( ( A4 != bot_bot(set(multiset(B))) )
        & condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4) ),
        aa(fun(B,nat),multiset(B),abs_multiset(B),aTP_Lamp_ahx(set(multiset(B)),fun(B,nat),A4)),
        zero_zero(multiset(B)) ) ).

% Sup_multiset_def
tff(fact_7049_count__Abs__multiset,axiom,
    ! [B: $tType,F: fun(B,nat)] :
      ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ft(fun(B,nat),fun(B,$o),F)))
     => ( aa(multiset(B),fun(B,nat),count(B),aa(fun(B,nat),multiset(B),abs_multiset(B),F)) = F ) ) ).

% count_Abs_multiset
tff(fact_7050_zero__multiset__def,axiom,
    ! [B: $tType] : zero_zero(multiset(B)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aTP_Lamp_am(B,nat)) ).

% zero_multiset_def
tff(fact_7051_Abs__multiset__inject,axiom,
    ! [B: $tType,X: fun(B,nat),Y: fun(B,nat)] :
      ( aa(set(fun(B,nat)),$o,member(fun(B,nat),X),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o)))
     => ( aa(set(fun(B,nat)),$o,member(fun(B,nat),Y),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o)))
       => ( ( aa(fun(B,nat),multiset(B),abs_multiset(B),X) = aa(fun(B,nat),multiset(B),abs_multiset(B),Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_multiset_inject
tff(fact_7052_Abs__multiset__induct,axiom,
    ! [B: $tType,Pa: fun(multiset(B),$o),X: multiset(B)] :
      ( ! [Y2: fun(B,nat)] :
          ( aa(set(fun(B,nat)),$o,member(fun(B,nat),Y2),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o)))
         => aa(multiset(B),$o,Pa,aa(fun(B,nat),multiset(B),abs_multiset(B),Y2)) )
     => aa(multiset(B),$o,Pa,X) ) ).

% Abs_multiset_induct
tff(fact_7053_Abs__multiset__cases,axiom,
    ! [B: $tType,X: multiset(B)] :
      ~ ! [Y2: fun(B,nat)] :
          ( ( X = aa(fun(B,nat),multiset(B),abs_multiset(B),Y2) )
         => ~ aa(set(fun(B,nat)),$o,member(fun(B,nat),Y2),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o))) ) ).

% Abs_multiset_cases
tff(fact_7054_plus__multiset__def,axiom,
    ! [B: $tType] : plus_plus(multiset(B)) = aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),fun(multiset(B),fun(multiset(B),multiset(B))),map_fun(multiset(B),fun(B,nat),fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),count(B),map_fun(multiset(B),fun(B,nat),fun(B,nat),multiset(B),count(B),abs_multiset(B))),aTP_Lamp_ahy(fun(B,nat),fun(fun(B,nat),fun(B,nat)))) ).

% plus_multiset_def
tff(fact_7055_minus__multiset__def,axiom,
    ! [B: $tType] : minus_minus(multiset(B)) = aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),fun(multiset(B),fun(multiset(B),multiset(B))),map_fun(multiset(B),fun(B,nat),fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),count(B),map_fun(multiset(B),fun(B,nat),fun(B,nat),multiset(B),count(B),abs_multiset(B))),aTP_Lamp_ahz(fun(B,nat),fun(fun(B,nat),fun(B,nat)))) ).

% minus_multiset_def
tff(fact_7056_Abs__multiset__inverse,axiom,
    ! [B: $tType,Y: fun(B,nat)] :
      ( aa(set(fun(B,nat)),$o,member(fun(B,nat),Y),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o)))
     => ( aa(multiset(B),fun(B,nat),count(B),aa(fun(B,nat),multiset(B),abs_multiset(B),Y)) = Y ) ) ).

% Abs_multiset_inverse
tff(fact_7057_add__mset__def,axiom,
    ! [B: $tType] : add_mset(B) = aa(fun(B,fun(fun(B,nat),fun(B,nat))),fun(B,fun(multiset(B),multiset(B))),map_fun(B,B,fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),id(B),map_fun(multiset(B),fun(B,nat),fun(B,nat),multiset(B),count(B),abs_multiset(B))),aTP_Lamp_aia(B,fun(fun(B,nat),fun(B,nat)))) ).

% add_mset_def
tff(fact_7058_subset__mset_Omono__cINF,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [F: fun(multiset(B),C),A4: fun(D,multiset(B)),I5: set(D)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(D),set(multiset(B)),image2(D,multiset(B),A4),I5))
           => ( ( I5 != bot_bot(set(D)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(D),set(multiset(B)),image2(D,multiset(B),A4),I5)))),aa(set(C),C,complete_Inf_Inf(C),aa(set(D),set(C),image2(D,C,aa(fun(D,multiset(B)),fun(D,C),aTP_Lamp_ahv(fun(multiset(B),C),fun(fun(D,multiset(B)),fun(D,C)),F),A4)),I5))) ) ) ) ) ).

% subset_mset.mono_cINF
tff(fact_7059_subset__mset_Omono__cInf,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [F: fun(multiset(B),C),A4: set(multiset(B))] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),A4)
           => ( ( A4 != bot_bot(set(multiset(B))) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(multiset(B)),set(C),image2(multiset(B),C,F),A4))) ) ) ) ) ).

% subset_mset.mono_cInf
tff(fact_7060_subset__mset_Obdd__below__UN,axiom,
    ! [C: $tType,B: $tType,I5: set(B),A4: fun(B,set(multiset(C)))] :
      ( aa(set(B),$o,finite_finite2(B),I5)
     => ( condit8119078960628432327_below(multiset(C),subseteq_mset(C),aa(set(set(multiset(C))),set(multiset(C)),complete_Sup_Sup(set(multiset(C))),aa(set(B),set(set(multiset(C))),image2(B,set(multiset(C)),A4),I5)))
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),I5)
           => condit8119078960628432327_below(multiset(C),subseteq_mset(C),aa(B,set(multiset(C)),A4,X3)) ) ) ) ).

% subset_mset.bdd_below_UN
tff(fact_7061_filter__mset__def,axiom,
    ! [B: $tType] : filter_mset(B) = aa(fun(fun(B,$o),fun(fun(B,nat),fun(B,nat))),fun(fun(B,$o),fun(multiset(B),multiset(B))),map_fun(fun(B,$o),fun(B,$o),fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),id(fun(B,$o)),map_fun(multiset(B),fun(B,nat),fun(B,nat),multiset(B),count(B),abs_multiset(B))),aTP_Lamp_aib(fun(B,$o),fun(fun(B,nat),fun(B,nat)))) ).

% filter_mset_def
tff(fact_7062_repeat__mset__def,axiom,
    ! [B: $tType] : repeat_mset(B) = aa(fun(nat,fun(fun(B,nat),fun(B,nat))),fun(nat,fun(multiset(B),multiset(B))),map_fun(nat,nat,fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),id(nat),map_fun(multiset(B),fun(B,nat),fun(B,nat),multiset(B),count(B),abs_multiset(B))),aTP_Lamp_aic(nat,fun(fun(B,nat),fun(B,nat)))) ).

% repeat_mset_def
tff(fact_7063_filter__mset__True,axiom,
    ! [B: $tType,M: multiset(B)] : aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_jq(B,$o)),M) = M ).

% filter_mset_True
tff(fact_7064_repeat__mset__distrib2,axiom,
    ! [B: $tType,N: nat,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),A4)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),B4)) ).

% repeat_mset_distrib2
tff(fact_7065_filter__union__mset,axiom,
    ! [B: $tType,Pa: fun(B,$o),M: multiset(B),N5: multiset(B)] : aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),M)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),N5)) ).

% filter_union_mset
tff(fact_7066_union__filter__mset__complement,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o),M: multiset(B)] :
      ( ! [X2: B] :
          ( aa(B,$o,Pa,X2)
        <=> ~ aa(B,$o,Q,X2) )
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),M)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Q),M)) = M ) ) ).

% union_filter_mset_complement
tff(fact_7067_filter__mset__False,axiom,
    ! [B: $tType,M: multiset(B)] : aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_hh(B,$o)),M) = zero_zero(multiset(B)) ).

% filter_mset_False
tff(fact_7068_set__mset__filter,axiom,
    ! [B: $tType,Pa: fun(B,$o),M: multiset(B)] : aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),M)) = aa(fun(B,$o),set(B),collect(B),aa(multiset(B),fun(B,$o),aTP_Lamp_aid(fun(B,$o),fun(multiset(B),fun(B,$o)),Pa),M)) ).

% set_mset_filter
tff(fact_7069_repeat__mset__0,axiom,
    ! [B: $tType,M: multiset(B)] : aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),zero_zero(nat)),M) = zero_zero(multiset(B)) ).

% repeat_mset_0
tff(fact_7070_repeat__mset__Suc,axiom,
    ! [B: $tType,N: nat,M: multiset(B)] : aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,suc,N)),M) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),M)) ).

% repeat_mset_Suc
tff(fact_7071_count__filter__mset,axiom,
    ! [B: $tType,Pa: fun(B,$o),M: multiset(B),A2: B] :
      aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),M)),A2) = $ite(aa(B,$o,Pa,A2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),A2),zero_zero(nat)) ).

% count_filter_mset
tff(fact_7072_mset__filter,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B)] : mset(B,filter2(B,Pa,Xs)) = aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),mset(B,Xs)) ).

% mset_filter
tff(fact_7073_filter__mset_Orep__eq,axiom,
    ! [B: $tType,X: fun(B,$o),Xa2: multiset(B),X4: B] :
      aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),X),Xa2)),X4) = $ite(aa(B,$o,X,X4),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Xa2),X4),zero_zero(nat)) ).

% filter_mset.rep_eq
tff(fact_7074_size__filter__mset__lesseq,axiom,
    ! [B: $tType,F: fun(B,$o),M: multiset(B)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),F),M))),aa(multiset(B),nat,size_size(multiset(B)),M)) ).

% size_filter_mset_lesseq
tff(fact_7075_repeat__mset__distrib,axiom,
    ! [B: $tType,Ma: nat,N: nat,A4: multiset(B)] : aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),A4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),Ma),A4)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),A4)) ).

% repeat_mset_distrib
tff(fact_7076_left__add__mult__distrib__mset,axiom,
    ! [B: $tType,I: nat,U: multiset(B),J: nat,K: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),I),U)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),J),U)),K)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib_mset
tff(fact_7077_repeat__mset__cancel1,axiom,
    ! [B: $tType,A2: nat,A4: multiset(B),B4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),A2),A4) = aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),A2),B4) )
    <=> ( ( A4 = B4 )
        | ( A2 = zero_zero(nat) ) ) ) ).

% repeat_mset_cancel1
tff(fact_7078_multiset__partition,axiom,
    ! [B: $tType,M: multiset(B),Pa: fun(B,$o)] : M = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),M)),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)),M)) ).

% multiset_partition
tff(fact_7079_filter__filter__mset,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o),M: multiset(B)] : aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Q),M)) = aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_pa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)),M) ).

% filter_filter_mset
tff(fact_7080_repeat__mset__eq__empty__iff,axiom,
    ! [B: $tType,N: nat,A4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),A4) = zero_zero(multiset(B)) )
    <=> ( ( N = zero_zero(nat) )
        | ( A4 = zero_zero(multiset(B)) ) ) ) ).

% repeat_mset_eq_empty_iff
tff(fact_7081_image__mset__If,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,$o),F: fun(C,B),G: fun(C,B),A4: multiset(C)] : aa(multiset(C),multiset(B),image_mset(C,B,aa(fun(C,B),fun(C,B),aa(fun(C,B),fun(fun(C,B),fun(C,B)),aTP_Lamp_aau(fun(C,$o),fun(fun(C,B),fun(fun(C,B),fun(C,B))),Pa),F),G)),A4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(C),multiset(B),image_mset(C,B,F),aa(multiset(C),multiset(C),aa(fun(C,$o),fun(multiset(C),multiset(C)),filter_mset(C),Pa),A4))),aa(multiset(C),multiset(B),image_mset(C,B,G),aa(multiset(C),multiset(C),aa(fun(C,$o),fun(multiset(C),multiset(C)),filter_mset(C),aTP_Lamp_aav(fun(C,$o),fun(C,$o),Pa)),A4))) ).

% image_mset_If
tff(fact_7082_mset__subseteq__add__iff1,axiom,
    ! [B: $tType,J: nat,I: nat,U: multiset(B),Ma: multiset(B),N: multiset(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),I),U)),Ma)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),J),U)),N))
      <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma)),N) ) ) ).

% mset_subseteq_add_iff1
tff(fact_7083_mset__subseteq__add__iff2,axiom,
    ! [B: $tType,I: nat,J: nat,U: multiset(B),Ma: multiset(B),N: multiset(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),I),U)),Ma)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),J),U)),N))
      <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Ma),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N)) ) ) ).

% mset_subseteq_add_iff2
tff(fact_7084_fold__mset__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,fun(B,B)),S2: B,M: multiset(C)] : fold_mset(C,B,F,S2,M) = finite_fold(C,B,aa(multiset(C),fun(C,fun(B,B)),aTP_Lamp_aie(fun(C,fun(B,B)),fun(multiset(C),fun(C,fun(B,B))),F),M),S2,aa(multiset(C),set(C),set_mset(C),M)) ).

% fold_mset_def
tff(fact_7085_set__mset__replicate__mset__subset,axiom,
    ! [B: $tType,N: nat,X: B] :
      aa(multiset(B),set(B),set_mset(B),replicate_mset(B,N,X)) = $ite(N = zero_zero(nat),bot_bot(set(B)),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) ).

% set_mset_replicate_mset_subset
tff(fact_7086_size__replicate__mset,axiom,
    ! [B: $tType,N: nat,M: B] : aa(multiset(B),nat,size_size(multiset(B)),replicate_mset(B,N,M)) = N ).

% size_replicate_mset
tff(fact_7087_replicate__mset__0,axiom,
    ! [B: $tType,X: B] : replicate_mset(B,zero_zero(nat),X) = zero_zero(multiset(B)) ).

% replicate_mset_0
tff(fact_7088_replicate__mset__eq__empty__iff,axiom,
    ! [B: $tType,N: nat,A2: B] :
      ( ( replicate_mset(B,N,A2) = zero_zero(multiset(B)) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_mset_eq_empty_iff
tff(fact_7089_count__replicate__mset,axiom,
    ! [B: $tType,N: nat,X: B,Y: B] :
      aa(B,nat,aa(multiset(B),fun(B,nat),count(B),replicate_mset(B,N,X)),Y) = $ite(Y = X,N,zero_zero(nat)) ).

% count_replicate_mset
tff(fact_7090_in__mset__fold__plus__iff,axiom,
    ! [B: $tType,X: B,M: multiset(B),NN: multiset(multiset(B))] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),fold_mset(multiset(B),multiset(B),plus_plus(multiset(B)),M,NN)))
    <=> ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),M))
        | ? [N11: multiset(B)] :
            ( aa(set(multiset(B)),$o,member(multiset(B),N11),aa(multiset(multiset(B)),set(multiset(B)),set_mset(multiset(B)),NN))
            & aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),N11)) ) ) ) ).

% in_mset_fold_plus_iff
tff(fact_7091_in__replicate__mset,axiom,
    ! [B: $tType,X: B,N: nat,Y: B] :
      ( aa(set(B),$o,member(B,X),aa(multiset(B),set(B),set_mset(B),replicate_mset(B,N,Y)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
        & ( X = Y ) ) ) ).

% in_replicate_mset
tff(fact_7092_sum__mset__replicate__mset,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [N: nat,A2: B] : comm_m7189776963980413722m_mset(B,replicate_mset(B,N,A2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),N)),A2) ) ).

% sum_mset_replicate_mset
tff(fact_7093_repeat__mset__distrib__add__mset,axiom,
    ! [B: $tType,N: nat,A2: B,A4: multiset(B)] : aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),A4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),replicate_mset(B,N,A2)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),N),A4)) ).

% repeat_mset_distrib_add_mset
tff(fact_7094_msubseteq__replicate__msetE,axiom,
    ! [B: $tType,A4: multiset(B),N: nat,A2: B] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A4),replicate_mset(B,N,A2))
     => ~ ! [M2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
           => ( A4 != replicate_mset(B,M2,A2) ) ) ) ).

% msubseteq_replicate_msetE
tff(fact_7095_replicate__mset__eq__iff,axiom,
    ! [B: $tType,Ma: nat,A2: B,N: nat,B2: B] :
      ( ( replicate_mset(B,Ma,A2) = replicate_mset(B,N,B2) )
    <=> ( ( ( Ma = zero_zero(nat) )
          & ( N = zero_zero(nat) ) )
        | ( ( Ma = N )
          & ( A2 = B2 ) ) ) ) ).

% replicate_mset_eq_iff
tff(fact_7096_union__fold__mset__add__mset,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4) = fold_mset(B,multiset(B),add_mset(B),A4,B4) ).

% union_fold_mset_add_mset
tff(fact_7097_comp__fun__commute_Ofold__mset__union,axiom,
    ! [C: $tType,B: $tType,F: fun(B,fun(C,C)),S2: C,M: multiset(B),N5: multiset(B)] :
      ( finite6289374366891150609ommute(B,C,F)
     => ( fold_mset(B,C,F,S2,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = fold_mset(B,C,F,fold_mset(B,C,F,S2,M),N5) ) ) ).

% comp_fun_commute.fold_mset_union
tff(fact_7098_image__mset__const__eq,axiom,
    ! [C: $tType,B: $tType,C2: B,M: multiset(C)] : aa(multiset(C),multiset(B),image_mset(C,B,aa(B,fun(C,B),aTP_Lamp_it(B,fun(C,B)),C2)),M) = replicate_mset(B,aa(multiset(C),nat,size_size(multiset(C)),M),C2) ).

% image_mset_const_eq
tff(fact_7099_filter__eq__replicate__mset,axiom,
    ! [B: $tType,X: B,D3: multiset(B)] : aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_af(B,fun(B,$o),X)),D3) = replicate_mset(B,aa(B,nat,aa(multiset(B),fun(B,nat),count(B),D3),X),X) ).

% filter_eq_replicate_mset
tff(fact_7100_replicate__mset__msubseteq__iff,axiom,
    ! [B: $tType,Ma: nat,A2: B,N: nat,B2: B] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),replicate_mset(B,Ma,A2)),replicate_mset(B,N,B2))
    <=> ( ( Ma = zero_zero(nat) )
        | ( ( A2 = B2 )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),N) ) ) ) ).

% replicate_mset_msubseteq_iff
tff(fact_7101_count__le__replicate__mset__subset__eq,axiom,
    ! [B: $tType,N: nat,M: multiset(B),X: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),M),X))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),replicate_mset(B,N,X)),M) ) ).

% count_le_replicate_mset_subset_eq
tff(fact_7102_sum__mset_Oeq__fold,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [M: multiset(B)] : comm_m7189776963980413722m_mset(B,M) = fold_mset(B,B,plus_plus(B),zero_zero(B),M) ) ).

% sum_mset.eq_fold
tff(fact_7103_sorted__list__of__multiset__def,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [M: multiset(B)] : linord6283353356039996273ltiset(B,M) = fold_mset(B,list(B),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),nil(B),M) ) ).

% sorted_list_of_multiset_def
tff(fact_7104_mult1E,axiom,
    ! [B: $tType,N5: multiset(B),M: multiset(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),N5),M)),mult1(B,R))
     => ~ ! [A3: B,M0: multiset(B)] :
            ( ( M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),M0) )
           => ! [K6: multiset(B)] :
                ( ( N5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M0),K6) )
               => ~ ! [B7: B] :
                      ( aa(set(B),$o,member(B,B7),aa(multiset(B),set(B),set_mset(B),K6))
                     => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B7),A3)),R) ) ) ) ) ).

% mult1E
tff(fact_7105_mult1I,axiom,
    ! [B: $tType,M: multiset(B),A2: B,M02: multiset(B),N5: multiset(B),K3: multiset(B),R: set(product_prod(B,B))] :
      ( ( M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),M02) )
     => ( ( N5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M02),K3) )
       => ( ! [B3: B] :
              ( aa(set(B),$o,member(B,B3),aa(multiset(B),set(B),set_mset(B),K3))
             => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B3),A2)),R) )
         => aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),N5),M)),mult1(B,R)) ) ) ) ).

% mult1I
tff(fact_7106_mult1__union,axiom,
    ! [B: $tType,B4: multiset(B),D3: multiset(B),R: set(product_prod(B,B)),C6: multiset(B)] :
      ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),B4),D3)),mult1(B,R))
     => aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),B4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),D3))),mult1(B,R)) ) ).

% mult1_union
tff(fact_7107_less__add,axiom,
    ! [B: $tType,N5: multiset(B),A2: B,M02: multiset(B),R: set(product_prod(B,B))] :
      ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),N5),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),M02))),mult1(B,R))
     => ( ? [M8: multiset(B)] :
            ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),M8),M02)),mult1(B,R))
            & ( N5 = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),M8) ) )
        | ? [K6: multiset(B)] :
            ( ! [B7: B] :
                ( aa(set(B),$o,member(B,B7),aa(multiset(B),set(B),set_mset(B),K6))
               => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B7),A2)),R) )
            & ( N5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M02),K6) ) ) ) ) ).

% less_add
tff(fact_7108_mult1__lessE,axiom,
    ! [B: $tType,N5: multiset(B),M: multiset(B),R: fun(B,fun(B,$o))] :
      ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),N5),M)),mult1(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),R))))
     => ( asymp(B,R)
       => ~ ! [A3: B,M0: multiset(B)] :
              ( ( M = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),M0) )
             => ! [K6: multiset(B)] :
                  ( ( N5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M0),K6) )
                 => ( ~ aa(set(B),$o,member(B,A3),aa(multiset(B),set(B),set_mset(B),K6))
                   => ~ ! [B7: B] :
                          ( aa(set(B),$o,member(B,B7),aa(multiset(B),set(B),set_mset(B),K6))
                         => aa(B,$o,aa(B,fun(B,$o),R,B7),A3) ) ) ) ) ) ) ).

% mult1_lessE
tff(fact_7109_one__step__implies__mult,axiom,
    ! [B: $tType,J4: multiset(B),K3: multiset(B),R: set(product_prod(B,B)),I5: multiset(B)] :
      ( ( J4 != zero_zero(multiset(B)) )
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),aa(multiset(B),set(B),set_mset(B),K3))
           => ? [Xa: B] :
                ( aa(set(B),$o,member(B,Xa),aa(multiset(B),set(B),set_mset(B),J4))
                & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X2),Xa)),R) ) )
       => aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I5),K3)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I5),J4))),mult(B,R)) ) ) ).

% one_step_implies_mult
tff(fact_7110_asymp__greater,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => asymp(B,aTP_Lamp_js(B,fun(B,$o))) ) ).

% asymp_greater
tff(fact_7111_prod__mset__minus,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [A2: B,A4: multiset(B)] :
          ( aa(set(B),$o,member(B,A2),aa(multiset(B),set(B),set_mset(B),A4))
         => ( ( A2 != zero_zero(B) )
           => ( comm_m9189036328036947845d_mset(B,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A2),zero_zero(multiset(B))))) = aa(B,B,aa(B,fun(B,B),divide_divide(B),comm_m9189036328036947845d_mset(B,A4)),A2) ) ) ) ) ).

% prod_mset_minus
tff(fact_7112_mult__implies__one__step,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),M: multiset(B),N5: multiset(B)] :
      ( trans(B,R)
     => ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),M),N5)),mult(B,R))
       => ? [I6: multiset(B),J5: multiset(B)] :
            ( ( N5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I6),J5) )
            & ? [K6: multiset(B)] :
                ( ( M = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I6),K6) )
                & ( J5 != zero_zero(multiset(B)) )
                & ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),aa(multiset(B),set(B),set_mset(B),K6))
                   => ? [Xa3: B] :
                        ( aa(set(B),$o,member(B,Xa3),aa(multiset(B),set(B),set_mset(B),J5))
                        & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X4),Xa3)),R) ) ) ) ) ) ) ).

% mult_implies_one_step
tff(fact_7113_prod__mset__empty,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ( comm_m9189036328036947845d_mset(B,zero_zero(multiset(B))) = one_one(B) ) ) ).

% prod_mset_empty
tff(fact_7114_prod__mset_Ounion,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [M: multiset(B),N5: multiset(B)] : comm_m9189036328036947845d_mset(B,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M),N5)) = aa(B,B,aa(B,fun(B,B),times_times(B),comm_m9189036328036947845d_mset(B,M)),comm_m9189036328036947845d_mset(B,N5)) ) ).

% prod_mset.union
tff(fact_7115_prod__mset__Un,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: multiset(B),B4: multiset(B)] : comm_m9189036328036947845d_mset(B,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),comm_m9189036328036947845d_mset(B,A4)),comm_m9189036328036947845d_mset(B,B4)) ) ).

% prod_mset_Un
tff(fact_7116_prod__mset__zero__iff,axiom,
    ! [B: $tType] :
      ( semidom(B)
     => ! [A4: multiset(B)] :
          ( ( comm_m9189036328036947845d_mset(B,A4) = zero_zero(B) )
        <=> aa(set(B),$o,member(B,zero_zero(B)),aa(multiset(B),set(B),set_mset(B),A4)) ) ) ).

% prod_mset_zero_iff
tff(fact_7117_prod__mset_Oneutral__const,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: multiset(C)] : comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aTP_Lamp_dm(C,B)),A4)) = one_one(B) ) ).

% prod_mset.neutral_const
tff(fact_7118_prod__mset__constant,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [C2: B,A4: multiset(C)] : comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aTP_Lamp_gn(B,fun(C,B),C2)),A4)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(multiset(C),nat,size_size(multiset(C)),A4)) ) ).

% prod_mset_constant
tff(fact_7119_trans__Restr,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),A4: set(B)] :
      ( trans(B,R)
     => trans(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4)))) ) ).

% trans_Restr
tff(fact_7120_prod__mset_Oswap,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,fun(D,B)),B4: multiset(D),A4: multiset(C)] : comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(multiset(D),fun(C,B),aTP_Lamp_aif(fun(C,fun(D,B)),fun(multiset(D),fun(C,B)),G),B4)),A4)) = comm_m9189036328036947845d_mset(B,aa(multiset(D),multiset(B),image_mset(D,B,aa(multiset(C),fun(D,B),aTP_Lamp_aig(fun(C,fun(D,B)),fun(multiset(C),fun(D,B)),G),A4)),B4)) ) ).

% prod_mset.swap
tff(fact_7121_prod__mset_Odistrib,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(C,B),H: fun(C,B),A4: multiset(C)] : comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_du(fun(C,B),fun(fun(C,B),fun(C,B)),G),H)),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,G),A4))),comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,H),A4))) ) ).

% prod_mset.distrib
tff(fact_7122_prod__mset_Oneutral,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: multiset(B)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),aa(multiset(B),set(B),set_mset(B),A4))
             => ( X2 = one_one(B) ) )
         => ( comm_m9189036328036947845d_mset(B,A4) = one_one(B) ) ) ) ).

% prod_mset.neutral
tff(fact_7123_is__unit__prod__mset__iff,axiom,
    ! [B: $tType] :
      ( algebraic_semidom(B)
     => ! [A4: multiset(B)] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),comm_m9189036328036947845d_mset(B,A4)),one_one(B))
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),aa(multiset(B),set(B),set_mset(B),A4))
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),X3),one_one(B)) ) ) ) ).

% is_unit_prod_mset_iff
tff(fact_7124_prod__mset_Oeq__fold,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [M: multiset(B)] : comm_m9189036328036947845d_mset(B,M) = fold_mset(B,B,times_times(B),one_one(B),M) ) ).

% prod_mset.eq_fold
tff(fact_7125_prod__mset__delta,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Y: C,C2: B,A4: multiset(C)] : comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(B,fun(C,B),aTP_Lamp_aih(C,fun(B,fun(C,B)),Y),C2)),A4)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(C,nat,aa(multiset(C),fun(C,nat),count(C),A4),Y)) ) ).

% prod_mset_delta
tff(fact_7126_prod__mset__delta_H,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Y: C,C2: B,A4: multiset(C)] : comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(B,fun(C,B),aTP_Lamp_aii(C,fun(B,fun(C,B)),Y),C2)),A4)) = aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(C,nat,aa(multiset(C),fun(C,nat),count(C),A4),Y)) ) ).

% prod_mset_delta'
tff(fact_7127_prod__mset__multiplicity,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [M: multiset(B)] : comm_m9189036328036947845d_mset(B,M) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7121269368397514597t_prod(B,B),aTP_Lamp_aij(multiset(B),fun(B,B),M)),aa(multiset(B),set(B),set_mset(B),M)) ) ).

% prod_mset_multiplicity
tff(fact_7128_prod__mset__diff,axiom,
    ! [B: $tType] :
      ( semidom_divide(B)
     => ! [B4: multiset(B),A4: multiset(B)] :
          ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B4),A4)
         => ( ~ aa(set(B),$o,member(B,zero_zero(B)),aa(multiset(B),set(B),set_mset(B),B4))
           => ( comm_m9189036328036947845d_mset(B,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),comm_m9189036328036947845d_mset(B,A4)),comm_m9189036328036947845d_mset(B,B4)) ) ) ) ) ).

% prod_mset_diff
tff(fact_7129_mult1__def,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : mult1(B,R) = aa(fun(product_prod(multiset(B),multiset(B)),$o),set(product_prod(multiset(B),multiset(B))),collect(product_prod(multiset(B),multiset(B))),aa(fun(multiset(B),fun(multiset(B),$o)),fun(product_prod(multiset(B),multiset(B)),$o),product_case_prod(multiset(B),multiset(B),$o),aTP_Lamp_aik(set(product_prod(B,B)),fun(multiset(B),fun(multiset(B),$o)),R))) ).

% mult1_def
tff(fact_7130_mult__cancel,axiom,
    ! [B: $tType,S2: set(product_prod(B,B)),X5: multiset(B),Z7: multiset(B),Y4: multiset(B)] :
      ( trans(B,S2)
     => ( irrefl(B,S2)
       => ( aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X5),Z7)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Y4),Z7))),mult(B,S2))
        <=> aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),X5),Y4)),mult(B,S2)) ) ) ) ).

% mult_cancel
tff(fact_7131_rel__fun__def,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,A4: fun(B,fun(D,$o)),B4: fun(C,fun(E,$o)),X4: fun(B,C),Xa: fun(D,E)] :
      ( aa(fun(D,E),$o,aa(fun(B,C),fun(fun(D,E),$o),bNF_rel_fun(B,D,C,E,A4,B4),X4),Xa)
    <=> ! [Xb4: B,Y5: D] :
          ( aa(D,$o,aa(B,fun(D,$o),A4,Xb4),Y5)
         => aa(E,$o,aa(C,fun(E,$o),B4,aa(B,C,X4,Xb4)),aa(D,E,Xa,Y5)) ) ) ).

% rel_fun_def
tff(fact_7132_rel__fun__eq__rel,axiom,
    ! [C: $tType,D: $tType,B: $tType,R2: fun(C,fun(D,$o)),X4: fun(B,C),Xa: fun(B,D)] :
      ( aa(fun(B,D),$o,aa(fun(B,C),fun(fun(B,D),$o),bNF_rel_fun(B,B,C,D,fequal(B),R2),X4),Xa)
    <=> ! [Xb4: B] : aa(D,$o,aa(C,fun(D,$o),R2,aa(B,C,X4,Xb4)),aa(B,D,Xa,Xb4)) ) ).

% rel_fun_eq_rel
tff(fact_7133_finite__set__of__finite__funs,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C),D2: C] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => aa(set(fun(B,C)),$o,finite_finite2(fun(B,C)),aa(fun(fun(B,C),$o),set(fun(B,C)),collect(fun(B,C)),aa(C,fun(fun(B,C),$o),aa(set(C),fun(C,fun(fun(B,C),$o)),aTP_Lamp_ail(set(B),fun(set(C),fun(C,fun(fun(B,C),$o))),A4),B4),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_7134_Collect__all__eq,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,fun(C,$o))] : aa(fun(B,$o),set(B),collect(B),aTP_Lamp_aim(fun(B,fun(C,$o)),fun(B,$o),Pa)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_yq(fun(B,fun(C,$o)),fun(C,set(B)),Pa)),top_top(set(C)))) ).

% Collect_all_eq
tff(fact_7135_Least__def,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Pa: fun(B,$o)] : ord_Least(B,Pa) = the(B,aTP_Lamp_ain(fun(B,$o),fun(B,$o),Pa)) ) ).

% Least_def
tff(fact_7136_Greatest__def,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Pa: fun(B,$o)] : order_Greatest(B,Pa) = the(B,aTP_Lamp_aio(fun(B,$o),fun(B,$o),Pa)) ) ).

% Greatest_def
tff(fact_7137_ord_OLeast__def,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o)),Pa: fun(B,$o)] : aa(fun(B,$o),B,least(B,Less_eq),Pa) = the(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aip(fun(B,fun(B,$o)),fun(fun(B,$o),fun(B,$o)),Less_eq),Pa)) ).

% ord.Least_def
tff(fact_7138_transfer__bforall__def,axiom,
    ! [B: $tType,X4: fun(B,$o),Xa: fun(B,$o)] :
      ( transfer_bforall(B,X4,Xa)
    <=> ! [Xb4: B] :
          ( aa(B,$o,X4,Xb4)
         => aa(B,$o,Xa,Xb4) ) ) ).

% transfer_bforall_def
tff(fact_7139_ord_OLeast_Ocong,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o))] : least(B,Less_eq) = least(B,Less_eq) ).

% ord.Least.cong
tff(fact_7140_subset__mset_OLeast__def,axiom,
    ! [B: $tType,Pa: fun(multiset(B),$o)] : aa(fun(multiset(B),$o),multiset(B),least(multiset(B),subseteq_mset(B)),Pa) = the(multiset(B),aTP_Lamp_aiq(fun(multiset(B),$o),fun(multiset(B),$o),Pa)) ).

% subset_mset.Least_def
tff(fact_7141_repeat__mset_Oabs__eq,axiom,
    ! [B: $tType,X: fun(B,nat),Xa2: nat] :
      ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),X),X)
     => ( aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),Xa2),aa(fun(B,nat),multiset(B),abs_multiset(B),X)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aa(nat,fun(B,nat),aTP_Lamp_air(fun(B,nat),fun(nat,fun(B,nat)),X),Xa2)) ) ) ).

% repeat_mset.abs_eq
tff(fact_7142_type__definition__multiset,axiom,
    ! [B: $tType] : type_definition(multiset(B),fun(B,nat),count(B),abs_multiset(B),aa(fun(fun(B,nat),$o),set(fun(B,nat)),collect(fun(B,nat)),aTP_Lamp_ahn(fun(B,nat),$o))) ).

% type_definition_multiset
tff(fact_7143_type__copy__obj__one__point__absE,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),S2: B] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ~ ! [X2: C] : S2 != aa(C,B,Abs,X2) ) ).

% type_copy_obj_one_point_absE
tff(fact_7144_eq__onp__live__step,axiom,
    ! [B: $tType,X: $o,Y: $o,Pa: fun(B,$o),A2: B] :
      ( ( (X)
      <=> (Y) )
     => ( ( aa(B,$o,aa(B,fun(B,$o),bNF_eq_onp(B,Pa),A2),A2)
          & (X) )
      <=> ( aa(B,$o,Pa,A2)
          & (Y) ) ) ) ).

% eq_onp_live_step
tff(fact_7145_eq__onp__True,axiom,
    ! [B: $tType] : bNF_eq_onp(B,aTP_Lamp_jq(B,$o)) = fequal(B) ).

% eq_onp_True
tff(fact_7146_eq__onp__def,axiom,
    ! [B: $tType,R2: fun(B,$o),X4: B,Xa: B] :
      ( aa(B,$o,aa(B,fun(B,$o),bNF_eq_onp(B,R2),X4),Xa)
    <=> ( aa(B,$o,R2,X4)
        & ( X4 = Xa ) ) ) ).

% eq_onp_def
tff(fact_7147_type__copy__map__comp0__undo,axiom,
    ! [F6: $tType,B: $tType,D: $tType,C: $tType,E: $tType,G4: $tType,Rep: fun(B,C),Abs: fun(C,B),Rep2: fun(D,E),Abs2: fun(E,D),Rep3: fun(F6,G4),Abs3: fun(G4,F6),M: fun(G4,E),M13: fun(C,E),M24: fun(G4,C)] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ( type_definition(D,E,Rep2,Abs2,top_top(set(E)))
       => ( type_definition(F6,G4,Rep3,Abs3,top_top(set(G4)))
         => ( ( aa(fun(F6,G4),fun(F6,D),aa(fun(G4,D),fun(fun(F6,G4),fun(F6,D)),comp(G4,D,F6),aa(fun(G4,E),fun(G4,D),aa(fun(E,D),fun(fun(G4,E),fun(G4,D)),comp(E,D,G4),Abs2),M)),Rep3) = aa(fun(F6,B),fun(F6,D),aa(fun(B,D),fun(fun(F6,B),fun(F6,D)),comp(B,D,F6),aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),aa(fun(C,E),fun(C,D),aa(fun(E,D),fun(fun(C,E),fun(C,D)),comp(E,D,C),Abs2),M13)),Rep)),aa(fun(F6,G4),fun(F6,B),aa(fun(G4,B),fun(fun(F6,G4),fun(F6,B)),comp(G4,B,F6),aa(fun(G4,C),fun(G4,B),aa(fun(C,B),fun(fun(G4,C),fun(G4,B)),comp(C,B,G4),Abs),M24)),Rep3)) )
           => ( aa(fun(G4,C),fun(G4,E),aa(fun(C,E),fun(fun(G4,C),fun(G4,E)),comp(C,E,G4),M13),M24) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_7148_rel__fun__eq__eq__onp,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,$o)] : bNF_rel_fun(B,B,C,C,fequal(B),bNF_eq_onp(C,Pa)) = bNF_eq_onp(fun(B,C),aTP_Lamp_ais(fun(C,$o),fun(fun(B,C),$o),Pa)) ).

% rel_fun_eq_eq_onp
tff(fact_7149_rel__fun__eq__onp__rel,axiom,
    ! [C: $tType,D: $tType,B: $tType,R2: fun(B,$o),S: fun(C,fun(D,$o)),X4: fun(B,C),Xa: fun(B,D)] :
      ( aa(fun(B,D),$o,aa(fun(B,C),fun(fun(B,D),$o),bNF_rel_fun(B,B,C,D,bNF_eq_onp(B,R2),S),X4),Xa)
    <=> ! [Xb4: B] :
          ( aa(B,$o,R2,Xb4)
         => aa(D,$o,aa(C,fun(D,$o),S,aa(B,C,X4,Xb4)),aa(B,D,Xa,Xb4)) ) ) ).

% rel_fun_eq_onp_rel
tff(fact_7150_zero__multiset_Orsp,axiom,
    ! [B: $tType] : aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),aTP_Lamp_am(B,nat)),aTP_Lamp_am(B,nat)) ).

% zero_multiset.rsp
tff(fact_7151_filter__mset_Orsp,axiom,
    ! [B: $tType] : aa(fun(fun(B,$o),fun(fun(B,nat),fun(B,nat))),$o,aa(fun(fun(B,$o),fun(fun(B,nat),fun(B,nat))),fun(fun(fun(B,$o),fun(fun(B,nat),fun(B,nat))),$o),bNF_rel_fun(fun(B,$o),fun(B,$o),fun(fun(B,nat),fun(B,nat)),fun(fun(B,nat),fun(B,nat)),fequal(fun(B,$o)),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)))),aTP_Lamp_aib(fun(B,$o),fun(fun(B,nat),fun(B,nat)))),aTP_Lamp_aib(fun(B,$o),fun(fun(B,nat),fun(B,nat)))) ).

% filter_mset.rsp
tff(fact_7152_type__copy__Rep__o__Abs,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B)] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ( aa(fun(C,B),fun(C,C),aa(fun(B,C),fun(fun(C,B),fun(C,C)),comp(B,C,C),Rep),Abs) = id(C) ) ) ).

% type_copy_Rep_o_Abs
tff(fact_7153_type__copy__Abs__o__Rep,axiom,
    ! [C: $tType,B: $tType,Rep: fun(B,C),Abs: fun(C,B)] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ( aa(fun(B,C),fun(B,B),aa(fun(C,B),fun(fun(B,C),fun(B,B)),comp(C,B,B),Abs),Rep) = id(B) ) ) ).

% type_copy_Abs_o_Rep
tff(fact_7154_add__mset_Orsp,axiom,
    ! [B: $tType] : aa(fun(B,fun(fun(B,nat),fun(B,nat))),$o,aa(fun(B,fun(fun(B,nat),fun(B,nat))),fun(fun(B,fun(fun(B,nat),fun(B,nat))),$o),bNF_rel_fun(B,B,fun(fun(B,nat),fun(B,nat)),fun(fun(B,nat),fun(B,nat)),fequal(B),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)))),aTP_Lamp_aia(B,fun(fun(B,nat),fun(B,nat)))),aTP_Lamp_aia(B,fun(fun(B,nat),fun(B,nat)))) ).

% add_mset.rsp
tff(fact_7155_plus__multiset_Orsp,axiom,
    ! [B: $tType] : aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),$o,aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),fun(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),$o),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(fun(B,nat),fun(B,nat)),fun(fun(B,nat),fun(B,nat)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)))),aTP_Lamp_ahy(fun(B,nat),fun(fun(B,nat),fun(B,nat)))),aTP_Lamp_ahy(fun(B,nat),fun(fun(B,nat),fun(B,nat)))) ).

% plus_multiset.rsp
tff(fact_7156_minus__multiset_Orsp,axiom,
    ! [B: $tType] : aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),$o,aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),fun(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),$o),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(fun(B,nat),fun(B,nat)),fun(fun(B,nat),fun(B,nat)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)))),aTP_Lamp_ahz(fun(B,nat),fun(fun(B,nat),fun(B,nat)))),aTP_Lamp_ahz(fun(B,nat),fun(fun(B,nat),fun(B,nat)))) ).

% minus_multiset.rsp
tff(fact_7157_repeat__mset_Orsp,axiom,
    ! [B: $tType] : aa(fun(nat,fun(fun(B,nat),fun(B,nat))),$o,aa(fun(nat,fun(fun(B,nat),fun(B,nat))),fun(fun(nat,fun(fun(B,nat),fun(B,nat))),$o),bNF_rel_fun(nat,nat,fun(fun(B,nat),fun(B,nat)),fun(fun(B,nat),fun(B,nat)),fequal(nat),bNF_rel_fun(fun(B,nat),fun(B,nat),fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)))),aTP_Lamp_aic(nat,fun(fun(B,nat),fun(B,nat)))),aTP_Lamp_aic(nat,fun(fun(B,nat),fun(B,nat)))) ).

% repeat_mset.rsp
tff(fact_7158_filter__mset_Oabs__eq,axiom,
    ! [B: $tType,X: fun(B,nat),Xa2: fun(B,$o)] :
      ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),X),X)
     => ( aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Xa2),aa(fun(B,nat),multiset(B),abs_multiset(B),X)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aa(fun(B,$o),fun(B,nat),aTP_Lamp_ait(fun(B,nat),fun(fun(B,$o),fun(B,nat)),X),Xa2)) ) ) ).

% filter_mset.abs_eq
tff(fact_7159_add__mset_Oabs__eq,axiom,
    ! [B: $tType,X: fun(B,nat),Xa2: B] :
      ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),X),X)
     => ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Xa2),aa(fun(B,nat),multiset(B),abs_multiset(B),X)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aa(B,fun(B,nat),aTP_Lamp_aiu(fun(B,nat),fun(B,fun(B,nat)),X),Xa2)) ) ) ).

% add_mset.abs_eq
tff(fact_7160_plus__multiset_Oabs__eq,axiom,
    ! [B: $tType,Xa2: fun(B,nat),X: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),Xa2),Xa2)
     => ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),X),X)
       => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(fun(B,nat),multiset(B),abs_multiset(B),Xa2)),aa(fun(B,nat),multiset(B),abs_multiset(B),X)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aa(fun(B,nat),fun(B,nat),aa(fun(B,nat),fun(fun(B,nat),fun(B,nat)),aTP_Lamp_ahy(fun(B,nat),fun(fun(B,nat),fun(B,nat))),Xa2),X)) ) ) ) ).

% plus_multiset.abs_eq
tff(fact_7161_minus__multiset_Oabs__eq,axiom,
    ! [B: $tType,Xa2: fun(B,nat),X: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),Xa2),Xa2)
     => ( aa(fun(B,nat),$o,aa(fun(B,nat),fun(fun(B,nat),$o),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o)),X),X)
       => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(fun(B,nat),multiset(B),abs_multiset(B),Xa2)),aa(fun(B,nat),multiset(B),abs_multiset(B),X)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aa(fun(B,nat),fun(B,nat),aa(fun(B,nat),fun(fun(B,nat),fun(B,nat)),aTP_Lamp_ahz(fun(B,nat),fun(fun(B,nat),fun(B,nat))),Xa2),X)) ) ) ) ).

% minus_multiset.abs_eq
tff(fact_7162_subset__mset_OGreatest__def,axiom,
    ! [B: $tType,Pa: fun(multiset(B),$o)] : aa(fun(multiset(B),$o),multiset(B),greatest(multiset(B),subseteq_mset(B)),Pa) = the(multiset(B),aTP_Lamp_aiv(fun(multiset(B),$o),fun(multiset(B),$o),Pa)) ).

% subset_mset.Greatest_def
tff(fact_7163_Inf__multiset_Oabs__eq,axiom,
    ! [B: $tType,X: set(fun(B,nat))] :
      ( aa(set(fun(B,nat)),$o,aa(set(fun(B,nat)),fun(set(fun(B,nat)),$o),bNF_rel_set(fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o))),X),X)
     => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(fun(B,nat)),set(multiset(B)),image2(fun(B,nat),multiset(B),abs_multiset(B)),X)) = aa(fun(B,nat),multiset(B),abs_multiset(B),aa(set(fun(B,nat)),fun(B,nat),aTP_Lamp_ahw(set(fun(B,nat)),fun(B,nat)),X)) ) ) ).

% Inf_multiset.abs_eq
tff(fact_7164_order_OGreatest_Ocong,axiom,
    ! [B: $tType,Less_eq: fun(B,fun(B,$o))] : greatest(B,Less_eq) = greatest(B,Less_eq) ).

% order.Greatest.cong
tff(fact_7165_fun_Oset__transfer,axiom,
    ! [C: $tType,D: $tType,B: $tType,R2: fun(C,fun(D,$o))] : aa(fun(fun(B,D),set(D)),$o,aa(fun(fun(B,C),set(C)),fun(fun(fun(B,D),set(D)),$o),bNF_rel_fun(fun(B,C),fun(B,D),set(C),set(D),bNF_rel_fun(B,B,C,D,fequal(B),R2),bNF_rel_set(C,D,R2)),aTP_Lamp_aiw(fun(B,C),set(C))),aTP_Lamp_aix(fun(B,D),set(D))) ).

% fun.set_transfer
tff(fact_7166_type__definition__integer,axiom,
    type_definition(code_integer,int,code_int_of_integer,code_integer_of_int,top_top(set(int))) ).

% type_definition_integer
tff(fact_7167_Inf__multiset_Orsp,axiom,
    ! [B: $tType] : aa(fun(set(fun(B,nat)),fun(B,nat)),$o,aa(fun(set(fun(B,nat)),fun(B,nat)),fun(fun(set(fun(B,nat)),fun(B,nat)),$o),bNF_rel_fun(set(fun(B,nat)),set(fun(B,nat)),fun(B,nat),fun(B,nat),bNF_rel_set(fun(B,nat),fun(B,nat),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o))),bNF_eq_onp(fun(B,nat),aTP_Lamp_ahn(fun(B,nat),$o))),aTP_Lamp_ahw(set(fun(B,nat)),fun(B,nat))),aTP_Lamp_ahw(set(fun(B,nat)),fun(B,nat))) ).

% Inf_multiset.rsp
tff(fact_7168_INF__parametric,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( complete_Inf(D)
     => ! [A4: fun(B,fun(C,$o))] : aa(fun(set(C),fun(fun(C,D),D)),$o,aa(fun(set(B),fun(fun(B,D),D)),fun(fun(set(C),fun(fun(C,D),D)),$o),bNF_rel_fun(set(B),set(C),fun(fun(B,D),D),fun(fun(C,D),D),bNF_rel_set(B,C,A4),bNF_rel_fun(fun(B,D),fun(C,D),D,D,bNF_rel_fun(B,C,D,D,A4,fequal(D)),fequal(D))),aTP_Lamp_aiy(set(B),fun(fun(B,D),D))),aTP_Lamp_aiz(set(C),fun(fun(C,D),D))) ) ).

% INF_parametric
tff(fact_7169_SUP__parametric,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( complete_Sup(D)
     => ! [R2: fun(B,fun(C,$o))] : aa(fun(set(C),fun(fun(C,D),D)),$o,aa(fun(set(B),fun(fun(B,D),D)),fun(fun(set(C),fun(fun(C,D),D)),$o),bNF_rel_fun(set(B),set(C),fun(fun(B,D),D),fun(fun(C,D),D),bNF_rel_set(B,C,R2),bNF_rel_fun(fun(B,D),fun(C,D),D,D,bNF_rel_fun(B,C,D,D,R2,fequal(D)),fequal(D))),aTP_Lamp_aja(set(B),fun(fun(B,D),D))),aTP_Lamp_ajb(set(C),fun(fun(C,D),D))) ) ).

% SUP_parametric
tff(fact_7170_Union__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] : aa(fun(set(set(C)),set(C)),$o,aa(fun(set(set(B)),set(B)),fun(fun(set(set(C)),set(C)),$o),bNF_rel_fun(set(set(B)),set(set(C)),set(B),set(C),bNF_rel_set(set(B),set(C),bNF_rel_set(B,C,A4)),bNF_rel_set(B,C,A4)),complete_Sup_Sup(set(B))),complete_Sup_Sup(set(C))) ).

% Union_transfer
tff(fact_7171_UNION__transfer,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,A4: fun(B,fun(C,$o)),B4: fun(D,fun(E,$o))] : aa(fun(set(C),fun(fun(C,set(E)),set(E))),$o,aa(fun(set(B),fun(fun(B,set(D)),set(D))),fun(fun(set(C),fun(fun(C,set(E)),set(E))),$o),bNF_rel_fun(set(B),set(C),fun(fun(B,set(D)),set(D)),fun(fun(C,set(E)),set(E)),bNF_rel_set(B,C,A4),bNF_rel_fun(fun(B,set(D)),fun(C,set(E)),set(D),set(E),bNF_rel_fun(B,C,set(D),set(E),A4,bNF_rel_set(D,E,B4)),bNF_rel_set(D,E,B4))),aTP_Lamp_ajc(set(B),fun(fun(B,set(D)),set(D)))),aTP_Lamp_ajd(set(C),fun(fun(C,set(E)),set(E)))) ).

% UNION_transfer
tff(fact_7172_Inf__multiset_Otransfer,axiom,
    ! [B: $tType] : aa(fun(set(multiset(B)),multiset(B)),$o,aa(fun(set(fun(B,nat)),fun(B,nat)),fun(fun(set(multiset(B)),multiset(B)),$o),bNF_rel_fun(set(fun(B,nat)),set(multiset(B)),fun(B,nat),multiset(B),bNF_rel_set(fun(B,nat),multiset(B),pcr_multiset(B,B,fequal(B))),pcr_multiset(B,B,fequal(B))),aTP_Lamp_ahw(set(fun(B,nat)),fun(B,nat))),complete_Inf_Inf(multiset(B))) ).

% Inf_multiset.transfer
tff(fact_7173_typedef__rep__transfer,axiom,
    ! [C: $tType,B: $tType,Rep: fun(B,C),Abs: fun(C,B),A4: set(C),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,A4)
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => aa(fun(B,C),$o,aa(fun(C,C),fun(fun(B,C),$o),bNF_rel_fun(C,B,C,C,T4,fequal(C)),aTP_Lamp_rv(C,C)),Rep) ) ) ).

% typedef_rep_transfer
tff(fact_7174_zero__multiset_Otransfer,axiom,
    ! [B: $tType] : aa(multiset(B),$o,aa(fun(B,nat),fun(multiset(B),$o),pcr_multiset(B,B,fequal(B)),aTP_Lamp_am(B,nat)),zero_zero(multiset(B))) ).

% zero_multiset.transfer
tff(fact_7175_multiset_Orep__transfer,axiom,
    ! [B: $tType,C: $tType,T4: fun(B,fun(C,$o))] : aa(fun(multiset(C),fun(C,nat)),$o,aa(fun(fun(B,nat),fun(B,nat)),fun(fun(multiset(C),fun(C,nat)),$o),bNF_rel_fun(fun(B,nat),multiset(C),fun(B,nat),fun(C,nat),pcr_multiset(B,C,T4),bNF_rel_fun(B,C,nat,nat,T4,fequal(nat))),aTP_Lamp_aje(fun(B,nat),fun(B,nat))),count(C)) ).

% multiset.rep_transfer
tff(fact_7176_add__mset_Otransfer,axiom,
    ! [B: $tType] : aa(fun(B,fun(multiset(B),multiset(B))),$o,aa(fun(B,fun(fun(B,nat),fun(B,nat))),fun(fun(B,fun(multiset(B),multiset(B))),$o),bNF_rel_fun(B,B,fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),fequal(B),bNF_rel_fun(fun(B,nat),multiset(B),fun(B,nat),multiset(B),pcr_multiset(B,B,fequal(B)),pcr_multiset(B,B,fequal(B)))),aTP_Lamp_aia(B,fun(fun(B,nat),fun(B,nat)))),add_mset(B)) ).

% add_mset.transfer
tff(fact_7177_plus__multiset_Otransfer,axiom,
    ! [B: $tType] : aa(fun(multiset(B),fun(multiset(B),multiset(B))),$o,aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),fun(fun(multiset(B),fun(multiset(B),multiset(B))),$o),bNF_rel_fun(fun(B,nat),multiset(B),fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),pcr_multiset(B,B,fequal(B)),bNF_rel_fun(fun(B,nat),multiset(B),fun(B,nat),multiset(B),pcr_multiset(B,B,fequal(B)),pcr_multiset(B,B,fequal(B)))),aTP_Lamp_ahy(fun(B,nat),fun(fun(B,nat),fun(B,nat)))),plus_plus(multiset(B))) ).

% plus_multiset.transfer
tff(fact_7178_minus__multiset_Otransfer,axiom,
    ! [B: $tType] : aa(fun(multiset(B),fun(multiset(B),multiset(B))),$o,aa(fun(fun(B,nat),fun(fun(B,nat),fun(B,nat))),fun(fun(multiset(B),fun(multiset(B),multiset(B))),$o),bNF_rel_fun(fun(B,nat),multiset(B),fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),pcr_multiset(B,B,fequal(B)),bNF_rel_fun(fun(B,nat),multiset(B),fun(B,nat),multiset(B),pcr_multiset(B,B,fequal(B)),pcr_multiset(B,B,fequal(B)))),aTP_Lamp_ahz(fun(B,nat),fun(fun(B,nat),fun(B,nat)))),minus_minus(multiset(B))) ).

% minus_multiset.transfer
tff(fact_7179_filter__mset_Otransfer,axiom,
    ! [B: $tType] : aa(fun(fun(B,$o),fun(multiset(B),multiset(B))),$o,aa(fun(fun(B,$o),fun(fun(B,nat),fun(B,nat))),fun(fun(fun(B,$o),fun(multiset(B),multiset(B))),$o),bNF_rel_fun(fun(B,$o),fun(B,$o),fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),fequal(fun(B,$o)),bNF_rel_fun(fun(B,nat),multiset(B),fun(B,nat),multiset(B),pcr_multiset(B,B,fequal(B)),pcr_multiset(B,B,fequal(B)))),aTP_Lamp_aib(fun(B,$o),fun(fun(B,nat),fun(B,nat)))),filter_mset(B)) ).

% filter_mset.transfer
tff(fact_7180_repeat__mset_Otransfer,axiom,
    ! [B: $tType] : aa(fun(nat,fun(multiset(B),multiset(B))),$o,aa(fun(nat,fun(fun(B,nat),fun(B,nat))),fun(fun(nat,fun(multiset(B),multiset(B))),$o),bNF_rel_fun(nat,nat,fun(fun(B,nat),fun(B,nat)),fun(multiset(B),multiset(B)),fequal(nat),bNF_rel_fun(fun(B,nat),multiset(B),fun(B,nat),multiset(B),pcr_multiset(B,B,fequal(B)),pcr_multiset(B,B,fequal(B)))),aTP_Lamp_aic(nat,fun(fun(B,nat),fun(B,nat)))),repeat_mset(B)) ).

% repeat_mset.transfer
tff(fact_7181_rel__pred__comp__def,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),Pa: fun(C,$o),X4: B] :
      ( rel_pred_comp(B,C,R2,Pa,X4)
    <=> ? [Y5: C] :
          ( aa(C,$o,aa(B,fun(C,$o),R2,X4),Y5)
          & aa(C,$o,Pa,Y5) ) ) ).

% rel_pred_comp_def
tff(fact_7182_sum__multiset__singleton,axiom,
    ! [B: $tType,A4: set(B)] : aa(set(B),multiset(B),aa(fun(B,multiset(B)),fun(set(B),multiset(B)),groups7311177749621191930dd_sum(B,multiset(B)),aTP_Lamp_agk(B,multiset(B))),A4) = mset_set(B,A4) ).

% sum_multiset_singleton
tff(fact_7183_count__mset__set_I3_J,axiom,
    ! [B: $tType,X: B,A4: set(B)] :
      ( ~ aa(set(B),$o,member(B,X),A4)
     => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset_set(B,A4)),X) = zero_zero(nat) ) ) ).

% count_mset_set(3)
tff(fact_7184_size__mset__set,axiom,
    ! [B: $tType,A4: set(B)] : aa(multiset(B),nat,size_size(multiset(B)),mset_set(B,A4)) = aa(set(B),nat,finite_card(B),A4) ).

% size_mset_set
tff(fact_7185_filter__mset__mset__set,axiom,
    ! [B: $tType,A4: set(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(multiset(B),multiset(B),aa(fun(B,$o),fun(multiset(B),multiset(B)),filter_mset(B),Pa),mset_set(B,A4)) = mset_set(B,aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa))) ) ) ).

% filter_mset_mset_set
tff(fact_7186_count__mset__set_I2_J,axiom,
    ! [B: $tType,A4: set(B),X: B] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset_set(B,A4)),X) = zero_zero(nat) ) ) ).

% count_mset_set(2)
tff(fact_7187_count__mset__set_I1_J,axiom,
    ! [B: $tType,A4: set(B),X: B] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,member(B,X),A4)
       => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset_set(B,A4)),X) = one_one(nat) ) ) ) ).

% count_mset_set(1)
tff(fact_7188_mset__set__upto__eq__mset__upto,axiom,
    ! [N: nat] : mset_set(nat,aa(nat,set(nat),set_ord_lessThan(nat),N)) = mset(nat,upt(zero_zero(nat),N)) ).

% mset_set_upto_eq_mset_upto
tff(fact_7189_count__mset__set_H,axiom,
    ! [B: $tType,A4: set(B),X: B] :
      aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset_set(B,A4)),X) = $ite(
        ( aa(set(B),$o,finite_finite2(B),A4)
        & aa(set(B),$o,member(B,X),A4) ),
        one_one(nat),
        zero_zero(nat) ) ).

% count_mset_set'
tff(fact_7190_count__mset__set__finite__iff,axiom,
    ! [B: $tType,S: set(B),A2: B] :
      ( aa(set(B),$o,finite_finite2(B),S)
     => ( aa(B,nat,aa(multiset(B),fun(B,nat),count(B),mset_set(B,S)),A2) = $ite(aa(set(B),$o,member(B,A2),S),one_one(nat),zero_zero(nat)) ) ) ).

% count_mset_set_finite_iff
tff(fact_7191_mset__set__Union,axiom,
    ! [B: $tType,A4: set(B),B4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(B),$o,finite_finite2(B),B4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = bot_bot(set(B)) )
         => ( mset_set(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),mset_set(B,A4)),mset_set(B,B4)) ) ) ) ) ).

% mset_set_Union
tff(fact_7192_relImage__Gr,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,B)),A4: set(B),F: fun(B,C)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),A4)))
     => ( bNF_Gr4221423524335903396lImage(B,C,R2,F) = relcomp(C,B,C,converse(B,C,bNF_Gr(B,C,A4,F)),relcomp(B,B,C,R2,bNF_Gr(B,C,A4,F))) ) ) ).

% relImage_Gr
tff(fact_7193_relInvImage__Gr,axiom,
    ! [B: $tType,C: $tType,R2: set(product_prod(B,B)),B4: set(B),A4: set(C),F: fun(C,B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R2),product_Sigma(B,B,B4,aTP_Lamp_zv(set(B),fun(B,set(B)),B4)))
     => ( bNF_Gr7122648621184425601vImage(C,B,A4,R2,F) = relcomp(C,B,C,bNF_Gr(C,B,A4,F),relcomp(B,B,C,R2,converse(C,B,bNF_Gr(C,B,A4,F)))) ) ) ).

% relInvImage_Gr
tff(fact_7194_pair__set__inverse,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,fun(B,$o))] : converse(C,B,aa(fun(product_prod(C,B),$o),set(product_prod(C,B)),collect(product_prod(C,B)),aa(fun(C,fun(B,$o)),fun(product_prod(C,B),$o),product_case_prod(C,B,$o),Pa))) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Pa))) ).

% pair_set_inverse
tff(fact_7195_converse__unfold,axiom,
    ! [B: $tType,C: $tType,R: set(product_prod(C,B))] : converse(C,B,R) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_ajf(set(product_prod(C,B)),fun(B,fun(C,$o)),R))) ).

% converse_unfold
tff(fact_7196_converse__Times,axiom,
    ! [B: $tType,C: $tType,A4: set(C),B4: set(B)] : converse(C,B,product_Sigma(C,B,A4,aTP_Lamp_vx(set(B),fun(C,set(B)),B4))) = product_Sigma(B,C,B4,aTP_Lamp_vu(set(C),fun(B,set(C)),A4)) ).

% converse_Times
tff(fact_7197_converse__INTER,axiom,
    ! [C: $tType,B: $tType,D: $tType,R: fun(D,set(product_prod(C,B))),S: set(D)] : converse(C,B,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Inf_Inf(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image2(D,set(product_prod(C,B)),R),S))) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Inf_Inf(set(product_prod(B,C))),aa(set(D),set(set(product_prod(B,C))),image2(D,set(product_prod(B,C)),aTP_Lamp_ajg(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(B,C))),R)),S)) ).

% converse_INTER
tff(fact_7198_converse__UNION,axiom,
    ! [C: $tType,B: $tType,D: $tType,R: fun(D,set(product_prod(C,B))),S: set(D)] : converse(C,B,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)),R),S))) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(D),set(set(product_prod(B,C))),image2(D,set(product_prod(B,C)),aTP_Lamp_ajg(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(B,C))),R)),S)) ).

% converse_UNION
tff(fact_7199_image2__Gr,axiom,
    ! [B: $tType,C: $tType,D: $tType,A4: set(D),F: fun(D,B),G: fun(D,C)] : bNF_Greatest_image2(D,B,C,A4,F,G) = relcomp(B,D,C,converse(D,B,bNF_Gr(D,B,A4,F)),bNF_Gr(D,C,A4,G)) ).

% image2_Gr
tff(fact_7200_Image__INT__eq,axiom,
    ! [C: $tType,B: $tType,D: $tType,R: set(product_prod(C,B)),A4: set(D),B4: fun(D,set(C))] :
      ( single_valued(B,C,converse(C,B,R))
     => ( ( A4 != bot_bot(set(D)) )
       => ( aa(set(C),set(B),image(C,B,R),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B4),A4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_aga(set(product_prod(C,B)),fun(fun(D,set(C)),fun(D,set(B))),R),B4)),A4)) ) ) ) ).

% Image_INT_eq
tff(fact_7201_stable__sort__key__def,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Sk: fun(fun(B,C),fun(list(B),list(B)))] :
          ( linord3483353639454293061rt_key(B,C,Sk)
        <=> ! [F5: fun(B,C),Xs3: list(B),K2: C] : filter2(B,aa(C,fun(B,$o),aTP_Lamp_pc(fun(B,C),fun(C,fun(B,$o)),F5),K2),aa(list(B),list(B),aa(fun(B,C),fun(list(B),list(B)),Sk,F5),Xs3)) = filter2(B,aa(C,fun(B,$o),aTP_Lamp_pc(fun(B,C),fun(C,fun(B,$o)),F5),K2),Xs3) ) ) ).

% stable_sort_key_def
tff(fact_7202_single__valued__inter1,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(B,C))] :
      ( single_valued(B,C,R2)
     => single_valued(B,C,aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),R2),S)) ) ).

% single_valued_inter1
tff(fact_7203_single__valued__inter2,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C)),S: set(product_prod(B,C))] :
      ( single_valued(B,C,R2)
     => single_valued(B,C,aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),set(product_prod(B,C))),inf_inf(set(product_prod(B,C))),S),R2)) ) ).

% single_valued_inter2
tff(fact_7204_bijective__alt,axiom,
    ! [C: $tType,B: $tType,R2: set(product_prod(B,C))] :
      ( bijective(B,C,R2)
    <=> ( single_valued(B,C,R2)
        & single_valued(C,B,converse(B,C,R2)) ) ) ).

% bijective_alt
tff(fact_7205_numeral__xor__num,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Ma: num,N: num] : aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = case_option(B,num,zero_zero(B),numeral_numeral(B),bit_un2480387367778600638or_num(Ma,N)) ) ).

% numeral_xor_num
tff(fact_7206_filterlim__base__iff,axiom,
    ! [B: $tType,C: $tType,D: $tType,E: $tType,I5: set(B),F3: fun(B,set(C)),F: fun(C,D),G5: fun(E,set(D)),J4: set(E)] :
      ( ( I5 != bot_bot(set(B)) )
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),I5)
           => ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),I5)
               => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),F3,I2)),aa(B,set(C),F3,J2))
                  | aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),F3,J2)),aa(B,set(C),F3,I2)) ) ) )
       => ( filterlim(C,D,F,aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(E),set(filter(D)),image2(E,filter(D),aTP_Lamp_ajh(fun(E,set(D)),fun(E,filter(D)),G5)),J4)),aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),aTP_Lamp_aeh(fun(B,set(C)),fun(B,filter(C)),F3)),I5)))
        <=> ! [X3: E] :
              ( aa(set(E),$o,member(E,X3),J4)
             => ? [Xa4: B] :
                  ( aa(set(B),$o,member(B,Xa4),I5)
                  & ! [Xb4: C] :
                      ( aa(set(C),$o,member(C,Xb4),aa(B,set(C),F3,Xa4))
                     => aa(set(D),$o,member(D,aa(C,D,F,Xb4)),aa(E,set(D),G5,X3)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_7207_filterlim__ident,axiom,
    ! [B: $tType,F3: filter(B)] : filterlim(B,B,aTP_Lamp_il(B,B),F3,F3) ).

% filterlim_ident
tff(fact_7208_filterlim__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,G: fun(B,C),F33: filter(C),F23: filter(B),F: fun(D,B),F12: filter(D)] :
      ( filterlim(B,C,G,F33,F23)
     => ( filterlim(D,B,F,F23,F12)
       => filterlim(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aji(fun(B,C),fun(fun(D,B),fun(D,C)),G),F),F33,F12) ) ) ).

% filterlim_compose
tff(fact_7209_filterlim__inf,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),F23: filter(C),F33: filter(C),F12: filter(B)] :
      ( filterlim(B,C,F,aa(filter(C),filter(C),aa(filter(C),fun(filter(C),filter(C)),inf_inf(filter(C)),F23),F33),F12)
    <=> ( filterlim(B,C,F,F23,F12)
        & filterlim(B,C,F,F33,F12) ) ) ).

% filterlim_inf
tff(fact_7210_filterlim__sequentially__Suc,axiom,
    ! [B: $tType,F: fun(nat,B),F3: filter(B)] :
      ( filterlim(nat,B,aTP_Lamp_qu(fun(nat,B),fun(nat,B),F),F3,at_top(nat))
    <=> filterlim(nat,B,F,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7211_filterlim__INF,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,C),G5: fun(D,filter(C)),B4: set(D),F3: filter(B)] :
      ( filterlim(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),G5),B4)),F3)
    <=> ! [X3: D] :
          ( aa(set(D),$o,member(D,X3),B4)
         => filterlim(B,C,F,aa(D,filter(C),G5,X3),F3) ) ) ).

% filterlim_INF
tff(fact_7212_filterlim__INF_H,axiom,
    ! [D: $tType,C: $tType,B: $tType,X: B,A4: set(B),F: fun(C,D),F3: filter(D),G5: fun(B,filter(C))] :
      ( aa(set(B),$o,member(B,X),A4)
     => ( filterlim(C,D,F,F3,aa(B,filter(C),G5,X))
       => filterlim(C,D,F,F3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),G5),A4))) ) ) ).

% filterlim_INF'
tff(fact_7213_filterlim__If,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),G5: filter(C),F3: filter(B),Pa: fun(B,$o),G: fun(B,C)] :
      ( filterlim(B,C,F,G5,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F3),principal(B,aa(fun(B,$o),set(B),collect(B),Pa))))
     => ( filterlim(B,C,G,G5,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F3),principal(B,aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)))))
       => filterlim(B,C,aa(fun(B,C),fun(B,C),aa(fun(B,$o),fun(fun(B,C),fun(B,C)),aTP_Lamp_ajj(fun(B,C),fun(fun(B,$o),fun(fun(B,C),fun(B,C))),F),Pa),G),G5,F3) ) ) ).

% filterlim_If
tff(fact_7214_filterlim__base,axiom,
    ! [C: $tType,B: $tType,F6: $tType,E: $tType,D: $tType,J4: set(B),I: fun(B,D),I5: set(D),F3: fun(D,set(E)),F: fun(E,F6),G5: fun(B,set(F6))] :
      ( ! [M2: B,X2: C] :
          ( aa(set(B),$o,member(B,M2),J4)
         => aa(set(D),$o,member(D,aa(B,D,I,M2)),I5) )
     => ( ! [M2: B,X2: E] :
            ( aa(set(B),$o,member(B,M2),J4)
           => ( aa(set(E),$o,member(E,X2),aa(D,set(E),F3,aa(B,D,I,M2)))
             => aa(set(F6),$o,member(F6,aa(E,F6,F,X2)),aa(B,set(F6),G5,M2)) ) )
       => filterlim(E,F6,F,aa(set(filter(F6)),filter(F6),complete_Inf_Inf(filter(F6)),aa(set(B),set(filter(F6)),image2(B,filter(F6),aTP_Lamp_ajk(fun(B,set(F6)),fun(B,filter(F6)),G5)),J4)),aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(D),set(filter(E)),image2(D,filter(E),aTP_Lamp_ajl(fun(D,set(E)),fun(D,filter(E)),F3)),I5))) ) ) ).

% filterlim_base
tff(fact_7215_xor__num__eq__None__iff,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Ma: num,N: num] :
          ( ( bit_un2480387367778600638or_num(Ma,N) = none(num) )
        <=> ( aa(B,B,aa(B,fun(B,B),bit_se5824344971392196577ns_xor(B),aa(num,B,numeral_numeral(B),Ma)),aa(num,B,numeral_numeral(B),N)) = zero_zero(B) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_7216_map__rec,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),Xs: list(C)] : aa(list(C),list(B),map(C,B,F),Xs) = aa(list(C),list(B),rec_list(list(B),C,nil(B),aTP_Lamp_ajm(fun(C,B),fun(C,fun(list(C),fun(list(B),list(B)))),F)),Xs) ).

% map_rec
tff(fact_7217_zipf__zip,axiom,
    ! [B: $tType,C: $tType,L1: list(B),L22: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),L1) = aa(list(C),nat,size_size(list(C)),L22) )
     => ( zipf(B,C,product_prod(B,C),product_Pair(B,C),L1,L22) = zip(B,C,L1,L22) ) ) ).

% zipf_zip
tff(fact_7218_rec__list__Cons__imp,axiom,
    ! [C: $tType,B: $tType,F: fun(list(B),C),F1: C,F22: fun(B,fun(list(B),fun(C,C))),X: B,Xs: list(B)] :
      ( ( F = rec_list(C,B,F1,F22) )
     => ( aa(list(B),C,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(C,C,aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),F22,X),Xs),aa(list(B),C,F,Xs)) ) ) ).

% rec_list_Cons_imp
tff(fact_7219_zipf_Osimps_I2_J,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,fun(D,B)),A2: C,As: list(C),B2: D,Bs: list(D)] : zipf(C,D,B,F,aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A2),As),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),B2),Bs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(D,B,aa(C,fun(D,B),F,A2),B2)),zipf(C,D,B,F,As,Bs)) ).

% zipf.simps(2)
tff(fact_7220_zipf_Osimps_I1_J,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(C,fun(D,B))] : zipf(C,D,B,F,nil(C),nil(D)) = nil(B) ).

% zipf.simps(1)
tff(fact_7221_rec__list__Nil__imp,axiom,
    ! [B: $tType,C: $tType,F: fun(list(B),C),F1: C,F22: fun(B,fun(list(B),fun(C,C)))] :
      ( ( F = rec_list(C,B,F1,F22) )
     => ( aa(list(B),C,F,nil(B)) = F1 ) ) ).

% rec_list_Nil_imp
tff(fact_7222_zipf_Oelims,axiom,
    ! [D: $tType,C: $tType,B: $tType,X: fun(C,fun(D,B)),Xa2: list(C),Xb: list(D),Y: list(B)] :
      ( ( zipf(C,D,B,X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = nil(C) )
         => ( ( Xb = nil(D) )
           => ( Y != nil(B) ) ) )
       => ( ! [A3: C,As2: list(C)] :
              ( ( Xa2 = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),As2) )
             => ! [B3: D,Bs2: list(D)] :
                  ( ( Xb = aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),B3),Bs2) )
                 => ( Y != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(D,B,aa(C,fun(D,B),X,A3),B3)),zipf(C,D,B,X,As2,Bs2)) ) ) )
         => ( ( ? [V4: C,Va: list(C)] : Xa2 = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V4),Va)
             => ( ( Xb = nil(D) )
               => ( Y != undefined(list(B)) ) ) )
           => ~ ( ( Xa2 = nil(C) )
               => ( ? [V4: D,Va: list(D)] : Xb = aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),V4),Va)
                 => ( Y != undefined(list(B)) ) ) ) ) ) ) ) ).

% zipf.elims
tff(fact_7223_list_Orec__o__map,axiom,
    ! [C: $tType,D: $tType,B: $tType,G: C,Ga: fun(D,fun(list(D),fun(C,C))),F: fun(B,D)] : aa(fun(list(B),list(D)),fun(list(B),C),aa(fun(list(D),C),fun(fun(list(B),list(D)),fun(list(B),C)),comp(list(D),C,list(B)),rec_list(C,D,G,Ga)),map(B,D,F)) = rec_list(C,B,G,aa(fun(B,D),fun(B,fun(list(B),fun(C,C))),aTP_Lamp_ajn(fun(D,fun(list(D),fun(C,C))),fun(fun(B,D),fun(B,fun(list(B),fun(C,C)))),Ga),F)) ).

% list.rec_o_map
tff(fact_7224_zipf_Opelims,axiom,
    ! [B: $tType,C: $tType,D: $tType,X: fun(C,fun(D,B)),Xa2: list(C),Xb: list(D),Y: list(B)] :
      ( ( zipf(C,D,B,X,Xa2,Xb) = Y )
     => ( accp(product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),zipf_rel(C,D,B),aa(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),aa(fun(C,fun(D,B)),fun(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D)))),product_Pair(fun(C,fun(D,B)),product_prod(list(C),list(D))),X),aa(list(D),product_prod(list(C),list(D)),aa(list(C),fun(list(D),product_prod(list(C),list(D))),product_Pair(list(C),list(D)),Xa2),Xb)))
       => ( ( ( Xa2 = nil(C) )
           => ( ( Xb = nil(D) )
             => ( ( Y = nil(B) )
               => ~ accp(product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),zipf_rel(C,D,B),aa(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),aa(fun(C,fun(D,B)),fun(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D)))),product_Pair(fun(C,fun(D,B)),product_prod(list(C),list(D))),X),aa(list(D),product_prod(list(C),list(D)),aa(list(C),fun(list(D),product_prod(list(C),list(D))),product_Pair(list(C),list(D)),nil(C)),nil(D)))) ) ) )
         => ( ! [A3: C,As2: list(C)] :
                ( ( Xa2 = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),As2) )
               => ! [B3: D,Bs2: list(D)] :
                    ( ( Xb = aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),B3),Bs2) )
                   => ( ( Y = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(D,B,aa(C,fun(D,B),X,A3),B3)),zipf(C,D,B,X,As2,Bs2)) )
                     => ~ accp(product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),zipf_rel(C,D,B),aa(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),aa(fun(C,fun(D,B)),fun(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D)))),product_Pair(fun(C,fun(D,B)),product_prod(list(C),list(D))),X),aa(list(D),product_prod(list(C),list(D)),aa(list(C),fun(list(D),product_prod(list(C),list(D))),product_Pair(list(C),list(D)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),As2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),B3),Bs2)))) ) ) )
           => ( ! [V4: C,Va: list(C)] :
                  ( ( Xa2 = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V4),Va) )
                 => ( ( Xb = nil(D) )
                   => ( ( Y = undefined(list(B)) )
                     => ~ accp(product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),zipf_rel(C,D,B),aa(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),aa(fun(C,fun(D,B)),fun(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D)))),product_Pair(fun(C,fun(D,B)),product_prod(list(C),list(D))),X),aa(list(D),product_prod(list(C),list(D)),aa(list(C),fun(list(D),product_prod(list(C),list(D))),product_Pair(list(C),list(D)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V4),Va)),nil(D)))) ) ) )
             => ~ ( ( Xa2 = nil(C) )
                 => ! [V4: D,Va: list(D)] :
                      ( ( Xb = aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),V4),Va) )
                     => ( ( Y = undefined(list(B)) )
                       => ~ accp(product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),zipf_rel(C,D,B),aa(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D))),aa(fun(C,fun(D,B)),fun(product_prod(list(C),list(D)),product_prod(fun(C,fun(D,B)),product_prod(list(C),list(D)))),product_Pair(fun(C,fun(D,B)),product_prod(list(C),list(D))),X),aa(list(D),product_prod(list(C),list(D)),aa(list(C),fun(list(D),product_prod(list(C),list(D))),product_Pair(list(C),list(D)),nil(C)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),V4),Va)))) ) ) ) ) ) ) ) ) ).

% zipf.pelims
tff(fact_7225_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_7226_finite__subsets__at__top__def,axiom,
    ! [B: $tType,A4: set(B)] : finite5375528669736107172at_top(B,A4) = aa(set(filter(set(B))),filter(set(B)),complete_Inf_Inf(filter(set(B))),aa(set(set(B)),set(filter(set(B))),image2(set(B),filter(set(B)),aTP_Lamp_ajp(set(B),fun(set(B),filter(set(B))),A4)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_ajq(set(B),fun(set(B),$o),A4)))) ).

% finite_subsets_at_top_def
tff(fact_7227_set__rec,axiom,
    ! [B: $tType,Xs: list(B)] : aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),rec_list(set(B),B,bot_bot(set(B)),aTP_Lamp_ajr(B,fun(list(B),fun(set(B),set(B))))),Xs) ).

% set_rec
tff(fact_7228_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_7229_the__dflt__None__empty,axiom,
    ! [B: $tType] : dflt_None_set(B,bot_bot(set(B))) = none(set(B)) ).

% the_dflt_None_empty
tff(fact_7230_the__dflt__None__nonempty,axiom,
    ! [B: $tType,S: set(B)] :
      ( ( S != bot_bot(set(B)) )
     => ( dflt_None_set(B,S) = aa(set(B),option(set(B)),some(set(B)),S) ) ) ).

% the_dflt_None_nonempty
tff(fact_7231_dflt__None__set__def,axiom,
    ! [B: $tType,S: set(B)] :
      dflt_None_set(B,S) = $ite(S = bot_bot(set(B)),none(set(B)),aa(set(B),option(set(B)),some(set(B)),S)) ).

% dflt_None_set_def
tff(fact_7232_the__dflt__None__set,axiom,
    ! [B: $tType,X: set(B)] : the_default(set(B),bot_bot(set(B)),dflt_None_set(B,X)) = X ).

% the_dflt_None_set
tff(fact_7233_curr__surj,axiom,
    ! [D: $tType,C: $tType,B: $tType,G: fun(B,fun(C,D)),A4: set(B),B4: set(C),C6: set(D)] :
      ( aa(set(fun(B,fun(C,D))),$o,member(fun(B,fun(C,D)),G),bNF_Wellorder_Func(B,fun(C,D),A4,bNF_Wellorder_Func(C,D,B4,C6)))
     => ? [X2: fun(product_prod(B,C),D)] :
          ( aa(set(fun(product_prod(B,C),D)),$o,member(fun(product_prod(B,C),D),X2),bNF_Wellorder_Func(product_prod(B,C),D,product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)),C6))
          & ( aa(fun(product_prod(B,C),D),fun(B,fun(C,D)),bNF_Wellorder_curr(B,C,D,A4),X2) = G ) ) ) ).

% curr_surj
tff(fact_7234_curr__def,axiom,
    ! [B: $tType,D: $tType,C: $tType,A4: set(B),F: fun(product_prod(B,C),D),X4: B] :
      aa(B,fun(C,D),aa(fun(product_prod(B,C),D),fun(B,fun(C,D)),bNF_Wellorder_curr(B,C,D,A4),F),X4) = $ite(aa(set(B),$o,member(B,X4),A4),aa(B,fun(C,D),aTP_Lamp_ly(fun(product_prod(B,C),D),fun(B,fun(C,D)),F),X4),undefined(fun(C,D))) ).

% curr_def
tff(fact_7235_curr__inj,axiom,
    ! [D: $tType,C: $tType,B: $tType,F1: fun(product_prod(B,C),D),A4: set(B),B4: set(C),C6: set(D),F22: fun(product_prod(B,C),D)] :
      ( aa(set(fun(product_prod(B,C),D)),$o,member(fun(product_prod(B,C),D),F1),bNF_Wellorder_Func(product_prod(B,C),D,product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)),C6))
     => ( aa(set(fun(product_prod(B,C),D)),$o,member(fun(product_prod(B,C),D),F22),bNF_Wellorder_Func(product_prod(B,C),D,product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)),C6))
       => ( ( aa(fun(product_prod(B,C),D),fun(B,fun(C,D)),bNF_Wellorder_curr(B,C,D,A4),F1) = aa(fun(product_prod(B,C),D),fun(B,fun(C,D)),bNF_Wellorder_curr(B,C,D,A4),F22) )
        <=> ( F1 = F22 ) ) ) ) ).

% curr_inj
tff(fact_7236_the__default_Osimps_I1_J,axiom,
    ! [B: $tType,Uu: B,X: B] : the_default(B,Uu,aa(B,option(B),some(B),X)) = X ).

% the_default.simps(1)
tff(fact_7237_the__default_Osimps_I2_J,axiom,
    ! [B: $tType,X: B] : the_default(B,X,none(B)) = X ).

% the_default.simps(2)
tff(fact_7238_curr__in,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(product_prod(B,C),D),A4: set(B),B4: set(C),C6: set(D)] :
      ( aa(set(fun(product_prod(B,C),D)),$o,member(fun(product_prod(B,C),D),F),bNF_Wellorder_Func(product_prod(B,C),D,product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)),C6))
     => aa(set(fun(B,fun(C,D))),$o,member(fun(B,fun(C,D)),aa(fun(product_prod(B,C),D),fun(B,fun(C,D)),bNF_Wellorder_curr(B,C,D,A4),F)),bNF_Wellorder_Func(B,fun(C,D),A4,bNF_Wellorder_Func(C,D,B4,C6))) ) ).

% curr_in
tff(fact_7239_bij__betw__curr,axiom,
    ! [B: $tType,C: $tType,D: $tType,A4: set(B),B4: set(C),C6: set(D)] : bij_betw(fun(product_prod(B,C),D),fun(B,fun(C,D)),bNF_Wellorder_curr(B,C,D,A4),bNF_Wellorder_Func(product_prod(B,C),D,product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4)),C6),bNF_Wellorder_Func(B,fun(C,D),A4,bNF_Wellorder_Func(C,D,B4,C6))) ).

% bij_betw_curr
tff(fact_7240_listrel__iff__nth,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C),R: set(product_prod(B,C))] :
      ( aa(set(product_prod(list(B),list(C))),$o,member(product_prod(list(B),list(C)),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),Xs),Ys)),listrel(B,C,R))
    <=> ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
        & ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Xs))
           => aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(nat,B,nth(B,Xs),N4)),aa(nat,C,nth(C,Ys),N4))),R) ) ) ) ).

% listrel_iff_nth
tff(fact_7241_bij__betw__add,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [A2: B,A4: set(B),B4: set(B)] :
          ( bij_betw(B,B,aa(B,fun(B,B),plus_plus(B),A2),A4,B4)
        <=> ( aa(set(B),set(B),image2(B,B,aa(B,fun(B,B),plus_plus(B),A2)),A4) = B4 ) ) ) ).

% bij_betw_add
tff(fact_7242_o__bij,axiom,
    ! [B: $tType,C: $tType,G: fun(C,B),F: fun(B,C)] :
      ( ( aa(fun(B,C),fun(B,B),aa(fun(C,B),fun(fun(B,C),fun(B,B)),comp(C,B,B),G),F) = id(B) )
     => ( ( aa(fun(C,B),fun(C,C),aa(fun(B,C),fun(fun(C,B),fun(C,C)),comp(B,C,C),F),G) = id(C) )
       => bij_betw(B,C,F,top_top(set(B)),top_top(set(C))) ) ) ).

% o_bij
tff(fact_7243_listrel__eq__len,axiom,
    ! [B: $tType,C: $tType,Xs: list(B),Ys: list(C),R: set(product_prod(B,C))] :
      ( aa(set(product_prod(list(B),list(C))),$o,member(product_prod(list(B),list(C)),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),Xs),Ys)),listrel(B,C,R))
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% listrel_eq_len
tff(fact_7244_sum_Oreindex__bij__betw,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [H: fun(B,C),S: set(B),T4: set(C),G: fun(C,D)] :
          ( bij_betw(B,C,H,S,T4)
         => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_ajs(fun(B,C),fun(fun(C,D),fun(B,D)),H),G)),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),G),T4) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_7245_bij__betw__iff__card,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => ( ? [F5: fun(B,C)] : bij_betw(B,C,F5,A4,B4)
        <=> ( aa(set(B),nat,finite_card(B),A4) = aa(set(C),nat,finite_card(C),B4) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_7246_finite__same__card__bij,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => ( ( aa(set(B),nat,finite_card(B),A4) = aa(set(C),nat,finite_card(C),B4) )
         => ? [H2: fun(B,C)] : bij_betw(B,C,H2,A4,B4) ) ) ) ).

% finite_same_card_bij
tff(fact_7247_prod_Oreindex__bij__betw,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [H: fun(B,C),S: set(B),T4: set(C),G: fun(C,D)] :
          ( bij_betw(B,C,H,S,T4)
         => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_ajt(fun(B,C),fun(fun(C,D),fun(B,D)),H),G)),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),G),T4) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_7248_bij__betw__funpow,axiom,
    ! [B: $tType,F: fun(B,B),S: set(B),N: nat] :
      ( bij_betw(B,B,F,S,S)
     => bij_betw(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),S,S) ) ).

% bij_betw_funpow
tff(fact_7249_bij__fn,axiom,
    ! [B: $tType,F: fun(B,B),N: nat] :
      ( bij_betw(B,B,F,top_top(set(B)),top_top(set(B)))
     => bij_betw(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N),F),top_top(set(B)),top_top(set(B))) ) ).

% bij_fn
tff(fact_7250_Schroeder__Bernstein,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),A4: set(B),B4: set(C),G: fun(C,B)] :
      ( inj_on(B,C,F,A4)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F),A4)),B4)
       => ( inj_on(C,B,G,B4)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(C),set(B),image2(C,B,G),B4)),A4)
           => ? [H2: fun(B,C)] : bij_betw(B,C,H2,A4,B4) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_7251_bij__betw__disjoint__Un,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),A4: set(B),C6: set(C),G: fun(B,C),B4: set(B),D3: set(C)] :
      ( bij_betw(B,C,F,A4,C6)
     => ( bij_betw(B,C,G,B4,D3)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) = bot_bot(set(B)) )
         => ( ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),C6),D3) = bot_bot(set(C)) )
           => bij_betw(B,C,aa(fun(B,C),fun(B,C),aa(set(B),fun(fun(B,C),fun(B,C)),aTP_Lamp_afl(fun(B,C),fun(set(B),fun(fun(B,C),fun(B,C))),F),A4),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),C6),D3)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7252_bij__betw__UNION__chain,axiom,
    ! [C: $tType,D: $tType,B: $tType,I5: set(B),A4: fun(B,set(C)),F: fun(C,D),A9: fun(B,set(D))] :
      ( ! [I2: B,J2: B] :
          ( aa(set(B),$o,member(B,I2),I5)
         => ( aa(set(B),$o,member(B,J2),I5)
           => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),A4,I2)),aa(B,set(C),A4,J2))
              | aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),A4,J2)),aa(B,set(C),A4,I2)) ) ) )
     => ( ! [I2: B] :
            ( aa(set(B),$o,member(B,I2),I5)
           => bij_betw(C,D,F,aa(B,set(C),A4,I2),aa(B,set(D),A9,I2)) )
       => bij_betw(C,D,F,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),A9),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_7253_infinite__imp__bij__betw2,axiom,
    ! [B: $tType,A4: set(B),A2: B] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ? [H2: fun(B,B)] : bij_betw(B,B,H2,A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7254_infinite__imp__bij__betw,axiom,
    ! [B: $tType,A4: set(B),A2: B] :
      ( ~ aa(set(B),$o,finite_finite2(B),A4)
     => ? [H2: fun(B,B)] : bij_betw(B,B,H2,A4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B))))) ) ).

% infinite_imp_bij_betw
tff(fact_7255_mono__bij__Inf,axiom,
    ! [C: $tType,B: $tType] :
      ( ( comple5582772986160207858norder(B)
        & comple5582772986160207858norder(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( bij_betw(B,C,F,top_top(set(B)),top_top(set(C)))
           => ( aa(B,C,F,aa(set(B),B,complete_Inf_Inf(B),A4)) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4)) ) ) ) ) ).

% mono_bij_Inf
tff(fact_7256_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [S5: set(B),T3: set(C),H: fun(B,C),S: set(B),T4: set(C),G: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),S5)
         => ( aa(set(C),$o,finite_finite2(C),T3)
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),S5)
                   => ( aa(C,D,G,aa(B,C,H,A3)) = zero_zero(D) ) )
               => ( ! [B3: C] :
                      ( aa(set(C),$o,member(C,B3),T3)
                     => ( aa(C,D,G,B3) = zero_zero(D) ) )
                 => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_ajs(fun(B,C),fun(fun(C,D),fun(B,D)),H),G)),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),G),T4) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7257_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [S5: set(B),T3: set(C),H: fun(B,C),S: set(B),T4: set(C),G: fun(C,D)] :
          ( aa(set(B),$o,finite_finite2(B),S5)
         => ( aa(set(C),$o,finite_finite2(C),T3)
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
             => ( ! [A3: B] :
                    ( aa(set(B),$o,member(B,A3),S5)
                   => ( aa(C,D,G,aa(B,C,H,A3)) = one_one(D) ) )
               => ( ! [B3: C] :
                      ( aa(set(C),$o,member(C,B3),T3)
                     => ( aa(C,D,G,B3) = one_one(D) ) )
                 => ( aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(fun(C,D),fun(B,D),aTP_Lamp_ajt(fun(B,C),fun(fun(C,D),fun(B,D)),H),G)),S) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),G),T4) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7258_bij__image__INT,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(B,C),B4: fun(D,set(B)),A4: set(D)] :
      ( bij_betw(B,C,F,top_top(set(B)),top_top(set(C)))
     => ( aa(set(B),set(C),image2(B,C,F),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),B4),A4))) = 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(B)),fun(D,set(C)),aTP_Lamp_afm(fun(B,C),fun(fun(D,set(B)),fun(D,set(C))),F),B4)),A4)) ) ) ).

% bij_image_INT
tff(fact_7259_listrel__iff__zip,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C),R: set(product_prod(B,C))] :
      ( aa(set(product_prod(list(B),list(C))),$o,member(product_prod(list(B),list(C)),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),Xs),Ys)),listrel(B,C,R))
    <=> ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
        & ! [X3: product_prod(B,C)] :
            ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X3),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,Xs,Ys)))
           => aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),R)),X3) ) ) ) ).

% listrel_iff_zip
tff(fact_7260_listrel__def,axiom,
    ! [C: $tType,B: $tType,X4: set(product_prod(B,C))] : listrel(B,C,X4) = aa(fun(product_prod(list(B),list(C)),$o),set(product_prod(list(B),list(C))),collect(product_prod(list(B),list(C))),aa(fun(list(B),fun(list(C),$o)),fun(product_prod(list(B),list(C)),$o),product_case_prod(list(B),list(C),$o),listrelp(B,C,aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),X4)))) ).

% listrel_def
tff(fact_7261_bij__betw__of__nat,axiom,
    ! [B: $tType] :
      ( semiring_char_0(B)
     => ! [N5: set(nat),A4: set(B)] :
          ( bij_betw(nat,B,semiring_1_of_nat(B),N5,A4)
        <=> ( aa(set(nat),set(B),image2(nat,B,semiring_1_of_nat(B)),N5) = A4 ) ) ) ).

% bij_betw_of_nat
tff(fact_7262_bij__betw__Suc,axiom,
    ! [M: set(nat),N5: set(nat)] :
      ( bij_betw(nat,nat,suc,M,N5)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M) = N5 ) ) ).

% bij_betw_Suc
tff(fact_7263_INF__bool__eq,axiom,
    ! [B: $tType] : aTP_Lamp_aju(set(B),fun(fun(B,$o),$o)) = ball(B) ).

% INF_bool_eq
tff(fact_7264_Collect__ball__eq,axiom,
    ! [B: $tType,C: $tType,A4: set(C),Pa: fun(B,fun(C,$o))] : aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),aTP_Lamp_ajv(set(C),fun(fun(B,fun(C,$o)),fun(B,$o)),A4),Pa)) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_yq(fun(B,fun(C,$o)),fun(C,set(B)),Pa)),A4)) ).

% Collect_ball_eq
tff(fact_7265_INTER__eq,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C)] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_ajw(fun(C,set(B)),fun(set(C),fun(B,$o)),B4),A4)) ).

% INTER_eq
tff(fact_7266_congruent2__def,axiom,
    ! [C: $tType,D: $tType,B: $tType,R1: set(product_prod(B,B)),R22: set(product_prod(C,C)),F: fun(B,fun(C,D))] :
      ( equiv_congruent2(B,C,D,R1,R22,F)
    <=> ! [X3: product_prod(B,B)] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X3),R1)
         => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(fun(B,fun(C,D)),fun(B,fun(B,$o)),aTP_Lamp_ajy(set(product_prod(C,C)),fun(fun(B,fun(C,D)),fun(B,fun(B,$o))),R22),F)),X3) ) ) ).

% congruent2_def
tff(fact_7267_irrefl__distinct,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( irrefl(B,R)
    <=> ! [X3: product_prod(B,B)] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X3),R)
         => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aTP_Lamp_ajz(B,fun(B,$o))),X3) ) ) ).

% irrefl_distinct
tff(fact_7268_congruent__def,axiom,
    ! [C: $tType,B: $tType,R: set(product_prod(B,B)),F: fun(B,C)] :
      ( equiv_congruent(B,C,R,F)
    <=> ! [X3: product_prod(B,B)] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X3),R)
         => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aTP_Lamp_afk(fun(B,C),fun(B,fun(B,$o)),F)),X3) ) ) ).

% congruent_def
tff(fact_7269_Ball__fold,axiom,
    ! [B: $tType,A4: set(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
           => aa(B,$o,Pa,X3) )
      <=> finite_fold(B,$o,aTP_Lamp_aka(fun(B,$o),fun(B,fun($o,$o)),Pa),$true,A4) ) ) ).

% Ball_fold
tff(fact_7270_set__relator__eq__onp,axiom,
    ! [B: $tType,Pa: fun(B,$o)] : bNF_rel_set(B,B,bNF_eq_onp(B,Pa)) = bNF_eq_onp(set(B),aTP_Lamp_akb(fun(B,$o),fun(set(B),$o),Pa)) ).

% set_relator_eq_onp
tff(fact_7271_Ball__comp__iff,axiom,
    ! [B: $tType,C: $tType,D: $tType,A4: fun(C,set(D)),F: fun(D,$o),G: fun(B,C),X4: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(C,$o),fun(fun(B,C),fun(B,$o)),comp(C,$o,B),aa(fun(D,$o),fun(C,$o),aTP_Lamp_akc(fun(C,set(D)),fun(fun(D,$o),fun(C,$o)),A4),F)),G),X4)
    <=> ! [Xa4: D] :
          ( aa(set(D),$o,member(D,Xa4),aa(B,set(D),aa(fun(B,C),fun(B,set(D)),aa(fun(C,set(D)),fun(fun(B,C),fun(B,set(D))),comp(C,set(D),B),A4),G),X4))
         => aa(D,$o,F,Xa4) ) ) ).

% Ball_comp_iff
tff(fact_7272_Inf__eq__Sup,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B)] : aa(set(B),B,complete_Inf_Inf(B),A4) = aa(set(B),B,complete_Sup_Sup(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_akd(set(B),fun(B,$o),A4))) ) ).

% Inf_eq_Sup
tff(fact_7273_Sup__eq__Inf,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(B)] : aa(set(B),B,complete_Sup_Sup(B),A4) = aa(set(B),B,complete_Inf_Inf(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ake(set(B),fun(B,$o),A4))) ) ).

% Sup_eq_Inf
tff(fact_7274_trans__join,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( trans(B,R)
    <=> ! [X3: product_prod(B,B)] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X3),R)
         => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aTP_Lamp_akg(set(product_prod(B,B)),fun(B,fun(B,$o)),R)),X3) ) ) ).

% trans_join
tff(fact_7275_takeWhile__append,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B),Ys: list(B)] :
      takeWhile(B,Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = $ite(
        ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Xs))
         => aa(B,$o,Pa,X3) ),
        aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),takeWhile(B,Pa,Ys)),
        takeWhile(B,Pa,Xs) ) ).

% takeWhile_append
tff(fact_7276_dropWhile__append,axiom,
    ! [B: $tType,Pa: fun(B,$o),Xs: list(B),Ys: list(B)] :
      dropWhile(B,Pa,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = $ite(
        ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(list(B),set(B),set2(B),Xs))
         => aa(B,$o,Pa,X3) ),
        dropWhile(B,Pa,Ys),
        aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),dropWhile(B,Pa,Xs)),Ys) ) ).

% dropWhile_append
tff(fact_7277_ex__bij__betw__finite__nat,axiom,
    ! [B: $tType,M: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),M)
     => ? [H2: fun(B,nat)] : bij_betw(B,nat,H2,M,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(B),nat,finite_card(B),M))) ) ).

% ex_bij_betw_finite_nat
tff(fact_7278_ex__bij__betw__nat__finite,axiom,
    ! [B: $tType,M: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),M)
     => ? [H2: fun(nat,B)] : bij_betw(nat,B,H2,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(B),nat,finite_card(B),M)),M) ) ).

% ex_bij_betw_nat_finite
tff(fact_7279_ex__bij__betw__nat__finite__1,axiom,
    ! [B: $tType,M: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),M)
     => ? [H2: fun(nat,B)] : bij_betw(nat,B,H2,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(B),nat,finite_card(B),M)),M) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_7280_Func__def,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] : bNF_Wellorder_Func(B,C,A4,B4) = aa(fun(fun(B,C),$o),set(fun(B,C)),collect(fun(B,C)),aa(set(C),fun(fun(B,C),$o),aTP_Lamp_akh(set(B),fun(set(C),fun(fun(B,C),$o)),A4),B4)) ).

% Func_def
tff(fact_7281_list__eq__iff__zip__eq,axiom,
    ! [B: $tType,Xs: list(B),Ys: list(B)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X3: product_prod(B,B)] :
            ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X3),aa(list(product_prod(B,B)),set(product_prod(B,B)),set2(product_prod(B,B)),zip(B,B,Xs,Ys)))
           => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),fequal(B)),X3) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_7282_bij__betw__nth,axiom,
    ! [B: $tType,Xs: list(B),A4: set(nat),B4: set(B)] :
      ( distinct(B,Xs)
     => ( ( A4 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(B),nat,size_size(list(B)),Xs)) )
       => ( ( B4 = aa(list(B),set(B),set2(B),Xs) )
         => bij_betw(nat,B,nth(B,Xs),A4,B4) ) ) ) ).

% bij_betw_nth
tff(fact_7283_sum_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comm_monoid_add(C)
        & ord(B) )
     => ! [H: fun(nat,B),Ma: nat,N: nat,G: fun(B,C)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,Ma,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,Ma),aa(nat,B,H,N)))
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,Ma),aa(nat,B,H,N))) = aa(set(nat),C,aa(fun(nat,C),fun(set(nat),C),groups7311177749621191930dd_sum(nat,C),aa(fun(nat,B),fun(nat,C),aa(fun(B,C),fun(fun(nat,B),fun(nat,C)),comp(B,C,nat),G),H)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ) ) ).

% sum.atLeastAtMost_reindex
tff(fact_7284_sum_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comm_monoid_add(C)
        & ord(B) )
     => ! [H: fun(nat,B),Ma: nat,N: nat,G: fun(B,C)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,Ma,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,Ma),aa(nat,B,H,N)))
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,Ma),aa(nat,B,H,N))) = aa(set(nat),C,aa(fun(nat,C),fun(set(nat),C),groups7311177749621191930dd_sum(nat,C),aa(fun(nat,B),fun(nat,C),aa(fun(B,C),fun(fun(nat,B),fun(nat,C)),comp(B,C,nat),G),H)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ) ) ).

% sum.atLeastLessThan_reindex
tff(fact_7285_prod_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comm_monoid_mult(C)
        & ord(B) )
     => ! [H: fun(nat,B),Ma: nat,N: nat,G: fun(B,C)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,Ma,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,Ma),aa(nat,B,H,N)))
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,Ma),aa(nat,B,H,N))) = aa(set(nat),C,aa(fun(nat,C),fun(set(nat),C),groups7121269368397514597t_prod(nat,C),aa(fun(nat,B),fun(nat,C),aa(fun(B,C),fun(fun(nat,B),fun(nat,C)),comp(B,C,nat),G),H)),set_or1337092689740270186AtMost(nat,Ma,N)) ) ) ) ).

% prod.atLeastAtMost_reindex
tff(fact_7286_prod_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comm_monoid_mult(C)
        & ord(B) )
     => ! [H: fun(nat,B),Ma: nat,N: nat,G: fun(B,C)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,Ma,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,Ma),aa(nat,B,H,N)))
         => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,Ma),aa(nat,B,H,N))) = aa(set(nat),C,aa(fun(nat,C),fun(set(nat),C),groups7121269368397514597t_prod(nat,C),aa(fun(nat,B),fun(nat,C),aa(fun(B,C),fun(fun(nat,B),fun(nat,C)),comp(B,C,nat),G),H)),set_or7035219750837199246ssThan(nat,Ma,N)) ) ) ) ).

% prod.atLeastLessThan_reindex
tff(fact_7287_concat__eq__concat__iff,axiom,
    ! [B: $tType,Xs: list(list(B)),Ys: list(list(B))] :
      ( ! [X2: product_prod(list(B),list(B))] :
          ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),X2),aa(list(product_prod(list(B),list(B))),set(product_prod(list(B),list(B))),set2(product_prod(list(B),list(B))),zip(list(B),list(B),Xs,Ys)))
         => aa(product_prod(list(B),list(B)),$o,aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_aki(list(B),fun(list(B),$o))),X2) )
     => ( ( aa(list(list(B)),nat,size_size(list(list(B))),Xs) = aa(list(list(B)),nat,size_size(list(list(B))),Ys) )
       => ( ( concat(B,Xs) = concat(B,Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_7288_concat__injective,axiom,
    ! [B: $tType,Xs: list(list(B)),Ys: list(list(B))] :
      ( ( concat(B,Xs) = concat(B,Ys) )
     => ( ( aa(list(list(B)),nat,size_size(list(list(B))),Xs) = aa(list(list(B)),nat,size_size(list(list(B))),Ys) )
       => ( ! [X2: product_prod(list(B),list(B))] :
              ( aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),X2),aa(list(product_prod(list(B),list(B))),set(product_prod(list(B),list(B))),set2(product_prod(list(B),list(B))),zip(list(B),list(B),Xs,Ys)))
             => aa(product_prod(list(B),list(B)),$o,aa(fun(list(B),fun(list(B),$o)),fun(product_prod(list(B),list(B)),$o),product_case_prod(list(B),list(B),$o),aTP_Lamp_aki(list(B),fun(list(B),$o))),X2) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_7289_list__all2__iff,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o)),Xs: list(B),Ys: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,Pa),Xs),Ys)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
        & ! [X3: product_prod(B,C)] :
            ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),X3),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),zip(B,C,Xs,Ys)))
           => aa(product_prod(B,C),$o,aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Pa),X3) ) ) ) ).

% list_all2_iff
tff(fact_7290_listrelp__listrel__eq,axiom,
    ! [C: $tType,B: $tType,R: set(product_prod(B,C)),X4: list(B),Xa: list(C)] :
      ( aa(list(C),$o,aa(list(B),fun(list(C),$o),listrelp(B,C,aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),R)),X4),Xa)
    <=> aa(set(product_prod(list(B),list(C))),$o,member(product_prod(list(B),list(C)),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),X4),Xa)),listrel(B,C,R)) ) ).

% listrelp_listrel_eq
tff(fact_7291_iteratesp_Omono,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [F: fun(B,B)] : aa(fun(fun(B,$o),fun(B,$o)),$o,order_mono(fun(B,$o),fun(B,$o)),aTP_Lamp_akj(fun(B,B),fun(fun(B,$o),fun(B,$o)),F)) ) ).

% iteratesp.mono
tff(fact_7292_effect__makeE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,F: fun(nat,B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: array(B),N3: nat] :
          ( heap_Time_effect(array(B),array_make(B,N,F),H,H4,R,N3)
         => ~ ( ( R = aa(product_prod(array(B),heap_ext(product_unit)),array(B),product_fst(array(B),heap_ext(product_unit)),array_alloc(B,aa(list(nat),list(B),map(nat,B,F),upt(zero_zero(nat),N)),H)) )
             => ( ( H4 = aa(product_prod(array(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(B),heap_ext(product_unit)),array_alloc(B,aa(list(nat),list(B),map(nat,B,F),upt(zero_zero(nat),N)),H)) )
               => ( ( array_get(B,H4,R) = aa(list(nat),list(B),map(nat,B,F),upt(zero_zero(nat),N)) )
                 => ( array_present(B,H4,R)
                   => ( ~ array_present(B,H,R)
                     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N3 ) ) ) ) ) ) ) ) ).

% effect_makeE
tff(fact_7293_Inter__eq,axiom,
    ! [B: $tType,A4: set(set(B))] : aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),A4) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_akk(set(set(B)),fun(B,$o),A4)) ).

% Inter_eq
tff(fact_7294_chain__singleton,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [X: B] : comple1602240252501008431_chain(B,ord_less_eq(B),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))) ) ).

% chain_singleton
tff(fact_7295_chain__compr,axiom,
    ! [B: $tType,Ord: fun(B,fun(B,$o)),A4: set(B),Pa: fun(B,$o)] :
      ( comple1602240252501008431_chain(B,Ord,A4)
     => comple1602240252501008431_chain(B,Ord,aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),A4),Pa))) ) ).

% chain_compr
tff(fact_7296_ccpo__Sup__least,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [A4: set(B),Z: B] :
          ( comple1602240252501008431_chain(B,ord_less_eq(B),A4)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Z) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),Z) ) ) ) ).

% ccpo_Sup_least
tff(fact_7297_ccpo__Sup__upper,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [A4: set(B),X: B] :
          ( comple1602240252501008431_chain(B,ord_less_eq(B),A4)
         => ( aa(set(B),$o,member(B,X),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ) ).

% ccpo_Sup_upper
tff(fact_7298_ccpo__Sup__mono,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [A4: set(B),B4: set(B)] :
          ( comple1602240252501008431_chain(B,ord_less_eq(B),A4)
         => ( comple1602240252501008431_chain(B,ord_less_eq(B),B4)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),A4)
                 => ? [Xa: B] :
                      ( aa(set(B),$o,member(B,Xa),B4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Xa) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),aa(set(B),B,complete_Sup_Sup(B),B4)) ) ) ) ) ).

% ccpo_Sup_mono
tff(fact_7299_Sup__int__def,axiom,
    ! [X5: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X5) = the(int,aTP_Lamp_akl(set(int),fun(int,$o),X5)) ).

% Sup_int_def
tff(fact_7300_Union__maximal__sets,axiom,
    ! [B: $tType,F8: set(set(B))] :
      ( aa(set(set(B)),$o,finite_finite2(set(B)),F8)
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_akm(set(set(B)),fun(set(B),$o),F8))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),F8) ) ) ).

% Union_maximal_sets
tff(fact_7301_Sup__Inf,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A4: set(set(B))] : aa(set(B),B,complete_Sup_Sup(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Inf_Inf(B)),A4)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Sup_Sup(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_akn(set(set(B)),fun(set(B),$o),A4)))) ) ).

% Sup_Inf
tff(fact_7302_Inf__Sup,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A4: set(set(B))] : aa(set(B),B,complete_Inf_Inf(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Sup_Sup(B)),A4)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Inf_Inf(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_akn(set(set(B)),fun(set(B),$o),A4)))) ) ).

% Inf_Sup
tff(fact_7303_Inf__filter__def,axiom,
    ! [B: $tType,S: set(filter(B))] : aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),S) = aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(fun(filter(B),$o),set(filter(B)),collect(filter(B)),aTP_Lamp_ako(set(filter(B)),fun(filter(B),$o),S))) ).

% Inf_filter_def
tff(fact_7304_effect__of__listE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: array(B),N3: nat] :
          ( heap_Time_effect(array(B),array_of_list(B,Xs),H,H4,R,N3)
         => ~ ( ( R = aa(product_prod(array(B),heap_ext(product_unit)),array(B),product_fst(array(B),heap_ext(product_unit)),array_alloc(B,Xs,H)) )
             => ( ( H4 = aa(product_prod(array(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(B),heap_ext(product_unit)),array_alloc(B,Xs,H)) )
               => ( ( array_get(B,H4,R) = Xs )
                 => ( array_present(B,H4,R)
                   => ( ~ array_present(B,H,R)
                     => ( N3 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(B),nat,size_size(list(B)),Xs)) ) ) ) ) ) ) ) ) ).

% effect_of_listE
tff(fact_7305_effect__newE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [N: nat,X: B,H: heap_ext(product_unit),H4: heap_ext(product_unit),R: array(B),N3: nat] :
          ( heap_Time_effect(array(B),array_new(B,N,X),H,H4,R,N3)
         => ~ ( ( R = aa(product_prod(array(B),heap_ext(product_unit)),array(B),product_fst(array(B),heap_ext(product_unit)),array_alloc(B,replicate(B,N,X),H)) )
             => ( ( H4 = aa(product_prod(array(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(B),heap_ext(product_unit)),array_alloc(B,replicate(B,N,X),H)) )
               => ( ( array_get(B,H4,R) = replicate(B,N,X) )
                 => ( array_present(B,H4,R)
                   => ( ~ array_present(B,H,R)
                     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N3 ) ) ) ) ) ) ) ) ).

% effect_newE
tff(fact_7306_Sup__Inf__le,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(set(B))] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Inf_Inf(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_akp(set(set(B)),fun(set(B),$o),A4))))),aa(set(B),B,complete_Inf_Inf(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Sup_Sup(B)),A4))) ) ).

% Sup_Inf_le
tff(fact_7307_Inf__Sup__le,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A4: set(set(B))] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Sup_Sup(B)),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Inf_Inf(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_akn(set(set(B)),fun(set(B),$o),A4))))) ) ).

% Inf_Sup_le
tff(fact_7308_finite__Inf__Sup,axiom,
    ! [B: $tType] :
      ( finite8700451911770168679attice(B)
     => ! [A4: set(set(B))] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Sup_Sup(B)),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(set(B)),set(B),image2(set(B),B,complete_Inf_Inf(B)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_akq(set(set(B)),fun(set(B),$o),A4))))) ) ).

% finite_Inf_Sup
tff(fact_7309_INF__SUP__set,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [G: fun(C,B),A4: set(set(C))] : aa(set(B),B,complete_Inf_Inf(B),aa(set(set(C)),set(B),image2(set(C),B,aTP_Lamp_akr(fun(C,B),fun(set(C),B),G)),A4)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(set(C)),set(B),image2(set(C),B,aTP_Lamp_aks(fun(C,B),fun(set(C),B),G)),aa(fun(set(C),$o),set(set(C)),collect(set(C)),aTP_Lamp_akt(set(set(C)),fun(set(C),$o),A4)))) ) ).

% INF_SUP_set
tff(fact_7310_SUP__INF__set,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [G: fun(C,B),A4: set(set(C))] : aa(set(B),B,complete_Sup_Sup(B),aa(set(set(C)),set(B),image2(set(C),B,aTP_Lamp_aks(fun(C,B),fun(set(C),B),G)),A4)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(set(C)),set(B),image2(set(C),B,aTP_Lamp_akr(fun(C,B),fun(set(C),B),G)),aa(fun(set(C),$o),set(set(C)),collect(set(C)),aTP_Lamp_akt(set(set(C)),fun(set(C),$o),A4)))) ) ).

% SUP_INF_set
tff(fact_7311_option__Inf__Sup,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [A4: set(set(option(B)))] : aa(option(B),$o,aa(option(B),fun(option(B),$o),ord_less_eq(option(B)),aa(set(option(B)),option(B),complete_Inf_Inf(option(B)),aa(set(set(option(B))),set(option(B)),image2(set(option(B)),option(B),complete_Sup_Sup(option(B))),A4))),aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(set(option(B))),set(option(B)),image2(set(option(B)),option(B),complete_Inf_Inf(option(B))),aa(fun(set(option(B)),$o),set(set(option(B))),collect(set(option(B))),aTP_Lamp_aku(set(set(option(B))),fun(set(option(B)),$o),A4))))) ) ).

% option_Inf_Sup
tff(fact_7312_subset__mset_OcSup__cInf,axiom,
    ! [B: $tType,S: set(multiset(B))] :
      ( ( S != bot_bot(set(multiset(B))) )
     => ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),S)
       => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),S) = aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aTP_Lamp_akv(set(multiset(B)),fun(multiset(B),$o),S))) ) ) ) ).

% subset_mset.cSup_cInf
tff(fact_7313_subset__mset_OcInf__cSup,axiom,
    ! [B: $tType,S: set(multiset(B))] :
      ( ( S != bot_bot(set(multiset(B))) )
     => ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),S)
       => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),S) = aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aTP_Lamp_akw(set(multiset(B)),fun(multiset(B),$o),S))) ) ) ) ).

% subset_mset.cInf_cSup
tff(fact_7314_in__chain__finite,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [A4: set(B)] :
          ( comple1602240252501008431_chain(B,ord_less_eq(B),A4)
         => ( aa(set(B),$o,finite_finite2(B),A4)
           => ( ( A4 != bot_bot(set(B)) )
             => aa(set(B),$o,member(B,aa(set(B),B,complete_Sup_Sup(B),A4)),A4) ) ) ) ) ).

% in_chain_finite
tff(fact_7315_iteratesp__def,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [X4: fun(B,B)] : comple7512665784863727008ratesp(B,X4) = complete_lattice_lfp(fun(B,$o),aTP_Lamp_akj(fun(B,B),fun(fun(B,$o),fun(B,$o)),X4)) ) ).

% iteratesp_def
tff(fact_7316_iteratesp_Osimps,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [F: fun(B,B),A2: B] :
          ( aa(B,$o,comple7512665784863727008ratesp(B,F),A2)
        <=> ( ? [X3: B] :
                ( ( A2 = aa(B,B,F,X3) )
                & aa(B,$o,comple7512665784863727008ratesp(B,F),X3) )
            | ? [M10: set(B)] :
                ( ( A2 = aa(set(B),B,complete_Sup_Sup(B),M10) )
                & comple1602240252501008431_chain(B,ord_less_eq(B),M10)
                & ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),M10)
                   => aa(B,$o,comple7512665784863727008ratesp(B,F),X3) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_7317_iteratesp_Ocases,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [F: fun(B,B),A2: B] :
          ( aa(B,$o,comple7512665784863727008ratesp(B,F),A2)
         => ( ! [X2: B] :
                ( ( A2 = aa(B,B,F,X2) )
               => ~ aa(B,$o,comple7512665784863727008ratesp(B,F),X2) )
           => ~ ! [M8: set(B)] :
                  ( ( A2 = aa(set(B),B,complete_Sup_Sup(B),M8) )
                 => ( comple1602240252501008431_chain(B,ord_less_eq(B),M8)
                   => ~ ! [X4: B] :
                          ( aa(set(B),$o,member(B,X4),M8)
                         => aa(B,$o,comple7512665784863727008ratesp(B,F),X4) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_7318_iteratesp_OSup,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [M: set(B),F: fun(B,B)] :
          ( comple1602240252501008431_chain(B,ord_less_eq(B),M)
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),M)
               => aa(B,$o,comple7512665784863727008ratesp(B,F),X2) )
           => aa(B,$o,comple7512665784863727008ratesp(B,F),aa(set(B),B,complete_Sup_Sup(B),M)) ) ) ) ).

% iteratesp.Sup
tff(fact_7319_execute__nth_I1_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,H: heap_ext(product_unit),A2: array(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,A2))
         => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,array_nth(B,A2,I)),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)),aa(nat,B,nth(B,array_get(B,H,A2)),I)),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_7320_Array__Time_Onth__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),I: nat] : array_nth(B,A2,I) = heap_Time_guard(B,aa(nat,fun(heap_ext(product_unit),$o),aTP_Lamp_akx(array(B),fun(nat,fun(heap_ext(product_unit),$o)),A2),I),aa(nat,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aky(array(B),fun(nat,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))),A2),I)) ) ).

% Array_Time.nth_def
tff(fact_7321_length__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H: heap_ext(product_unit),A2: array(B)] : array_length(B,H,A2) = aa(list(B),nat,size_size(list(B)),array_get(B,H,A2)) ) ).

% length_def
tff(fact_7322_fails__array__nth,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [P2: array(B),I: nat,H: heap_ext(product_unit)] :
          ( time_fails(B,array_nth(B,P2,I),H)
        <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,P2)) ) ) ).

% fails_array_nth
tff(fact_7323_fails__array__swap,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,P2: array(B),H: heap_ext(product_unit)] :
          ( time_fails(B,array_swap(B,I,X,P2),H)
        <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,P2)) ) ) ).

% fails_array_swap
tff(fact_7324_fails__array__upd,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,P2: array(B),H: heap_ext(product_unit)] :
          ( time_fails(array(B),array_upd(B,I,X,P2),H)
        <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,P2)) ) ) ).

% fails_array_upd
tff(fact_7325_execute__upd_I2_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H: heap_ext(product_unit),A2: array(B),I: nat,X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),array_length(B,H,A2)),I)
         => ( aa(heap_ext(product_unit),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(B),array_upd(B,I,X,A2)),H) = none(product_prod(array(B),product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_upd(2)
tff(fact_7326_fails__array__map__entry,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: fun(B,B),P2: array(B),H: heap_ext(product_unit)] :
          ( time_fails(array(B),array_map_entry(B,I,X,P2),H)
        <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,P2)) ) ) ).

% fails_array_map_entry
tff(fact_7327_execute__map__entry_I2_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H: heap_ext(product_unit),A2: array(B),I: nat,F: fun(B,B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),array_length(B,H,A2)),I)
         => ( aa(heap_ext(product_unit),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(B),array_map_entry(B,I,F,A2)),H) = none(product_prod(array(B),product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_map_entry(2)
tff(fact_7328_execute__nth_I2_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H: heap_ext(product_unit),A2: array(B),I: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),array_length(B,H,A2)),I)
         => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,array_nth(B,A2,I)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_nth(2)
tff(fact_7329_execute__swap_I2_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H: heap_ext(product_unit),A2: array(B),I: nat,X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),array_length(B,H,A2)),I)
         => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,array_swap(B,I,X,A2)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_swap(2)
tff(fact_7330_Array__Time_Olength__alloc,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Xs: list(B),H: heap_ext(product_unit),Ys: list(B)] : array_length(B,aa(product_prod(array(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(B),heap_ext(product_unit)),array_alloc(B,Xs,H)),aa(product_prod(array(B),heap_ext(product_unit)),array(B),product_fst(array(B),heap_ext(product_unit)),array_alloc(B,Ys,H))) = aa(list(B),nat,size_size(list(B)),Xs) ) ).

% Array_Time.length_alloc
tff(fact_7331_effect__nthI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,H: heap_ext(product_unit),A2: array(B),H4: heap_ext(product_unit),R: B,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,A2))
         => ( ( H4 = H )
           => ( ( R = aa(nat,B,nth(B,array_get(B,H,A2)),I) )
             => ( ( N = one_one(nat) )
               => heap_Time_effect(B,array_nth(B,A2,I),H,H4,R,N) ) ) ) ) ) ).

% effect_nthI
tff(fact_7332_effect__nthE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),I: nat,H: heap_ext(product_unit),H4: heap_ext(product_unit),R: B,N: nat] :
          ( heap_Time_effect(B,array_nth(B,A2,I),H,H4,R,N)
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,A2))
             => ( ( R = aa(nat,B,nth(B,array_get(B,H,A2)),I) )
               => ( ( H4 = H )
                 => ( N != one_one(nat) ) ) ) ) ) ) ).

% effect_nthE
tff(fact_7333_swap__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,A2: array(B)] : array_swap(B,I,X,A2) = heap_Time_guard(B,aa(array(B),fun(heap_ext(product_unit),$o),aTP_Lamp_akz(nat,fun(array(B),fun(heap_ext(product_unit),$o)),I),A2),aa(array(B),fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(B,fun(array(B),fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ala(nat,fun(B,fun(array(B),fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))))),I),X),A2)) ) ).

% swap_def
tff(fact_7334_map__entry__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,F: fun(B,B),A2: array(B)] : array_map_entry(B,I,F,A2) = heap_Time_guard(array(B),aa(array(B),fun(heap_ext(product_unit),$o),aTP_Lamp_akz(nat,fun(array(B),fun(heap_ext(product_unit),$o)),I),A2),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(fun(B,B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_alb(nat,fun(fun(B,B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))))),I),F),A2)) ) ).

% map_entry_def
tff(fact_7335_effect__updI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,H: heap_ext(product_unit),A2: array(B),H4: heap_ext(product_unit),V2: B,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,A2))
         => ( ( H4 = array_update(B,A2,I,V2,H) )
           => ( ( N = one_one(nat) )
             => heap_Time_effect(array(B),array_upd(B,I,V2,A2),H,H4,A2,N) ) ) ) ) ).

% effect_updI
tff(fact_7336_effect__updE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,V2: B,A2: array(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: array(B),N: nat] :
          ( heap_Time_effect(array(B),array_upd(B,I,V2,A2),H,H4,R,N)
         => ~ ( ( R = A2 )
             => ( ( H4 = array_update(B,A2,I,V2,H) )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,A2))
                 => ( N != one_one(nat) ) ) ) ) ) ) ).

% effect_updE
tff(fact_7337_upd__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,X: B,A2: array(B)] : array_upd(B,I,X,A2) = heap_Time_guard(array(B),aa(array(B),fun(heap_ext(product_unit),$o),aTP_Lamp_akz(nat,fun(array(B),fun(heap_ext(product_unit),$o)),I),A2),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(B,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_alc(nat,fun(B,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))))),I),X),A2)) ) ).

% upd_def
tff(fact_7338_execute__upd_I1_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [I: nat,H: heap_ext(product_unit),A2: array(B),X: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),array_length(B,H,A2))
         => ( aa(heap_ext(product_unit),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(B),array_upd(B,I,X,A2)),H) = aa(product_prod(array(B),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_Pair(array(B),product_prod(heap_ext(product_unit),nat)),A2),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(B,A2,I,X,H)),one_one(nat)))) ) ) ) ).

% execute_upd(1)
tff(fact_7339_freeze__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B)] : array_freeze(B,A2) = heap_Time_heap(list(B),aTP_Lamp_ald(array(B),fun(heap_ext(product_unit),product_prod(list(B),product_prod(heap_ext(product_unit),nat))),A2)) ) ).

% freeze_def
tff(fact_7340_execute__freeze,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(list(B),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(list(B),array_freeze(B,A2)),H) = aa(product_prod(list(B),product_prod(heap_ext(product_unit),nat)),option(product_prod(list(B),product_prod(heap_ext(product_unit),nat))),some(product_prod(list(B),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(list(B),product_prod(heap_ext(product_unit),nat)),aa(list(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(B),product_prod(heap_ext(product_unit),nat))),product_Pair(list(B),product_prod(heap_ext(product_unit),nat)),array_get(B,H,A2)),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(B,H,A2))))) ) ).

% execute_freeze
tff(fact_7341_time__array__freeze,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [P2: array(B),H: heap_ext(product_unit)] : time_time(list(B),array_freeze(B,P2),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),array_length(B,H,P2)),one_one(nat)) ) ).

% time_array_freeze
tff(fact_7342_effect__freezeE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: list(B),N: nat] :
          ( heap_Time_effect(list(B),array_freeze(B,A2),H,H4,R,N)
         => ~ ( ( H4 = H )
             => ( ( R = array_get(B,H,A2) )
               => ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),array_length(B,H,A2)),one_one(nat)) ) ) ) ) ) ).

% effect_freezeE
tff(fact_7343_effect__freezeI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H4: heap_ext(product_unit),H: heap_ext(product_unit),R: list(B),A2: array(B),N: nat] :
          ( ( H4 = H )
         => ( ( R = array_get(B,H,A2) )
           => ( ( N = array_length(B,H,A2) )
             => heap_Time_effect(list(B),array_freeze(B,A2),H,H4,R,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ) ) ).

% effect_freezeI
tff(fact_7344_flat__lub__def,axiom,
    ! [B: $tType,B2: B,A4: set(B)] :
      partial_flat_lub(B,B2,A4) = $ite(aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(B),set(B),insert(B,B2),bot_bot(set(B)))),B2,the(B,aa(set(B),fun(B,$o),aTP_Lamp_ale(B,fun(set(B),fun(B,$o)),B2),A4))) ).

% flat_lub_def
tff(fact_7345_execute__len,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),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(B,A2)),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(B,H,A2)),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_7346_fails__array__len,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [P2: array(B),H: heap_ext(product_unit)] : ~ time_fails(nat,array_len(B,P2),H) ) ).

% fails_array_len
tff(fact_7347_time__array__len,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [P2: array(B),H: heap_ext(product_unit)] : time_time(nat,array_len(B,P2),H) = one_one(nat) ) ).

% time_array_len
tff(fact_7348_len__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B)] : array_len(B,A2) = heap_Time_tap(nat,aTP_Lamp_alf(array(B),fun(heap_ext(product_unit),nat),A2)) ) ).

% len_def
tff(fact_7349_effect__lengthI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H4: heap_ext(product_unit),H: heap_ext(product_unit),R: nat,A2: array(B),N: nat] :
          ( ( H4 = H )
         => ( ( R = array_length(B,H,A2) )
           => ( ( N = one_one(nat) )
             => heap_Time_effect(nat,array_len(B,A2),H,H4,R,N) ) ) ) ) ).

% effect_lengthI
tff(fact_7350_effect__lengthE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),H: heap_ext(product_unit),H4: heap_ext(product_unit),R: nat,N: nat] :
          ( heap_Time_effect(nat,array_len(B,A2),H,H4,R,N)
         => ~ ( ( R = array_length(B,H4,A2) )
             => ( ( H4 = H )
               => ( N != one_one(nat) ) ) ) ) ) ).

% effect_lengthE
tff(fact_7351_len_H__def,axiom,
    ! [C: $tType,B: $tType] :
      ( ( semiring_1(B)
        & heap(C) )
     => ! [A2: array(C)] : array_len2(C,B,A2) = heap_Time_bind(nat,B,array_len(C,A2),aTP_Lamp_alg(nat,heap_Time_Heap(B))) ) ).

% len'_def
tff(fact_7352_minus__coset__filter,axiom,
    ! [B: $tType,A4: set(B),Xs: list(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),coset(B,Xs)) = aa(list(B),set(B),set2(B),filter2(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4),Xs)) ).

% minus_coset_filter
tff(fact_7353_union__coset__filter,axiom,
    ! [B: $tType,Xs: list(B),A4: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),coset(B,Xs)),A4) = coset(B,filter2(B,aTP_Lamp_jr(set(B),fun(B,$o),A4),Xs)) ).

% union_coset_filter
tff(fact_7354_multiset_Odomain,axiom,
    ! [B: $tType,C: $tType,T4: fun(B,fun(C,$o)),X4: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aa(fun(fun(B,nat),fun(multiset(C),$o)),fun(fun(B,nat),$o),domainp(fun(B,nat),multiset(C)),pcr_multiset(B,C,T4)),X4)
    <=> ? [Y5: fun(C,nat)] :
          ( aa(fun(C,nat),$o,aa(fun(B,nat),fun(fun(C,nat),$o),bNF_rel_fun(B,C,nat,nat,T4,fequal(nat)),X4),Y5)
          & aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aTP_Lamp_alh(fun(C,nat),fun(C,$o),Y5))) ) ) ).

% multiset.domain
tff(fact_7355_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( 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 = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = none(num) )
               => ~ 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),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa2 = aa(num,num,bit0,N2) )
                 => ( ( Y = aa(num,option(num),some(num),one2) )
                   => ~ 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),one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa2 = aa(num,num,bit1,N2) )
                   => ( ( Y = none(num) )
                     => ~ 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),one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M2: num] :
                    ( ( X = aa(num,num,bit0,M2) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M2)) )
                       => ~ 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,M2)),one2)) ) ) )
               => ( ! [M2: num] :
                      ( ( X = aa(num,num,bit0,M2) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M2,N2)) )
                           => ~ 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,M2)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M2: num] :
                        ( ( X = aa(num,num,bit0,M2) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M2,N2)) )
                             => ~ 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,M2)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M2: num] :
                          ( ( X = aa(num,num,bit1,M2) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M2)) )
                             => ~ 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,M2)),one2)) ) ) )
                     => ( ! [M2: num] :
                            ( ( X = aa(num,num,bit1,M2) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit0,N2) )
                               => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_jb(num,option(num)),bit_and_not_num(M2,N2)) )
                                 => ~ 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,M2)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M2: num] :
                              ( ( X = aa(num,num,bit1,M2) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N2) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M2,N2)) )
                                   => ~ 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,M2)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_7356_multiset_Odomain__eq,axiom,
    ! [B: $tType,X4: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aa(fun(fun(B,nat),fun(multiset(B),$o)),fun(fun(B,nat),$o),domainp(fun(B,nat),multiset(B)),pcr_multiset(B,B,fequal(B))),X4)
    <=> aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ft(fun(B,nat),fun(B,$o),X4))) ) ).

% multiset.domain_eq
tff(fact_7357_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( 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 = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = aa(num,option(num),some(num),one2) )
               => ~ 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),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa2 = aa(num,num,bit0,N2) )
                 => ( ( Y = none(num) )
                   => ~ 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),one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa2 = aa(num,num,bit1,N2) )
                   => ( ( Y = aa(num,option(num),some(num),one2) )
                     => ~ 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),one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M2: num] :
                    ( ( X = aa(num,num,bit0,M2) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = none(num) )
                       => ~ 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,M2)),one2)) ) ) )
               => ( ! [M2: num] :
                      ( ( X = aa(num,num,bit0,M2) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M2,N2)) )
                           => ~ 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,M2)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M2: num] :
                        ( ( X = aa(num,num,bit0,M2) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M2,N2)) )
                             => ~ 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,M2)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M2: num] :
                          ( ( X = aa(num,num,bit1,M2) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),one2) )
                             => ~ 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,M2)),one2)) ) ) )
                     => ( ! [M2: num] :
                            ( ( X = aa(num,num,bit1,M2) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit0,N2) )
                               => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M2,N2)) )
                                 => ~ 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,M2)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M2: num] :
                              ( ( X = aa(num,num,bit1,M2) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N2) )
                                 => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_jb(num,option(num)),bit_un7362597486090784418nd_num(M2,N2)) )
                                   => ~ 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,M2)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_7358_multiset_Odomain__par__left__total,axiom,
    ! [C: $tType,B: $tType,T4: fun(B,fun(C,$o)),P: fun(fun(B,nat),$o)] :
      ( left_total(fun(B,nat),fun(C,nat),bNF_rel_fun(B,C,nat,nat,T4,fequal(nat)))
     => ( aa(fun(fun(C,nat),$o),$o,aa(fun(fun(B,nat),$o),fun(fun(fun(C,nat),$o),$o),bNF_rel_fun(fun(B,nat),fun(C,nat),$o,$o,bNF_rel_fun(B,C,nat,nat,T4,fequal(nat)),fequal($o)),P),aTP_Lamp_ali(fun(C,nat),$o))
       => ( aa(fun(fun(B,nat),fun(multiset(C),$o)),fun(fun(B,nat),$o),domainp(fun(B,nat),multiset(C)),pcr_multiset(B,C,T4)) = P ) ) ) ).

% multiset.domain_par_left_total
tff(fact_7359_multiset_Odomain__par,axiom,
    ! [C: $tType,B: $tType,T4: fun(B,fun(C,$o)),DT: fun(B,$o),DS: fun(nat,$o),P23: fun(fun(B,nat),$o)] :
      ( ( aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),T4) = DT )
     => ( ( aa(fun(nat,fun(nat,$o)),fun(nat,$o),domainp(nat,nat),fequal(nat)) = DS )
       => ( left_unique(B,C,T4)
         => ( aa(fun(fun(C,nat),$o),$o,aa(fun(fun(B,nat),$o),fun(fun(fun(C,nat),$o),$o),bNF_rel_fun(fun(B,nat),fun(C,nat),$o,$o,bNF_rel_fun(B,C,nat,nat,T4,fequal(nat)),fequal($o)),P23),aTP_Lamp_ali(fun(C,nat),$o))
           => ( aa(fun(fun(B,nat),fun(multiset(C),$o)),fun(fun(B,nat),$o),domainp(fun(B,nat),multiset(C)),pcr_multiset(B,C,T4)) = aa(fun(fun(B,nat),$o),fun(fun(B,nat),$o),aa(fun(fun(B,nat),$o),fun(fun(fun(B,nat),$o),fun(fun(B,nat),$o)),inf_inf(fun(fun(B,nat),$o)),aa(fun(nat,$o),fun(fun(B,nat),$o),basic_pred_fun(B,nat,DT),DS)),P23) ) ) ) ) ) ).

% multiset.domain_par
tff(fact_7360_mono__transfer,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType] :
      ( ( order(C)
        & order(E)
        & order(D)
        & order(B) )
     => ! [A4: fun(B,fun(C,$o)),B4: fun(D,fun(E,$o))] :
          ( bi_total(B,C,A4)
         => ( aa(fun(C,fun(C,$o)),$o,aa(fun(B,fun(B,$o)),fun(fun(C,fun(C,$o)),$o),bNF_rel_fun(B,C,fun(B,$o),fun(C,$o),A4,bNF_rel_fun(B,C,$o,$o,A4,fequal($o))),ord_less_eq(B)),ord_less_eq(C))
           => ( aa(fun(E,fun(E,$o)),$o,aa(fun(D,fun(D,$o)),fun(fun(E,fun(E,$o)),$o),bNF_rel_fun(D,E,fun(D,$o),fun(E,$o),B4,bNF_rel_fun(D,E,$o,$o,B4,fequal($o))),ord_less_eq(D)),ord_less_eq(E))
             => aa(fun(fun(C,E),$o),$o,aa(fun(fun(B,D),$o),fun(fun(fun(C,E),$o),$o),bNF_rel_fun(fun(B,D),fun(C,E),$o,$o,bNF_rel_fun(B,C,D,E,A4,B4),fequal($o)),order_mono(B,D)),order_mono(C,E)) ) ) ) ) ).

% mono_transfer
tff(fact_7361_pred__fun__def,axiom,
    ! [C: $tType,B: $tType,A4: fun(B,$o),B4: fun(C,$o),X4: fun(B,C)] :
      ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,A4),B4),X4)
    <=> ! [Xa4: B] :
          ( aa(B,$o,A4,Xa4)
         => aa(C,$o,B4,aa(B,C,X4,Xa4)) ) ) ).

% pred_fun_def
tff(fact_7362_fun_Opred__mono,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),Paa: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(fun(B,$o),fun(fun(B,$o),$o),ord_less_eq(fun(B,$o)),Pa),Paa)
     => aa(fun(fun(C,B),$o),$o,aa(fun(fun(C,B),$o),fun(fun(fun(C,B),$o),$o),ord_less_eq(fun(fun(C,B),$o)),aa(fun(B,$o),fun(fun(C,B),$o),basic_pred_fun(C,B,aTP_Lamp_alj(C,$o)),Pa)),aa(fun(B,$o),fun(fun(C,B),$o),basic_pred_fun(C,B,aTP_Lamp_alj(C,$o)),Paa)) ) ).

% fun.pred_mono
tff(fact_7363_fun_Opred__True,axiom,
    ! [C: $tType,B: $tType,X4: fun(B,C)] : aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),aTP_Lamp_alj(C,$o)),X4) ).

% fun.pred_True
tff(fact_7364_int_Obi__total,axiom,
    bi_total(product_prod(nat,nat),int,pcr_int) ).

% int.bi_total
tff(fact_7365_typedef__left__unique,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),A4: set(C),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,A4)
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => left_unique(C,B,T4) ) ) ).

% typedef_left_unique
tff(fact_7366_pred__fun__True__id,axiom,
    ! [B: $tType,C: $tType,D: $tType,P2: fun(C,$o),F: fun(D,C)] :
      ( nO_MATCH(fun(B,B),fun(C,$o),id(B),P2)
     => ( aa(fun(D,C),$o,aa(fun(C,$o),fun(fun(D,C),$o),basic_pred_fun(D,C,aTP_Lamp_alk(D,$o)),P2),F)
      <=> aa(fun(D,$o),$o,aa(fun($o,$o),fun(fun(D,$o),$o),basic_pred_fun(D,$o,aTP_Lamp_alk(D,$o)),id($o)),aa(fun(D,C),fun(D,$o),aa(fun(C,$o),fun(fun(D,C),fun(D,$o)),comp(C,$o,D),P2),F)) ) ) ).

% pred_fun_True_id
tff(fact_7367_fun_Omap__cong__pred,axiom,
    ! [D: $tType,C: $tType,B: $tType,X: fun(B,C),Ya: fun(B,C),F: fun(C,D),G: fun(C,D)] :
      ( ( X = Ya )
     => ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),aa(fun(C,D),fun(C,$o),aTP_Lamp_all(fun(C,D),fun(fun(C,D),fun(C,$o)),F),G)),Ya)
       => ( aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),F),X) = aa(fun(B,C),fun(B,D),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),G),Ya) ) ) ) ).

% fun.map_cong_pred
tff(fact_7368_fun_Opred__mono__strong,axiom,
    ! [C: $tType,B: $tType,Pa: fun(C,$o),X: fun(B,C),Paa: fun(C,$o)] :
      ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Pa),X)
     => ( ! [Z2: C] :
            ( aa(set(C),$o,member(C,Z2),aa(set(B),set(C),image2(B,C,X),top_top(set(B))))
           => ( aa(C,$o,Pa,Z2)
             => aa(C,$o,Paa,Z2) ) )
       => aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Paa),X) ) ) ).

% fun.pred_mono_strong
tff(fact_7369_fun_Opred__cong,axiom,
    ! [C: $tType,B: $tType,X: fun(B,C),Ya: fun(B,C),Pa: fun(C,$o),Paa: fun(C,$o)] :
      ( ( X = Ya )
     => ( ! [Z2: C] :
            ( aa(set(C),$o,member(C,Z2),aa(set(B),set(C),image2(B,C,Ya),top_top(set(B))))
           => ( aa(C,$o,Pa,Z2)
            <=> aa(C,$o,Paa,Z2) ) )
       => ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Pa),X)
        <=> aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Paa),Ya) ) ) ) ).

% fun.pred_cong
tff(fact_7370_Domainp__cr__rat,axiom,
    ! [X4: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat),X4)
    <=> ( aa(product_prod(int,int),int,product_snd(int,int),X4) != zero_zero(int) ) ) ).

% Domainp_cr_rat
tff(fact_7371_fun_Opred__rel,axiom,
    ! [C: $tType,B: $tType,Pa: fun(C,$o),X: fun(B,C)] :
      ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Pa),X)
    <=> aa(fun(B,C),$o,aa(fun(B,C),fun(fun(B,C),$o),bNF_rel_fun(B,B,C,C,fequal(B),bNF_eq_onp(C,Pa)),X),X) ) ).

% fun.pred_rel
tff(fact_7372_fun_ODomainp__rel,axiom,
    ! [B: $tType,D: $tType,C: $tType,R2: fun(C,fun(D,$o))] : aa(fun(fun(B,C),fun(fun(B,D),$o)),fun(fun(B,C),$o),domainp(fun(B,C),fun(B,D)),bNF_rel_fun(B,B,C,D,fequal(B),R2)) = aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),aa(fun(C,fun(D,$o)),fun(C,$o),domainp(C,D),R2)) ).

% fun.Domainp_rel
tff(fact_7373_fun_Opred__map,axiom,
    ! [C: $tType,D: $tType,B: $tType,Q: fun(C,$o),F: fun(D,C),X: fun(B,D)] :
      ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Q),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),F),X))
    <=> aa(fun(B,D),$o,aa(fun(D,$o),fun(fun(B,D),$o),basic_pred_fun(B,D,aTP_Lamp_jq(B,$o)),aa(fun(D,C),fun(D,$o),aa(fun(C,$o),fun(fun(D,C),fun(D,$o)),comp(C,$o,D),Q),F)),X) ) ).

% fun.pred_map
tff(fact_7374_fun_Opred__set,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,$o),X4: fun(B,C)] :
      ( aa(fun(B,C),$o,aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Pa),X4)
    <=> ! [Xa4: C] :
          ( aa(set(C),$o,member(C,Xa4),aa(set(B),set(C),image2(B,C,X4),top_top(set(B))))
         => aa(C,$o,Pa,Xa4) ) ) ).

% fun.pred_set
tff(fact_7375_fun_Opred__transfer,axiom,
    ! [B: $tType,C: $tType,D: $tType,R2: fun(B,fun(C,$o))] : aa(fun(fun(C,$o),fun(fun(D,C),$o)),$o,aa(fun(fun(B,$o),fun(fun(D,B),$o)),fun(fun(fun(C,$o),fun(fun(D,C),$o)),$o),bNF_rel_fun(fun(B,$o),fun(C,$o),fun(fun(D,B),$o),fun(fun(D,C),$o),bNF_rel_fun(B,C,$o,$o,R2,fequal($o)),bNF_rel_fun(fun(D,B),fun(D,C),$o,$o,bNF_rel_fun(D,D,B,C,fequal(D),R2),fequal($o))),basic_pred_fun(D,B,aTP_Lamp_alk(D,$o))),basic_pred_fun(D,C,aTP_Lamp_alk(D,$o))) ).

% fun.pred_transfer
tff(fact_7376_fun_Orel__eq__onp,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,$o)] : bNF_rel_fun(B,B,C,C,fequal(B),bNF_eq_onp(C,Pa)) = bNF_eq_onp(fun(B,C),aa(fun(C,$o),fun(fun(B,C),$o),basic_pred_fun(B,C,aTP_Lamp_jq(B,$o)),Pa)) ).

% fun.rel_eq_onp
tff(fact_7377_Inter__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] :
      ( bi_unique(B,C,A4)
     => ( bi_total(B,C,A4)
       => aa(fun(set(set(C)),set(C)),$o,aa(fun(set(set(B)),set(B)),fun(fun(set(set(C)),set(C)),$o),bNF_rel_fun(set(set(B)),set(set(C)),set(B),set(C),bNF_rel_set(set(B),set(C),bNF_rel_set(B,C,A4)),bNF_rel_set(B,C,A4)),complete_Inf_Inf(set(B))),complete_Inf_Inf(set(C))) ) ) ).

% Inter_transfer
tff(fact_7378_eventually__INF__base,axiom,
    ! [C: $tType,B: $tType,B4: set(B),F3: fun(B,filter(C)),Pa: fun(C,$o)] :
      ( ( B4 != bot_bot(set(B)) )
     => ( ! [A3: B] :
            ( aa(set(B),$o,member(B,A3),B4)
           => ! [B3: B] :
                ( aa(set(B),$o,member(B,B3),B4)
               => ? [X4: B] :
                    ( aa(set(B),$o,member(B,X4),B4)
                    & aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),aa(B,filter(C),F3,X4)),aa(filter(C),filter(C),aa(filter(C),fun(filter(C),filter(C)),inf_inf(filter(C)),aa(B,filter(C),F3,A3)),aa(B,filter(C),F3,B3))) ) ) )
       => ( eventually(C,Pa,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),B4)))
        <=> ? [X3: B] :
              ( aa(set(B),$o,member(B,X3),B4)
              & eventually(C,Pa,aa(B,filter(C),F3,X3)) ) ) ) ) ).

% eventually_INF_base
tff(fact_7379_eventually__const,axiom,
    ! [B: $tType,F3: filter(B),Pa: $o] :
      ( ( F3 != bot_bot(filter(B)) )
     => ( eventually(B,aTP_Lamp_vi($o,fun(B,$o),(Pa)),F3)
      <=> (Pa) ) ) ).

% eventually_const
tff(fact_7380_eventually__sequentially__Suc,axiom,
    ! [Pa: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_ahc(fun(nat,$o),fun(nat,$o),Pa),at_top(nat))
    <=> eventually(nat,Pa,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7381_eventually__sequentially__seg,axiom,
    ! [Pa: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_alm(fun(nat,$o),fun(nat,fun(nat,$o)),Pa),K),at_top(nat))
    <=> eventually(nat,Pa,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7382_eventually__inf__principal,axiom,
    ! [B: $tType,Pa: fun(B,$o),F3: filter(B),S2: set(B)] :
      ( eventually(B,Pa,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F3),principal(B,S2)))
    <=> eventually(B,aa(set(B),fun(B,$o),aTP_Lamp_aln(fun(B,$o),fun(set(B),fun(B,$o)),Pa),S2),F3) ) ).

% eventually_inf_principal
tff(fact_7383_eventually__ex,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,fun(C,$o)),F3: filter(B)] :
      ( eventually(B,aTP_Lamp_ut(fun(B,fun(C,$o)),fun(B,$o),Pa),F3)
    <=> ? [Y7: fun(B,C)] : eventually(B,aa(fun(B,C),fun(B,$o),aTP_Lamp_alo(fun(B,fun(C,$o)),fun(fun(B,C),fun(B,$o)),Pa),Y7),F3) ) ).

% eventually_ex
tff(fact_7384_filterlim__at__top__at__top,axiom,
    ! [C: $tType,B: $tType] :
      ( ( linorder(B)
        & linorder(C) )
     => ! [Q: fun(B,$o),F: fun(B,C),Pa: fun(C,$o),G: fun(C,B)] :
          ( ! [X2: B,Y2: B] :
              ( aa(B,$o,Q,X2)
             => ( aa(B,$o,Q,Y2)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y2)) ) ) )
         => ( ! [X2: C] :
                ( aa(C,$o,Pa,X2)
               => ( aa(B,C,F,aa(C,B,G,X2)) = X2 ) )
           => ( ! [X2: C] :
                  ( aa(C,$o,Pa,X2)
                 => aa(B,$o,Q,aa(C,B,G,X2)) )
             => ( eventually(B,Q,at_top(B))
               => ( eventually(C,Pa,at_top(C))
                 => filterlim(B,C,F,at_top(C),at_top(B)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_7385_eventually__ball__finite__distrib,axiom,
    ! [B: $tType,C: $tType,A4: set(B),Pa: fun(C,fun(B,$o)),Net: filter(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( eventually(C,aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_alp(set(B),fun(fun(C,fun(B,$o)),fun(C,$o)),A4),Pa),Net)
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
           => eventually(C,aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Pa),X3),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_7386_eventually__ball__finite,axiom,
    ! [B: $tType,C: $tType,A4: set(B),Pa: fun(C,fun(B,$o)),Net: filter(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A4)
           => eventually(C,aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Pa),X2),Net) )
       => eventually(C,aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_alp(set(B),fun(fun(C,fun(B,$o)),fun(C,$o)),A4),Pa),Net) ) ) ).

% eventually_ball_finite
tff(fact_7387_eventually__INF1,axiom,
    ! [C: $tType,B: $tType,I: B,I5: set(B),Pa: fun(C,$o),F3: fun(B,filter(C))] :
      ( aa(set(B),$o,member(B,I),I5)
     => ( eventually(C,Pa,aa(B,filter(C),F3,I))
       => eventually(C,Pa,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),I5))) ) ) ).

% eventually_INF1
tff(fact_7388_filterlim__mono__eventually,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),F3: filter(C),G5: filter(B),F9: filter(C),G6: filter(B),F2: fun(B,C)] :
      ( filterlim(B,C,F,F3,G5)
     => ( aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),F3),F9)
       => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G6),G5)
         => ( eventually(B,aa(fun(B,C),fun(B,$o),aTP_Lamp_alq(fun(B,C),fun(fun(B,C),fun(B,$o)),F),F2),G6)
           => filterlim(B,C,F2,F9,G6) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_7389_filterlim__iff,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),F23: filter(C),F12: filter(B)] :
      ( filterlim(B,C,F,F23,F12)
    <=> ! [P7: fun(C,$o)] :
          ( eventually(C,P7,F23)
         => eventually(B,aa(fun(C,$o),fun(B,$o),aTP_Lamp_adp(fun(B,C),fun(fun(C,$o),fun(B,$o)),F),P7),F12) ) ) ).

% filterlim_iff
tff(fact_7390_filterlim__cong,axiom,
    ! [B: $tType,C: $tType,F12: filter(B),F13: filter(B),F23: filter(C),F24: filter(C),F: fun(C,B),G: fun(C,B)] :
      ( ( F12 = F13 )
     => ( ( F23 = F24 )
       => ( eventually(C,aa(fun(C,B),fun(C,$o),aTP_Lamp_alr(fun(C,B),fun(fun(C,B),fun(C,$o)),F),G),F23)
         => ( filterlim(C,B,F,F12,F23)
          <=> filterlim(C,B,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_7391_eventually__compose__filterlim,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),F3: filter(B),F: fun(C,B),G5: filter(C)] :
      ( eventually(B,Pa,F3)
     => ( filterlim(C,B,F,F3,G5)
       => eventually(C,aa(fun(C,B),fun(C,$o),aTP_Lamp_als(fun(B,$o),fun(fun(C,B),fun(C,$o)),Pa),F),G5) ) ) ).

% eventually_compose_filterlim
tff(fact_7392_typedef__bi__unique,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),A4: set(C),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,A4)
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => bi_unique(C,B,T4) ) ) ).

% typedef_bi_unique
tff(fact_7393_False__imp__not__eventually,axiom,
    ! [B: $tType,Pa: fun(B,$o),Net: filter(B)] :
      ( ! [X2: B] : ~ aa(B,$o,Pa,X2)
     => ( ( Net != bot_bot(filter(B)) )
       => ~ eventually(B,Pa,Net) ) ) ).

% False_imp_not_eventually
tff(fact_7394_eventually__const__iff,axiom,
    ! [B: $tType,Pa: $o,F3: filter(B)] :
      ( eventually(B,aTP_Lamp_vi($o,fun(B,$o),(Pa)),F3)
    <=> ( (Pa)
        | ( F3 = bot_bot(filter(B)) ) ) ) ).

% eventually_const_iff
tff(fact_7395_trivial__limit__def,axiom,
    ! [B: $tType,F3: filter(B)] :
      ( ( F3 = bot_bot(filter(B)) )
    <=> eventually(B,aTP_Lamp_hh(B,$o),F3) ) ).

% trivial_limit_def
tff(fact_7396_eventually__gt__at__bot,axiom,
    ! [B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [C2: B] : eventually(B,aTP_Lamp_alt(B,fun(B,$o),C2),at_bot(B)) ) ).

% eventually_gt_at_bot
tff(fact_7397_eventually__at__bot__not__equal,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [C2: B] : eventually(B,aTP_Lamp_alu(B,fun(B,$o),C2),at_bot(B)) ) ).

% eventually_at_bot_not_equal
tff(fact_7398_not__eventually__impI,axiom,
    ! [B: $tType,Pa: fun(B,$o),F3: filter(B),Q: fun(B,$o)] :
      ( eventually(B,Pa,F3)
     => ( ~ eventually(B,Q,F3)
       => ~ eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aba(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),F3) ) ) ).

% not_eventually_impI
tff(fact_7399_eventually__conj__iff,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o),F3: filter(B)] :
      ( eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_gd(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),F3)
    <=> ( eventually(B,Pa,F3)
        & eventually(B,Q,F3) ) ) ).

% eventually_conj_iff
tff(fact_7400_eventually__rev__mp,axiom,
    ! [B: $tType,Pa: fun(B,$o),F3: filter(B),Q: fun(B,$o)] :
      ( eventually(B,Pa,F3)
     => ( eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aba(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),F3)
       => eventually(B,Q,F3) ) ) ).

% eventually_rev_mp
tff(fact_7401_eventually__subst,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o),F3: filter(B)] :
      ( eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_alv(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),F3)
     => ( eventually(B,Pa,F3)
      <=> eventually(B,Q,F3) ) ) ).

% eventually_subst
tff(fact_7402_eventually__elim2,axiom,
    ! [B: $tType,Pa: fun(B,$o),F3: filter(B),Q: fun(B,$o),R2: fun(B,$o)] :
      ( eventually(B,Pa,F3)
     => ( eventually(B,Q,F3)
       => ( ! [I2: B] :
              ( aa(B,$o,Pa,I2)
             => ( aa(B,$o,Q,I2)
               => aa(B,$o,R2,I2) ) )
         => eventually(B,R2,F3) ) ) ) ).

% eventually_elim2
tff(fact_7403_eventually__conj,axiom,
    ! [B: $tType,Pa: fun(B,$o),F3: filter(B),Q: fun(B,$o)] :
      ( eventually(B,Pa,F3)
     => ( eventually(B,Q,F3)
       => eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_gd(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),F3) ) ) ).

% eventually_conj
tff(fact_7404_eventually__True,axiom,
    ! [B: $tType,F3: filter(B)] : eventually(B,aTP_Lamp_jq(B,$o),F3) ).

% eventually_True
tff(fact_7405_eventually__mp,axiom,
    ! [B: $tType,Pa: fun(B,$o),Q: fun(B,$o),F3: filter(B)] :
      ( eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aba(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q),F3)
     => ( eventually(B,Pa,F3)
       => eventually(B,Q,F3) ) ) ).

% eventually_mp
tff(fact_7406_eventually__frequently__const__simps_I3_J,axiom,
    ! [B: $tType,Pa: fun(B,$o),C6: $o,F3: filter(B)] :
      ( eventually(B,aa($o,fun(B,$o),aTP_Lamp_alw(fun(B,$o),fun($o,fun(B,$o)),Pa),(C6)),F3)
    <=> ( eventually(B,Pa,F3)
        | (C6) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_7407_eventually__frequently__const__simps_I4_J,axiom,
    ! [B: $tType,C6: $o,Pa: fun(B,$o),F3: filter(B)] :
      ( eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_alx($o,fun(fun(B,$o),fun(B,$o)),(C6)),Pa),F3)
    <=> ( (C6)
        | eventually(B,Pa,F3) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_7408_eventually__frequently__const__simps_I6_J,axiom,
    ! [B: $tType,C6: $o,Pa: fun(B,$o),F3: filter(B)] :
      ( eventually(B,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aly($o,fun(fun(B,$o),fun(B,$o)),(C6)),Pa),F3)
    <=> ( (C6)
       => eventually(B,Pa,F3) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_7409_eventually__le__at__bot,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B] : eventually(B,aa(B,fun(B,$o),aTP_Lamp_kh(B,fun(B,$o)),C2),at_bot(B)) ) ).

% eventually_le_at_bot
tff(fact_7410_eventually__at__bot__linorder,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o)] :
          ( eventually(B,Pa,at_bot(B))
        <=> ? [N11: B] :
            ! [N4: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N4),N11)
             => aa(B,$o,Pa,N4) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_7411_eventually__all__finite,axiom,
    ! [B: $tType,C: $tType] :
      ( finite_finite(B)
     => ! [Pa: fun(C,fun(B,$o)),Net: filter(C)] :
          ( ! [Y2: B] : eventually(C,aa(B,fun(C,$o),aTP_Lamp_alz(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Pa),Y2),Net)
         => eventually(C,aTP_Lamp_ama(fun(C,fun(B,$o)),fun(C,$o),Pa),Net) ) ) ).

% eventually_all_finite
tff(fact_7412_eventually__sequentially,axiom,
    ! [Pa: fun(nat,$o)] :
      ( eventually(nat,Pa,at_top(nat))
    <=> ? [N11: nat] :
        ! [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),N4)
         => aa(nat,$o,Pa,N4) ) ) ).

% eventually_sequentially
tff(fact_7413_eventually__sequentiallyI,axiom,
    ! [C2: nat,Pa: fun(nat,$o)] :
      ( ! [X2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X2)
         => aa(nat,$o,Pa,X2) )
     => eventually(nat,Pa,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_7414_eventually__at__top__linorder,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o)] :
          ( eventually(B,Pa,at_top(B))
        <=> ? [N11: B] :
            ! [N4: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N11),N4)
             => aa(B,$o,Pa,N4) ) ) ) ).

% eventually_at_top_linorder
tff(fact_7415_eventually__at__top__linorderI,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B,Pa: fun(B,$o)] :
          ( ! [X2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),X2)
             => aa(B,$o,Pa,X2) )
         => eventually(B,Pa,at_top(B)) ) ) ).

% eventually_at_top_linorderI
tff(fact_7416_eventually__ge__at__top,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [C2: B] : eventually(B,aa(B,fun(B,$o),ord_less_eq(B),C2),at_top(B)) ) ).

% eventually_ge_at_top
tff(fact_7417_le__sequentially,axiom,
    ! [F3: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F3),at_top(nat))
    <=> ! [N11: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),F3) ) ).

% le_sequentially
tff(fact_7418_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_fl(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_7419_eventually__at__top__not__equal,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [C2: B] : eventually(B,aTP_Lamp_amb(B,fun(B,$o),C2),at_top(B)) ) ).

% eventually_at_top_not_equal
tff(fact_7420_eventually__gt__at__top,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [C2: B] : eventually(B,aa(B,fun(B,$o),ord_less(B),C2),at_top(B)) ) ).

% eventually_gt_at_top
tff(fact_7421_le__principal,axiom,
    ! [B: $tType,F3: filter(B),A4: set(B)] :
      ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F3),principal(B,A4))
    <=> eventually(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4),F3) ) ).

% le_principal
tff(fact_7422_filterlim__principal,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),S: set(C),F3: filter(B)] :
      ( filterlim(B,C,F,principal(C,S),F3)
    <=> eventually(B,aa(set(C),fun(B,$o),aTP_Lamp_ads(fun(B,C),fun(set(C),fun(B,$o)),F),S),F3) ) ).

% filterlim_principal
tff(fact_7423_eventually__all__ge__at__top,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Pa: fun(B,$o)] :
          ( eventually(B,Pa,at_top(B))
         => eventually(B,aTP_Lamp_amc(fun(B,$o),fun(B,$o),Pa),at_top(B)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7424_filterlim__at__top,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),F3: filter(B)] :
          ( filterlim(B,C,F,at_top(C),F3)
        <=> ! [Z8: C] : eventually(B,aa(C,fun(B,$o),aTP_Lamp_amd(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ).

% filterlim_at_top
tff(fact_7425_filterlim__at__top__ge,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),F3: filter(B),C2: C] :
          ( filterlim(B,C,F,at_top(C),F3)
        <=> ! [Z8: C] :
              ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),C2),Z8)
             => eventually(B,aa(C,fun(B,$o),aTP_Lamp_amd(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_7426_filterlim__at__top__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),F3: filter(B),G: fun(B,C)] :
          ( filterlim(B,C,F,at_top(C),F3)
         => ( eventually(B,aa(fun(B,C),fun(B,$o),aTP_Lamp_ame(fun(B,C),fun(fun(B,C),fun(B,$o)),F),G),F3)
           => filterlim(B,C,G,at_top(C),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_7427_filterlim__at__top__dense,axiom,
    ! [C: $tType,B: $tType] :
      ( unboun7993243217541854897norder(C)
     => ! [F: fun(B,C),F3: filter(B)] :
          ( filterlim(B,C,F,at_top(C),F3)
        <=> ! [Z8: C] : eventually(B,aa(C,fun(B,$o),aTP_Lamp_amf(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_7428_eventually__INF__finite,axiom,
    ! [B: $tType,C: $tType,A4: set(B),Pa: fun(C,$o),F3: fun(B,filter(C))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( eventually(C,Pa,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),A4)))
      <=> ? [Q7: fun(B,fun(C,$o))] :
            ( ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),A4)
               => eventually(C,aa(B,fun(C,$o),Q7,X3),aa(B,filter(C),F3,X3)) )
            & ! [Y5: C] :
                ( ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),A4)
                   => aa(C,$o,aa(B,fun(C,$o),Q7,X3),Y5) )
               => aa(C,$o,Pa,Y5) ) ) ) ) ).

% eventually_INF_finite
tff(fact_7429_filterlim__at__bot,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),F3: filter(B)] :
          ( filterlim(B,C,F,at_bot(C),F3)
        <=> ! [Z8: C] : eventually(B,aa(C,fun(B,$o),aTP_Lamp_amg(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ).

% filterlim_at_bot
tff(fact_7430_filterlim__at__bot__le,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [F: fun(B,C),F3: filter(B),C2: C] :
          ( filterlim(B,C,F,at_bot(C),F3)
        <=> ! [Z8: C] :
              ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Z8),C2)
             => eventually(B,aa(C,fun(B,$o),aTP_Lamp_amg(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7431_filterlim__at__bot__dense,axiom,
    ! [C: $tType,B: $tType] :
      ( ( dense_linorder(C)
        & no_bot(C) )
     => ! [F: fun(B,C),F3: filter(B)] :
          ( filterlim(B,C,F,at_bot(C),F3)
        <=> ! [Z8: C] : eventually(B,aa(C,fun(B,$o),aTP_Lamp_amh(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_7432_filterlim__at__top__gt,axiom,
    ! [C: $tType,B: $tType] :
      ( unboun7993243217541854897norder(C)
     => ! [F: fun(B,C),F3: filter(B),C2: C] :
          ( filterlim(B,C,F,at_top(C),F3)
        <=> ! [Z8: C] :
              ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),C2),Z8)
             => eventually(B,aa(C,fun(B,$o),aTP_Lamp_ami(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7433_eventually__INF,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),F3: fun(C,filter(B)),B4: set(C)] :
      ( eventually(B,Pa,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F3),B4)))
    <=> ? [X7: set(C)] :
          ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),X7),B4)
          & aa(set(C),$o,finite_finite2(C),X7)
          & eventually(B,Pa,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F3),X7))) ) ) ).

% eventually_INF
tff(fact_7434_filterlim__at__bot__lt,axiom,
    ! [C: $tType,B: $tType] :
      ( unboun7993243217541854897norder(C)
     => ! [F: fun(B,C),F3: filter(B),C2: C] :
          ( filterlim(B,C,F,at_bot(C),F3)
        <=> ! [Z8: C] :
              ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),Z8),C2)
             => eventually(B,aa(C,fun(B,$o),aTP_Lamp_amj(fun(B,C),fun(C,fun(B,$o)),F),Z8),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7435_filterlim__finite__subsets__at__top,axiom,
    ! [C: $tType,B: $tType,F: fun(B,set(C)),A4: set(C),F3: filter(B)] :
      ( filterlim(B,set(C),F,finite5375528669736107172at_top(C,A4),F3)
    <=> ! [X7: set(C)] :
          ( ( aa(set(C),$o,finite_finite2(C),X7)
            & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),X7),A4) )
         => eventually(B,aa(set(C),fun(B,$o),aa(set(C),fun(set(C),fun(B,$o)),aTP_Lamp_amk(fun(B,set(C)),fun(set(C),fun(set(C),fun(B,$o))),F),A4),X7),F3) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_7436_map__filter__on__comp,axiom,
    ! [B: $tType,D: $tType,C: $tType,G: fun(C,B),Y4: set(C),X5: set(B),F3: filter(C),F: fun(B,D)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(C),set(B),image2(C,B,G),Y4)),X5)
     => ( eventually(C,aTP_Lamp_al(set(C),fun(C,$o),Y4),F3)
       => ( map_filter_on(B,D,X5,F,map_filter_on(C,B,Y4,G,F3)) = map_filter_on(C,D,Y4,aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),F),G),F3) ) ) ) ).

% map_filter_on_comp
tff(fact_7437_right__total__Inter__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] :
      ( bi_unique(B,C,A4)
     => ( right_total(B,C,A4)
       => aa(fun(set(set(C)),set(C)),$o,aa(fun(set(set(B)),set(B)),fun(fun(set(set(C)),set(C)),$o),bNF_rel_fun(set(set(B)),set(set(C)),set(B),set(C),bNF_rel_set(set(B),set(C),bNF_rel_set(B,C,A4)),bNF_rel_set(B,C,A4)),aTP_Lamp_aml(fun(B,fun(C,$o)),fun(set(set(B)),set(B)),A4)),complete_Inf_Inf(set(C))) ) ) ).

% right_total_Inter_transfer
tff(fact_7438_int_Oright__total,axiom,
    right_total(product_prod(nat,nat),int,pcr_int) ).

% int.right_total
tff(fact_7439_typedef__right__total,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),A4: set(C),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,A4)
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => right_total(C,B,T4) ) ) ).

% typedef_right_total
tff(fact_7440_eventually__map__filter__on,axiom,
    ! [C: $tType,B: $tType,X5: set(B),F3: filter(B),Pa: fun(C,$o),F: fun(B,C)] :
      ( eventually(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),X5),F3)
     => ( eventually(C,Pa,map_filter_on(B,C,X5,F,F3))
      <=> eventually(B,aa(fun(B,C),fun(B,$o),aa(fun(C,$o),fun(fun(B,C),fun(B,$o)),aTP_Lamp_amm(set(B),fun(fun(C,$o),fun(fun(B,C),fun(B,$o))),X5),Pa),F),F3) ) ) ).

% eventually_map_filter_on
tff(fact_7441_right__total__Collect__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] :
      ( right_total(B,C,A4)
     => aa(fun(fun(C,$o),set(C)),$o,aa(fun(fun(B,$o),set(B)),fun(fun(fun(C,$o),set(C)),$o),bNF_rel_fun(fun(B,$o),fun(C,$o),set(B),set(C),bNF_rel_fun(B,C,$o,$o,A4,fequal($o)),bNF_rel_set(B,C,A4)),aTP_Lamp_amo(fun(B,fun(C,$o)),fun(fun(B,$o),set(B)),A4)),collect(C)) ) ).

% right_total_Collect_transfer
tff(fact_7442_right__total__Compl__transfer,axiom,
    ! [B: $tType,C: $tType,A4: fun(B,fun(C,$o))] :
      ( bi_unique(B,C,A4)
     => ( right_total(B,C,A4)
       => aa(fun(set(C),set(C)),$o,aa(fun(set(B),set(B)),fun(fun(set(C),set(C)),$o),bNF_rel_fun(set(B),set(C),set(B),set(C),bNF_rel_set(B,C,A4),bNF_rel_set(B,C,A4)),aTP_Lamp_amp(fun(B,fun(C,$o)),fun(set(B),set(B)),A4)),uminus_uminus(set(C))) ) ) ).

% right_total_Compl_transfer
tff(fact_7443_right__total__fun__eq__transfer,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,A4: fun(B,fun(C,$o)),B4: fun(D,fun(E,$o))] :
      ( right_total(B,C,A4)
     => ( bi_unique(D,E,B4)
       => aa(fun(fun(C,E),fun(fun(C,E),$o)),$o,aa(fun(fun(B,D),fun(fun(B,D),$o)),fun(fun(fun(C,E),fun(fun(C,E),$o)),$o),bNF_rel_fun(fun(B,D),fun(C,E),fun(fun(B,D),$o),fun(fun(C,E),$o),bNF_rel_fun(B,C,D,E,A4,B4),bNF_rel_fun(fun(B,D),fun(C,E),$o,$o,bNF_rel_fun(B,C,D,E,A4,B4),fequal($o))),aTP_Lamp_amq(fun(B,fun(C,$o)),fun(fun(B,D),fun(fun(B,D),$o)),A4)),fequal(fun(C,E))) ) ) ).

% right_total_fun_eq_transfer
tff(fact_7444_vimage__right__total__transfer,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,B4: fun(B,fun(C,$o)),A4: fun(D,fun(E,$o))] :
      ( bi_unique(B,C,B4)
     => ( right_total(D,E,A4)
       => aa(fun(fun(E,C),fun(set(C),set(E))),$o,aa(fun(fun(D,B),fun(set(B),set(D))),fun(fun(fun(E,C),fun(set(C),set(E))),$o),bNF_rel_fun(fun(D,B),fun(E,C),fun(set(B),set(D)),fun(set(C),set(E)),bNF_rel_fun(D,E,B,C,A4,B4),bNF_rel_fun(set(B),set(C),set(D),set(E),bNF_rel_set(B,C,B4),bNF_rel_set(D,E,A4))),aTP_Lamp_amr(fun(D,fun(E,$o)),fun(fun(D,B),fun(set(B),set(D))),A4)),vimage(E,C)) ) ) ).

% vimage_right_total_transfer
tff(fact_7445_Sup__filter__def,axiom,
    ! [B: $tType,S: set(filter(B))] : aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),S) = abs_filter(B,aTP_Lamp_ams(set(filter(B)),fun(fun(B,$o),$o),S)) ).

% Sup_filter_def
tff(fact_7446_sorted__insort__insert__key,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [F: fun(C,B),Xs: list(C),X: C] :
          ( sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),Xs))
         => sorted_wrt(B,ord_less_eq(B),aa(list(C),list(B),map(C,B,F),linord329482645794927042rt_key(C,B,F,X,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_7447_sup__filter__def,axiom,
    ! [B: $tType,F3: filter(B),F9: filter(B)] : aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F3),F9) = abs_filter(B,aa(filter(B),fun(fun(B,$o),$o),aTP_Lamp_amt(filter(B),fun(filter(B),fun(fun(B,$o),$o)),F3),F9)) ).

% sup_filter_def
tff(fact_7448_bot__filter__def,axiom,
    ! [B: $tType] : bot_bot(filter(B)) = abs_filter(B,aTP_Lamp_amu(fun(B,$o),$o)) ).

% bot_filter_def
tff(fact_7449_top__filter__def,axiom,
    ! [B: $tType] : top_top(filter(B)) = abs_filter(B,fAll(B)) ).

% top_filter_def
tff(fact_7450_insort__insert__triv,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Xs: list(B)] :
          ( aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
         => ( linord329482645794927042rt_key(B,B,aTP_Lamp_hr(B,B),X,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_7451_principal__def,axiom,
    ! [B: $tType,S: set(B)] : principal(B,S) = abs_filter(B,aa(set(B),fun(fun(B,$o),$o),ball(B),S)) ).

% principal_def
tff(fact_7452_map__filter__on__def,axiom,
    ! [B: $tType,C: $tType,X5: set(C),F: fun(C,B),F3: filter(C)] : map_filter_on(C,B,X5,F,F3) = abs_filter(B,aa(filter(C),fun(fun(B,$o),$o),aa(fun(C,B),fun(filter(C),fun(fun(B,$o),$o)),aTP_Lamp_amw(set(C),fun(fun(C,B),fun(filter(C),fun(fun(B,$o),$o))),X5),F),F3)) ).

% map_filter_on_def
tff(fact_7453_set__insort__insert,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Xs: list(B)] : aa(list(B),set(B),set2(B),linord329482645794927042rt_key(B,B,aTP_Lamp_hr(B,B),X,Xs)) = aa(set(B),set(B),insert(B,X),aa(list(B),set(B),set2(B),Xs)) ) ).

% set_insort_insert
tff(fact_7454_sorted__insort__insert,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B),X: B] :
          ( sorted_wrt(B,ord_less_eq(B),Xs)
         => sorted_wrt(B,ord_less_eq(B),linord329482645794927042rt_key(B,B,aTP_Lamp_hr(B,B),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_7455_insort__insert__insort,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [X: B,Xs: list(B)] :
          ( ~ aa(set(B),$o,member(B,X),aa(list(B),set(B),set2(B),Xs))
         => ( linord329482645794927042rt_key(B,B,aTP_Lamp_hr(B,B),X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),X),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_7456_inf__filter__def,axiom,
    ! [B: $tType,F3: filter(B),F9: filter(B)] : aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F3),F9) = abs_filter(B,aa(filter(B),fun(fun(B,$o),$o),aTP_Lamp_amx(filter(B),fun(filter(B),fun(fun(B,$o),$o)),F3),F9)) ).

% inf_filter_def
tff(fact_7457_subset__mset_OcINF__inf__distrib,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,multiset(C)),G: fun(B,multiset(C))] :
      ( ( A4 != bot_bot(set(B)) )
     => ( condit8119078960628432327_below(multiset(C),subseteq_mset(C),aa(set(B),set(multiset(C)),image2(B,multiset(C),F),A4))
       => ( condit8119078960628432327_below(multiset(C),subseteq_mset(C),aa(set(B),set(multiset(C)),image2(B,multiset(C),G),A4))
         => ( aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),inter_mset(C),aa(set(multiset(C)),multiset(C),complete_Inf_Inf(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),F),A4))),aa(set(multiset(C)),multiset(C),complete_Inf_Inf(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),G),A4))) = aa(set(multiset(C)),multiset(C),complete_Inf_Inf(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),aa(fun(B,multiset(C)),fun(B,multiset(C)),aTP_Lamp_amy(fun(B,multiset(C)),fun(fun(B,multiset(C)),fun(B,multiset(C))),F),G)),A4)) ) ) ) ) ).

% subset_mset.cINF_inf_distrib
tff(fact_7458_admissible__chfin,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [Pa: fun(B,$o)] :
          ( ! [S6: set(B)] :
              ( comple1602240252501008431_chain(B,ord_less_eq(B),S6)
             => aa(set(B),$o,finite_finite2(B),S6) )
         => comple1908693960933563346ssible(B,complete_Sup_Sup(B),ord_less_eq(B),Pa) ) ) ).

% admissible_chfin
tff(fact_7459_subset__mset_Obdd__below__image__inf,axiom,
    ! [B: $tType,C: $tType,F: fun(C,multiset(B)),G: fun(C,multiset(B)),A4: set(C)] :
      ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),aa(fun(C,multiset(B)),fun(C,multiset(B)),aTP_Lamp_amz(fun(C,multiset(B)),fun(fun(C,multiset(B)),fun(C,multiset(B))),F),G)),A4))
    <=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))
        & condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),A4)) ) ) ).

% subset_mset.bdd_below_image_inf
tff(fact_7460_admissible__True,axiom,
    ! [B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,$o))] : comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_jq(B,$o)) ).

% admissible_True
tff(fact_7461_admissible__conj,axiom,
    ! [B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,$o)),Pa: fun(B,$o),Q: fun(B,$o)] :
      ( comple1908693960933563346ssible(B,Lub,Ord,Pa)
     => ( comple1908693960933563346ssible(B,Lub,Ord,Q)
       => comple1908693960933563346ssible(B,Lub,Ord,aa(fun(B,$o),fun(B,$o),aTP_Lamp_gd(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)) ) ) ).

% admissible_conj
tff(fact_7462_admissible__const,axiom,
    ! [B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,$o)),Ta: $o] : comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_vi($o,fun(B,$o),(Ta))) ).

% admissible_const
tff(fact_7463_admissible__all,axiom,
    ! [B: $tType,C: $tType,Lub: fun(set(C),C),Ord: fun(C,fun(C,$o)),Pa: fun(C,fun(B,$o))] :
      ( ! [Y2: B] : comple1908693960933563346ssible(C,Lub,Ord,aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Pa),Y2))
     => comple1908693960933563346ssible(C,Lub,Ord,aTP_Lamp_ana(fun(C,fun(B,$o)),fun(C,$o),Pa)) ) ).

% admissible_all
tff(fact_7464_inter__subset__eq__union,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)) ).

% inter_subset_eq_union
tff(fact_7465_inter__union__distrib__left,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4)),C6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),C6)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6)) ).

% inter_union_distrib_left
tff(fact_7466_inter__union__distrib__right,axiom,
    ! [B: $tType,C6: multiset(B),A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),A4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),B4)) ).

% inter_union_distrib_right
tff(fact_7467_inter__mset__empty__distrib__right,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6)) = zero_zero(multiset(B)) )
    <=> ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4) = zero_zero(multiset(B)) )
        & ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),C6) = zero_zero(multiset(B)) ) ) ) ).

% inter_mset_empty_distrib_right
tff(fact_7468_inter__mset__empty__distrib__left,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),C6) = zero_zero(multiset(B)) )
    <=> ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),C6) = zero_zero(multiset(B)) )
        & ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),B4),C6) = zero_zero(multiset(B)) ) ) ) ).

% inter_mset_empty_distrib_left
tff(fact_7469_admissible__ball,axiom,
    ! [B: $tType,C: $tType,A4: set(B),Lub: fun(set(C),C),Ord: fun(C,fun(C,$o)),Pa: fun(C,fun(B,$o))] :
      ( ! [Y2: B] :
          ( aa(set(B),$o,member(B,Y2),A4)
         => comple1908693960933563346ssible(C,Lub,Ord,aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Pa),Y2)) )
     => comple1908693960933563346ssible(C,Lub,Ord,aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_alp(set(B),fun(fun(C,fun(B,$o)),fun(C,$o)),A4),Pa)) ) ).

% admissible_ball
tff(fact_7470_multiset__union__diff__commute,axiom,
    ! [B: $tType,B4: multiset(B),C6: multiset(B),A4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),B4),C6) = zero_zero(multiset(B)) )
     => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),C6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),A4),C6)),B4) ) ) ).

% multiset_union_diff_commute
tff(fact_7471_size__Diff__subset__Int,axiom,
    ! [B: $tType,M: multiset(B),M7: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),M7)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(B),nat,size_size(multiset(B)),M)),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),M),M7))) ).

% size_Diff_subset_Int
tff(fact_7472_admissible__disj,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [Pa: fun(B,$o),Q: fun(B,$o)] :
          ( comple1908693960933563346ssible(B,complete_Sup_Sup(B),ord_less_eq(B),Pa)
         => ( comple1908693960933563346ssible(B,complete_Sup_Sup(B),ord_less_eq(B),Q)
           => comple1908693960933563346ssible(B,complete_Sup_Sup(B),ord_less_eq(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_anb(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Pa),Q)) ) ) ) ).

% admissible_disj
tff(fact_7473_subset__mset_Omono__inf,axiom,
    ! [C: $tType,B: $tType] :
      ( semilattice_inf(C)
     => ! [F: fun(multiset(B),C),A4: multiset(B),B4: multiset(B)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(multiset(B),C,F,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4))),aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(multiset(B),C,F,A4)),aa(multiset(B),C,F,B4))) ) ) ).

% subset_mset.mono_inf
tff(fact_7474_filtercomap__def,axiom,
    ! [B: $tType,C: $tType,F: fun(B,C),F3: filter(C)] : filtercomap(B,C,F,F3) = abs_filter(B,aa(filter(C),fun(fun(B,$o),$o),aTP_Lamp_anc(fun(B,C),fun(filter(C),fun(fun(B,$o),$o)),F),F3)) ).

% filtercomap_def
tff(fact_7475_multp__def,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),M: multiset(B),N5: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),M),N5)
    <=> aa(set(product_prod(multiset(B),multiset(B))),$o,member(product_prod(multiset(B),multiset(B)),aa(multiset(B),product_prod(multiset(B),multiset(B)),aa(multiset(B),fun(multiset(B),product_prod(multiset(B),multiset(B))),product_Pair(multiset(B),multiset(B)),M),N5)),mult(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),R)))) ) ).

% multp_def
tff(fact_7476_eventually__filtercomapI,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),F3: filter(B),F: fun(C,B)] :
      ( eventually(B,Pa,F3)
     => eventually(C,aa(fun(C,B),fun(C,$o),aTP_Lamp_als(fun(B,$o),fun(fun(C,B),fun(C,$o)),Pa),F),filtercomap(C,B,F,F3)) ) ).

% eventually_filtercomapI
tff(fact_7477_filtercomap__filtercomap,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(B,C),G: fun(C,D),F3: filter(D)] : filtercomap(B,C,F,filtercomap(C,D,G,F3)) = filtercomap(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_and(fun(B,C),fun(fun(C,D),fun(B,D)),F),G),F3) ).

% filtercomap_filtercomap
tff(fact_7478_filtercomap__ident,axiom,
    ! [B: $tType,F3: filter(B)] : filtercomap(B,B,aTP_Lamp_il(B,B),F3) = F3 ).

% filtercomap_ident
tff(fact_7479_filtercomap__INF,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(B,C),F3: fun(D,filter(C)),B4: set(D)] : filtercomap(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),F3),B4))) = aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(D),set(filter(B)),image2(D,filter(B),aa(fun(D,filter(C)),fun(D,filter(B)),aTP_Lamp_ane(fun(B,C),fun(fun(D,filter(C)),fun(D,filter(B))),F),F3)),B4)) ).

% filtercomap_INF
tff(fact_7480_eventually__filtercomap__at__top__linorder,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Pa: fun(B,$o),F: fun(B,C)] :
          ( eventually(B,Pa,filtercomap(B,C,F,at_top(C)))
        <=> ? [N11: C] :
            ! [X3: B] :
              ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),N11),aa(B,C,F,X3))
             => aa(B,$o,Pa,X3) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_7481_eventually__filtercomap__at__bot__linorder,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Pa: fun(B,$o),F: fun(B,C)] :
          ( eventually(B,Pa,filtercomap(B,C,F,at_bot(C)))
        <=> ? [N11: C] :
            ! [X3: B] :
              ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X3)),N11)
             => aa(B,$o,Pa,X3) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_7482_one__step__implies__multp,axiom,
    ! [B: $tType,J4: multiset(B),K3: multiset(B),R: fun(B,fun(B,$o)),I5: multiset(B)] :
      ( ( J4 != zero_zero(multiset(B)) )
     => ( ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),aa(multiset(B),set(B),set_mset(B),K3))
           => ? [Xa: B] :
                ( aa(set(B),$o,member(B,Xa),aa(multiset(B),set(B),set_mset(B),J4))
                & aa(B,$o,aa(B,fun(B,$o),R,X2),Xa) ) )
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I5),K3)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I5),J4)) ) ) ).

% one_step_implies_multp
tff(fact_7483_filtercomap__SUP,axiom,
    ! [B: $tType,D: $tType,C: $tType,F: fun(B,D),F3: fun(C,filter(D)),B4: set(C)] : aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),aa(fun(C,filter(D)),fun(C,filter(B)),aTP_Lamp_anf(fun(B,D),fun(fun(C,filter(D)),fun(C,filter(B))),F),F3)),B4))),filtercomap(B,D,F,aa(set(filter(D)),filter(D),complete_Sup_Sup(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),F3),B4)))) ).

% filtercomap_SUP
tff(fact_7484_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),Ta)
       => aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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),A2),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta))),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_7485_subset__mset_OSUP__sup__distrib,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,multiset(C)),G: fun(B,multiset(C))] :
      ( ( A4 != bot_bot(set(B)) )
     => ( condit8047198070973881523_above(multiset(C),subseteq_mset(C),aa(set(B),set(multiset(C)),image2(B,multiset(C),F),A4))
       => ( condit8047198070973881523_above(multiset(C),subseteq_mset(C),aa(set(B),set(multiset(C)),image2(B,multiset(C),G),A4))
         => ( aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),union_mset(C),aa(set(multiset(C)),multiset(C),complete_Sup_Sup(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),F),A4))),aa(set(multiset(C)),multiset(C),complete_Sup_Sup(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),G),A4))) = aa(set(multiset(C)),multiset(C),complete_Sup_Sup(multiset(C)),aa(set(B),set(multiset(C)),image2(B,multiset(C),aa(fun(B,multiset(C)),fun(B,multiset(C)),aTP_Lamp_ang(fun(B,multiset(C)),fun(fun(B,multiset(C)),fun(B,multiset(C))),F),G)),A4)) ) ) ) ) ).

% subset_mset.SUP_sup_distrib
tff(fact_7486_subset__mset_Obdd__above__image__sup,axiom,
    ! [B: $tType,C: $tType,F: fun(C,multiset(B)),G: fun(C,multiset(B)),A4: set(C)] :
      ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),aa(fun(C,multiset(B)),fun(C,multiset(B)),aTP_Lamp_anh(fun(C,multiset(B)),fun(fun(C,multiset(B)),fun(C,multiset(B))),F),G)),A4))
    <=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))
        & condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),A4)) ) ) ).

% subset_mset.bdd_above_image_sup
tff(fact_7487_size__Un__Int,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(B),nat,size_size(multiset(B)),A4)),aa(multiset(B),nat,size_size(multiset(B)),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4))),aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4))) ).

% size_Un_Int
tff(fact_7488_union__mset__def,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),B4),A4)) ).

% union_mset_def
tff(fact_7489_sup__union__distrib__left,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B),C6: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4)),C6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),C6)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B4),C6)) ).

% sup_union_distrib_left
tff(fact_7490_union__sup__distrib__right,axiom,
    ! [B: $tType,C6: multiset(B),A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),A4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C6),B4)) ).

% union_sup_distrib_right
tff(fact_7491_union__diff__inter__eq__sup,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4) ).

% union_diff_inter_eq_sup
tff(fact_7492_union__diff__sup__eq__inter,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] : aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A4),B4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4)) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4) ).

% union_diff_sup_eq_inter
tff(fact_7493_size__Un__disjoint,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] :
      ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4) = zero_zero(multiset(B)) )
     => ( aa(multiset(B),nat,size_size(multiset(B)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(B),nat,size_size(multiset(B)),A4)),aa(multiset(B),nat,size_size(multiset(B)),B4)) ) ) ).

% size_Un_disjoint
tff(fact_7494_subset__mset_Omono__sup,axiom,
    ! [C: $tType,B: $tType] :
      ( semilattice_sup(C)
     => ! [F: fun(multiset(B),C),A4: multiset(B),B4: multiset(B)] :
          ( aa(fun(multiset(B),C),$o,mono(multiset(B),C,subseteq_mset(B)),F)
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(multiset(B),C,F,A4)),aa(multiset(B),C,F,B4))),aa(multiset(B),C,F,aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4))) ) ) ).

% subset_mset.mono_sup
tff(fact_7495_sum__mset_Ounion__disjoint,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: multiset(B),B4: multiset(B),G: fun(B,C)] :
          ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A4),B4) = zero_zero(multiset(B)) )
         => ( comm_m7189776963980413722m_mset(C,aa(multiset(B),multiset(C),image_mset(B,C,G),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A4),B4))) = aa(C,C,aa(C,fun(C,C),plus_plus(C),comm_m7189776963980413722m_mset(C,aa(multiset(B),multiset(C),image_mset(B,C,G),A4))),comm_m7189776963980413722m_mset(C,aa(multiset(B),multiset(C),image_mset(B,C,G),B4))) ) ) ) ).

% sum_mset.union_disjoint
tff(fact_7496_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S2),Ta)
       => aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),$o,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),A2),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta))),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_7497_euclidean__size__times__nonunit,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),B2)),aa(B,nat,euclid6346220572633701492n_size(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7498_euclidean__size__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,nat,euclid6346220572633701492n_size(code_integer),X) = aa(int,nat,euclid6346220572633701492n_size(int),aa(code_integer,int,code_int_of_integer,X)) ).

% euclidean_size_integer.rep_eq
tff(fact_7499_euclidean__size__of__nat,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat] : aa(B,nat,euclid6346220572633701492n_size(B),aa(nat,B,semiring_1_of_nat(B),N)) = N ) ).

% euclidean_size_of_nat
tff(fact_7500_size__0,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ( aa(B,nat,euclid6346220572633701492n_size(B),zero_zero(B)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_7501_euclidean__size__eq__0__iff,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B] :
          ( ( aa(B,nat,euclid6346220572633701492n_size(B),B2) = zero_zero(nat) )
        <=> ( B2 = zero_zero(B) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_7502_euclidean__size__1,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ( aa(B,nat,euclid6346220572633701492n_size(B),one_one(B)) = one_one(nat) ) ) ).

% euclidean_size_1
tff(fact_7503_euclidean__size__greater__0__iff,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,euclid6346220572633701492n_size(B),B2))
        <=> ( B2 != zero_zero(B) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7504_euclidean__size__integer__def,axiom,
    euclid6346220572633701492n_size(code_integer) = aa(fun(int,nat),fun(code_integer,nat),map_fun(code_integer,int,nat,nat,code_int_of_integer,id(nat)),euclid6346220572633701492n_size(int)) ).

% euclidean_size_integer_def
tff(fact_7505_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( aa(B,nat,euclid6346220572633701492n_size(B),A2) = aa(B,nat,euclid6346220572633701492n_size(B),B2) )
           => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_7506_euclidean__size__unit,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,nat,euclid6346220572633701492n_size(B),A2) = aa(B,nat,euclid6346220572633701492n_size(B),one_one(B)) ) ) ) ).

% euclidean_size_unit
tff(fact_7507_euclidean__size__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,nat,euclid6346220572633701492n_size(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,nat,euclid6346220572633701492n_size(int),X) ).

% euclidean_size_integer.abs_eq
tff(fact_7508_euclidean__size__natural_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),euclid6346220572633701492n_size(nat)),euclid6346220572633701492n_size(nat)) ).

% euclidean_size_natural.rsp
tff(fact_7509_euclidean__size__integer_Orsp,axiom,
    aa(fun(int,nat),$o,aa(fun(int,nat),fun(fun(int,nat),$o),bNF_rel_fun(int,int,nat,nat,fequal(int),fequal(nat)),euclid6346220572633701492n_size(int)),euclid6346220572633701492n_size(int)) ).

% euclidean_size_integer.rsp
tff(fact_7510_unit__iff__euclidean__size,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
        <=> ( ( aa(B,nat,euclid6346220572633701492n_size(B),A2) = aa(B,nat,euclid6346220572633701492n_size(B),one_one(B)) )
            & ( A2 != zero_zero(B) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_7511_size__mult__mono,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,euclid6346220572633701492n_size(B),A2)),aa(B,nat,euclid6346220572633701492n_size(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))) ) ) ).

% size_mult_mono
tff(fact_7512_size__mult__mono_H,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,euclid6346220572633701492n_size(B),A2)),aa(B,nat,euclid6346220572633701492n_size(B),aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2))) ) ) ).

% size_mult_mono'
tff(fact_7513_euclidean__size__times__unit,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( aa(B,nat,euclid6346220572633701492n_size(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,nat,euclid6346220572633701492n_size(B),B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_7514_dvd__proper__imp__size__less,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2)
           => ( ( B2 != zero_zero(B) )
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),A2)),aa(B,nat,euclid6346220572633701492n_size(B),B2)) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7515_dvd__imp__size__le,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [A2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => ( ( B2 != zero_zero(B) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,euclid6346220572633701492n_size(B),A2)),aa(B,nat,euclid6346220572633701492n_size(B),B2)) ) ) ) ).

% dvd_imp_size_le
tff(fact_7516_mod__size__less,axiom,
    ! [B: $tType] :
      ( euclid3725896446679973847miring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2))),aa(B,nat,euclid6346220572633701492n_size(B),B2)) ) ) ).

% mod_size_less
tff(fact_7517_divmod__cases,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [B2: B,A2: B] :
          ( ( ( B2 != zero_zero(B) )
           => ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = zero_zero(B) )
             => ( A2 != aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(B) )
             => ! [Q4: B,R5: B] :
                  ( ( aa(B,B,euclid7384307370059645450egment(B),R5) = aa(B,B,euclid7384307370059645450egment(B),B2) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),R5)),aa(B,nat,euclid6346220572633701492n_size(B),B2))
                   => ( ( R5 != zero_zero(B) )
                     => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = Q4 )
                       => ( ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = R5 )
                         => ( A2 != aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Q4),B2)),R5) ) ) ) ) ) ) )
           => ( B2 = zero_zero(B) ) ) ) ) ).

% divmod_cases
tff(fact_7518_mod__eqI,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [B2: B,R: B,Q2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ( aa(B,B,euclid7384307370059645450egment(B),R) = aa(B,B,euclid7384307370059645450egment(B),B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),R)),aa(B,nat,euclid6346220572633701492n_size(B),B2))
             => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Q2),B2)),R) = A2 )
               => ( aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2) = R ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7519_division__segment__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,euclid7384307370059645450egment(code_integer),X)) = aa(int,int,euclid7384307370059645450egment(int),aa(code_integer,int,code_int_of_integer,X)) ).

% division_segment_integer.rep_eq
tff(fact_7520_abs__division__segment,axiom,
    ! [K: int] : aa(int,int,abs_abs(int),aa(int,int,euclid7384307370059645450egment(int),K)) = one_one(int) ).

% abs_division_segment
tff(fact_7521_division__segment__1,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ( aa(B,B,euclid7384307370059645450egment(B),one_one(B)) = one_one(B) ) ) ).

% division_segment_1
tff(fact_7522_division__segment__numeral,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [K: num] : aa(B,B,euclid7384307370059645450egment(B),aa(num,B,numeral_numeral(B),K)) = one_one(B) ) ).

% division_segment_numeral
tff(fact_7523_division__segment__of__nat,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [N: nat] : aa(B,B,euclid7384307370059645450egment(B),aa(nat,B,semiring_1_of_nat(B),N)) = one_one(B) ) ).

% division_segment_of_nat
tff(fact_7524_division__segment__euclidean__size,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,euclid7384307370059645450egment(B),A2)),aa(nat,B,semiring_1_of_nat(B),aa(B,nat,euclid6346220572633701492n_size(B),A2))) = A2 ) ).

% division_segment_euclidean_size
tff(fact_7525_division__segment__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),euclid7384307370059645450egment(int)),euclid7384307370059645450egment(int)) ).

% division_segment_integer.rsp
tff(fact_7526_division__segment__natural_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),euclid7384307370059645450egment(nat)),euclid7384307370059645450egment(nat)) ).

% division_segment_natural.rsp
tff(fact_7527_is__unit__division__segment,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [A2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,euclid7384307370059645450egment(B),A2)),one_one(B)) ) ).

% is_unit_division_segment
tff(fact_7528_division__segment__nat__def,axiom,
    ! [N: nat] : aa(nat,nat,euclid7384307370059645450egment(nat),N) = one_one(nat) ).

% division_segment_nat_def
tff(fact_7529_division__segment__not__0,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [A2: B] : aa(B,B,euclid7384307370059645450egment(B),A2) != zero_zero(B) ) ).

% division_segment_not_0
tff(fact_7530_division__segment__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,euclid7384307370059645450egment(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,euclid7384307370059645450egment(int),X)) ).

% division_segment_integer.abs_eq
tff(fact_7531_division__segment__mult,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [A2: B,B2: B] :
          ( ( A2 != zero_zero(B) )
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,euclid7384307370059645450egment(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,euclid7384307370059645450egment(B),A2)),aa(B,B,euclid7384307370059645450egment(B),B2)) ) ) ) ) ).

% division_segment_mult
tff(fact_7532_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,int,euclid7384307370059645450egment(int),K) = aa(int,int,sgn_sgn(int),K) ) ) ).

% division_segment_eq_sgn
tff(fact_7533_division__segment__integer__def,axiom,
    euclid7384307370059645450egment(code_integer) = 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),euclid7384307370059645450egment(int)) ).

% division_segment_integer_def
tff(fact_7534_division__segment__mod,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [B2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),A2)
           => ( aa(B,B,euclid7384307370059645450egment(B),aa(B,B,aa(B,fun(B,B),modulo_modulo(B),A2),B2)) = aa(B,B,euclid7384307370059645450egment(B),B2) ) ) ) ) ).

% division_segment_mod
tff(fact_7535_of__nat__euclidean__size,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [A2: B] : aa(nat,B,semiring_1_of_nat(B),aa(B,nat,euclid6346220572633701492n_size(B),A2)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),aa(B,B,euclid7384307370059645450egment(B),A2)) ) ).

% of_nat_euclidean_size
tff(fact_7536_division__segment__int__def,axiom,
    ! [K: int] :
      aa(int,int,euclid7384307370059645450egment(int),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ).

% division_segment_int_def
tff(fact_7537_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [A2: B,B2: B] :
          ( ( aa(B,B,euclid7384307370059645450egment(B),A2) = aa(B,B,euclid7384307370059645450egment(B),B2) )
         => ( ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = zero_zero(B) )
          <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),A2)),aa(B,nat,euclid6346220572633701492n_size(B),B2))
              | ( B2 = zero_zero(B) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7538_div__eqI,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [B2: B,R: B,Q2: B,A2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ( aa(B,B,euclid7384307370059645450egment(B),R) = aa(B,B,euclid7384307370059645450egment(B),B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),R)),aa(B,nat,euclid6346220572633701492n_size(B),B2))
             => ( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Q2),B2)),R) = A2 )
               => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2) = Q2 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7539_div__bounded,axiom,
    ! [B: $tType] :
      ( euclid3128863361964157862miring(B)
     => ! [B2: B,R: B,Q2: B] :
          ( ( B2 != zero_zero(B) )
         => ( ( aa(B,B,euclid7384307370059645450egment(B),R) = aa(B,B,euclid7384307370059645450egment(B),B2) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,euclid6346220572633701492n_size(B),R)),aa(B,nat,euclid6346220572633701492n_size(B),B2))
             => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Q2),B2)),R)),B2) = Q2 ) ) ) ) ) ).

% div_bounded
tff(fact_7540_ran__add,axiom,
    ! [C: $tType,B: $tType,F: fun(B,option(C)),G: fun(B,option(C))] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,C,F)),dom(B,C,G)) = bot_bot(set(B)) )
     => ( ran(B,C,map_add(B,C,F,G)) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),ran(B,C,F)),ran(B,C,G)) ) ) ).

% ran_add
tff(fact_7541_coinduct3__lemma,axiom,
    ! [B: $tType,X5: set(B),F: fun(set(B),set(B))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),aa(set(B),set(B),F,complete_lattice_lfp(set(B),aa(fun(set(B),set(B)),fun(set(B),set(B)),aTP_Lamp_ani(set(B),fun(fun(set(B),set(B)),fun(set(B),set(B))),X5),F))))
     => ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),complete_lattice_lfp(set(B),aa(fun(set(B),set(B)),fun(set(B),set(B)),aTP_Lamp_ani(set(B),fun(fun(set(B),set(B)),fun(set(B),set(B))),X5),F))),aa(set(B),set(B),F,complete_lattice_lfp(set(B),aa(fun(set(B),set(B)),fun(set(B),set(B)),aTP_Lamp_ani(set(B),fun(fun(set(B),set(B)),fun(set(B),set(B))),X5),F)))) ) ) ).

% coinduct3_lemma
tff(fact_7542_restrict__map__self,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : restrict_map(B,C,Ma,dom(B,C,Ma)) = Ma ).

% restrict_map_self
tff(fact_7543_dom__empty,axiom,
    ! [C: $tType,B: $tType] : dom(B,C,aTP_Lamp_jv(B,option(C))) = bot_bot(set(B)) ).

% dom_empty
tff(fact_7544_map__update__eta__repair_I1_J,axiom,
    ! [C: $tType,B: $tType,K: B,V2: C,Ma: fun(B,option(C))] : dom(B,C,aa(fun(B,option(C)),fun(B,option(C)),aa(C,fun(fun(B,option(C)),fun(B,option(C))),aTP_Lamp_anj(B,fun(C,fun(fun(B,option(C)),fun(B,option(C)))),K),V2),Ma)) = aa(set(B),set(B),insert(B,K),dom(B,C,Ma)) ).

% map_update_eta_repair(1)
tff(fact_7545_dom__const_H,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C)] : dom(B,C,aTP_Lamp_ank(fun(B,C),fun(B,option(C)),F)) = top_top(set(B)) ).

% dom_const'
tff(fact_7546_restrict__map__inv,axiom,
    ! [B: $tType,C: $tType,F: fun(B,option(C)),X4: B] : aa(B,option(C),restrict_map(B,C,F,aa(set(B),set(B),uminus_uminus(set(B)),dom(B,C,F))),X4) = none(C) ).

% restrict_map_inv
tff(fact_7547_dom__map__of__zip,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ys: list(C)] :
      ( ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) )
     => ( dom(B,C,map_of(B,C,zip(B,C,Xs,Ys))) = aa(list(B),set(B),set2(B),Xs) ) ) ).

% dom_map_of_zip
tff(fact_7548_dom__map__upds,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),Xs: list(B),Ys: list(C)] : dom(B,C,map_upds(B,C,Ma,Xs,Ys)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),take(B,aa(list(C),nat,size_size(list(C)),Ys),Xs))),dom(B,C,Ma)) ).

% dom_map_upds
tff(fact_7549_set__to__map__dom,axiom,
    ! [C: $tType,B: $tType,S: set(product_prod(B,C))] : dom(B,C,set_to_map(B,C,S)) = aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),S) ).

% set_to_map_dom
tff(fact_7550_def__Collect__coinduct,axiom,
    ! [B: $tType,A4: set(B),Pa: fun(set(B),fun(B,$o)),A2: B,X5: set(B)] :
      ( ( A4 = complete_lattice_gfp(set(B),aTP_Lamp_anl(fun(set(B),fun(B,$o)),fun(set(B),set(B)),Pa)) )
     => ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),aTP_Lamp_anl(fun(set(B),fun(B,$o)),fun(set(B),set(B)),Pa))
       => ( aa(set(B),$o,member(B,A2),X5)
         => ( ! [Z2: B] :
                ( aa(set(B),$o,member(B,Z2),X5)
               => aa(B,$o,aa(set(B),fun(B,$o),Pa,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),X5),A4)),Z2) )
           => aa(set(B),$o,member(B,A2),A4) ) ) ) ) ).

% def_Collect_coinduct
tff(fact_7551_gfp__fun__UnI2,axiom,
    ! [B: $tType,F: fun(set(B),set(B)),A2: B,X5: set(B)] :
      ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
     => ( aa(set(B),$o,member(B,A2),complete_lattice_gfp(set(B),F))
       => aa(set(B),$o,member(B,A2),aa(set(B),set(B),F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),X5),complete_lattice_gfp(set(B),F)))) ) ) ).

% gfp_fun_UnI2
tff(fact_7552_dom__if,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),F: fun(B,option(C)),G: fun(B,option(C))] : dom(B,C,aa(fun(B,option(C)),fun(B,option(C)),aa(fun(B,option(C)),fun(fun(B,option(C)),fun(B,option(C))),aTP_Lamp_anm(fun(B,$o),fun(fun(B,option(C)),fun(fun(B,option(C)),fun(B,option(C)))),Pa),F),G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,C,F)),aa(fun(B,$o),set(B),collect(B),Pa))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,C,G)),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ae(fun(B,$o),fun(B,$o),Pa)))) ).

% dom_if
tff(fact_7553_weak__coinduct__image,axiom,
    ! [B: $tType,C: $tType,A2: B,X5: set(B),G: fun(B,C),F: fun(set(C),set(C))] :
      ( aa(set(B),$o,member(B,A2),X5)
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G),X5)),aa(set(C),set(C),F,aa(set(B),set(C),image2(B,C,G),X5)))
       => aa(set(C),$o,member(C,aa(B,C,G,A2)),complete_lattice_gfp(set(C),F)) ) ) ).

% weak_coinduct_image
tff(fact_7554_gfp__def,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] : complete_lattice_gfp(B,F) = aa(set(B),B,complete_Sup_Sup(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ann(fun(B,B),fun(B,$o),F))) ) ).

% gfp_def
tff(fact_7555_le__map__dom__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( preorder(C)
     => ! [Ma: fun(B,option(C)),M3: fun(B,option(C))] :
          ( aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),Ma),M3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),dom(B,C,Ma)),dom(B,C,M3)) ) ) ).

% le_map_dom_mono
tff(fact_7556_gfp__gfp,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,fun(B,B))] :
          ( ! [X2: B,Y2: B,W2: B,Z2: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),W2),Z2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),F,X2),W2)),aa(B,B,aa(B,fun(B,B),F,Y2),Z2)) ) )
         => ( complete_lattice_gfp(B,aTP_Lamp_ano(fun(B,fun(B,B)),fun(B,B),F)) = complete_lattice_gfp(B,aTP_Lamp_aed(fun(B,fun(B,B)),fun(B,B),F)) ) ) ) ).

% gfp_gfp
tff(fact_7557_gfp__least,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),X5: B] :
          ( ! [U3: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U3),aa(B,B,F,U3))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U3),X5) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_gfp(B,F)),X5) ) ) ).

% gfp_least
tff(fact_7558_gfp__upperbound,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X5: B,F: fun(B,B)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),aa(B,B,F,X5))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),complete_lattice_gfp(B,F)) ) ) ).

% gfp_upperbound
tff(fact_7559_gfp__mono,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),G: fun(B,B)] :
          ( ! [Z6: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,F,Z6)),aa(B,B,G,Z6))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_gfp(B,F)),complete_lattice_gfp(B,G)) ) ) ).

% gfp_mono
tff(fact_7560_weak__coinduct,axiom,
    ! [B: $tType,A2: B,X5: set(B),F: fun(set(B),set(B))] :
      ( aa(set(B),$o,member(B,A2),X5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),aa(set(B),set(B),F,X5))
       => aa(set(B),$o,member(B,A2),complete_lattice_gfp(set(B),F)) ) ) ).

% weak_coinduct
tff(fact_7561_gfp__eqI,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(B,B),X: B] :
          ( aa(fun(B,B),$o,order_mono(B,B),F3)
         => ( ( aa(B,B,F3,X) = X )
           => ( ! [Z2: B] :
                  ( ( aa(B,B,F3,Z2) = Z2 )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z2),X) )
             => ( complete_lattice_gfp(B,F3) = X ) ) ) ) ) ).

% gfp_eqI
tff(fact_7562_finite__map__to__set,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] :
      ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),map_to_set(B,C,Ma))
    <=> aa(set(B),$o,finite_finite2(B),dom(B,C,Ma)) ) ).

% finite_map_to_set
tff(fact_7563_map__to__set__dom,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : dom(B,C,Ma) = aa(set(product_prod(B,C)),set(B),image2(product_prod(B,C),B,product_fst(B,C)),map_to_set(B,C,Ma)) ).

% map_to_set_dom
tff(fact_7564_map__add__left__comm,axiom,
    ! [C: $tType,B: $tType,A4: fun(B,option(C)),B4: fun(B,option(C)),C6: fun(B,option(C))] :
      ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,C,A4)),dom(B,C,B4)) = bot_bot(set(B)) )
     => ( map_add(B,C,A4,map_add(B,C,B4,C6)) = map_add(B,C,B4,map_add(B,C,A4,C6)) ) ) ).

% map_add_left_comm
tff(fact_7565_restrict__map__eq_I1_J,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B)),A4: set(C),K: C] :
      ( ( aa(C,option(B),restrict_map(C,B,Ma,A4),K) = none(B) )
    <=> ~ aa(set(C),$o,member(C,K),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),dom(C,B,Ma)),A4)) ) ).

% restrict_map_eq(1)
tff(fact_7566_dom__def,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : dom(B,C,Ma) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_anp(fun(B,option(C)),fun(B,$o),Ma)) ).

% dom_def
tff(fact_7567_nempty__dom,axiom,
    ! [C: $tType,B: $tType,E2: fun(B,option(C))] :
      ( ~ ! [X4: B] : aa(B,option(C),E2,X4) = none(C)
     => ~ ! [M2: B] : ~ aa(set(B),$o,member(B,M2),dom(B,C,E2)) ) ).

% nempty_dom
tff(fact_7568_dom__map__option,axiom,
    ! [C: $tType,D: $tType,B: $tType,F: fun(B,fun(D,C)),Ma: fun(B,option(D))] : dom(B,C,aa(fun(B,option(D)),fun(B,option(C)),aTP_Lamp_anq(fun(B,fun(D,C)),fun(fun(B,option(D)),fun(B,option(C))),F),Ma)) = dom(B,D,Ma) ).

% dom_map_option
tff(fact_7569_gfp__const,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ta: B] : complete_lattice_gfp(B,aTP_Lamp_adz(B,fun(B,B),Ta)) = Ta ) ).

% gfp_const
tff(fact_7570_map__dom__ran__finite,axiom,
    ! [C: $tType,B: $tType,M: fun(B,option(C))] :
      ( aa(set(B),$o,finite_finite2(B),dom(B,C,M))
     => aa(set(C),$o,finite_finite2(C),ran(B,C,M)) ) ).

% map_dom_ran_finite
tff(fact_7571_map__card__eq__iff,axiom,
    ! [C: $tType,B: $tType,M: fun(B,option(C)),X: B,Y: B] :
      ( aa(set(B),$o,finite_finite2(B),dom(B,C,M))
     => ( ( aa(set(B),nat,finite_card(B),dom(B,C,M)) = aa(set(C),nat,finite_card(C),ran(B,C,M)) )
       => ( aa(set(B),$o,member(B,X),dom(B,C,M))
         => ( ( aa(B,option(C),M,X) = aa(B,option(C),M,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% map_card_eq_iff
tff(fact_7572_def__gfp__unfold,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: B,F: fun(B,B)] :
          ( ( A4 = complete_lattice_gfp(B,F) )
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => ( A4 = aa(B,B,F,A4) ) ) ) ) ).

% def_gfp_unfold
tff(fact_7573_gfp__fixpoint,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,B,F,complete_lattice_gfp(B,F)) = complete_lattice_gfp(B,F) ) ) ) ).

% gfp_fixpoint
tff(fact_7574_gfp__unfold,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( complete_lattice_gfp(B,F) = aa(B,B,F,complete_lattice_gfp(B,F)) ) ) ) ).

% gfp_unfold
tff(fact_7575_gfp__rolling,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(C)
        & comple6319245703460814977attice(B) )
     => ! [G: fun(B,C),F: fun(C,B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),G)
         => ( aa(fun(C,B),$o,order_mono(C,B),F)
           => ( aa(B,C,G,complete_lattice_gfp(B,aa(fun(C,B),fun(B,B),aTP_Lamp_aea(fun(B,C),fun(fun(C,B),fun(B,B)),G),F))) = complete_lattice_gfp(C,aa(fun(C,B),fun(C,C),aTP_Lamp_aeb(fun(B,C),fun(fun(C,B),fun(C,C)),G),F)) ) ) ) ) ).

% gfp_rolling
tff(fact_7576_finite__set__of__finite__maps,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),B4)
       => aa(set(fun(B,option(C))),$o,finite_finite2(fun(B,option(C))),aa(fun(fun(B,option(C)),$o),set(fun(B,option(C))),collect(fun(B,option(C))),aa(set(C),fun(fun(B,option(C)),$o),aTP_Lamp_anr(set(B),fun(set(C),fun(fun(B,option(C)),$o)),A4),B4))) ) ) ).

% finite_set_of_finite_maps
tff(fact_7577_card__map__to__set,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : aa(set(product_prod(B,C)),nat,finite_card(product_prod(B,C)),map_to_set(B,C,Ma)) = aa(set(B),nat,finite_card(B),dom(B,C,Ma)) ).

% card_map_to_set
tff(fact_7578_coinduct__set,axiom,
    ! [B: $tType,F: fun(set(B),set(B)),A2: B,X5: set(B)] :
      ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
     => ( aa(set(B),$o,member(B,A2),X5)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),aa(set(B),set(B),F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),X5),complete_lattice_gfp(set(B),F))))
         => aa(set(B),$o,member(B,A2),complete_lattice_gfp(set(B),F)) ) ) ) ).

% coinduct_set
tff(fact_7579_def__coinduct__set,axiom,
    ! [B: $tType,A4: set(B),F: fun(set(B),set(B)),A2: B,X5: set(B)] :
      ( ( A4 = complete_lattice_gfp(set(B),F) )
     => ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
       => ( aa(set(B),$o,member(B,A2),X5)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),aa(set(B),set(B),F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),X5),A4)))
           => aa(set(B),$o,member(B,A2),A4) ) ) ) ) ).

% def_coinduct_set
tff(fact_7580_coinduct,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),X5: B] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),aa(B,B,F,aa(B,B,aa(B,fun(B,B),sup_sup(B),X5),complete_lattice_gfp(B,F))))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),complete_lattice_gfp(B,F)) ) ) ) ).

% coinduct
tff(fact_7581_def__coinduct,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: B,F: fun(B,B),X5: B] :
          ( ( A4 = complete_lattice_gfp(B,F) )
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),aa(B,B,F,aa(B,B,aa(B,fun(B,B),sup_sup(B),X5),A4)))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),A4) ) ) ) ) ).

% def_coinduct
tff(fact_7582_coinduct__lemma,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [X5: B,F: fun(B,B)] :
          ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X5),aa(B,B,F,aa(B,B,aa(B,fun(B,B),sup_sup(B),X5),complete_lattice_gfp(B,F))))
         => ( aa(fun(B,B),$o,order_mono(B,B),F)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),X5),complete_lattice_gfp(B,F))),aa(B,B,F,aa(B,B,aa(B,fun(B,B),sup_sup(B),X5),complete_lattice_gfp(B,F)))) ) ) ) ).

% coinduct_lemma
tff(fact_7583_gfp__ordinal__induct,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),Pa: fun(B,$o)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( ! [S6: B] :
                ( aa(B,$o,Pa,S6)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_gfp(B,F)),S6)
                 => aa(B,$o,Pa,aa(B,B,F,S6)) ) )
           => ( ! [M8: set(B)] :
                  ( ! [X4: B] :
                      ( aa(set(B),$o,member(B,X4),M8)
                     => aa(B,$o,Pa,X4) )
                 => aa(B,$o,Pa,aa(set(B),B,complete_Inf_Inf(B),M8)) )
             => aa(B,$o,Pa,complete_lattice_gfp(B,F)) ) ) ) ) ).

% gfp_ordinal_induct
tff(fact_7584_gfp__funpow,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),N: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( complete_lattice_gfp(B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,suc,N)),F)) = complete_lattice_gfp(B,F) ) ) ) ).

% gfp_funpow
tff(fact_7585_inj__on__map__the,axiom,
    ! [C: $tType,B: $tType,D3: set(B),Ma: fun(B,option(C))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),D3),dom(B,C,Ma))
     => ( inj_on(B,option(C),Ma,D3)
       => inj_on(B,C,aa(fun(B,option(C)),fun(B,C),aa(fun(option(C),C),fun(fun(B,option(C)),fun(B,C)),comp(option(C),C,B),the2(C)),Ma),D3) ) ) ).

% inj_on_map_the
tff(fact_7586_lfp__le__gfp,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B)] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),complete_lattice_lfp(B,F)),complete_lattice_gfp(B,F)) ) ) ).

% lfp_le_gfp
tff(fact_7587_finite__Map__induct,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),Pa: fun(fun(B,option(C)),$o)] :
      ( aa(set(B),$o,finite_finite2(B),dom(B,C,Ma))
     => ( aa(fun(B,option(C)),$o,Pa,aTP_Lamp_jv(B,option(C)))
       => ( ! [K4: B,V4: C,M2: fun(B,option(C))] :
              ( aa(set(B),$o,finite_finite2(B),dom(B,C,M2))
             => ( ~ aa(set(B),$o,member(B,K4),dom(B,C,M2))
               => ( aa(fun(B,option(C)),$o,Pa,M2)
                 => aa(fun(B,option(C)),$o,Pa,fun_upd(B,option(C),M2,K4,aa(C,option(C),some(C),V4))) ) ) )
         => aa(fun(B,option(C)),$o,Pa,Ma) ) ) ) ).

% finite_Map_induct
tff(fact_7588_ran__is__image,axiom,
    ! [B: $tType,C: $tType,M: fun(C,option(B))] : ran(C,B,M) = aa(set(C),set(B),image2(C,B,aa(fun(C,option(B)),fun(C,B),aa(fun(option(B),B),fun(fun(C,option(B)),fun(C,B)),comp(option(B),B,C),the2(B)),M)),dom(C,B,M)) ).

% ran_is_image
tff(fact_7589_map__add__distinct__le,axiom,
    ! [C: $tType,B: $tType] :
      ( preorder(C)
     => ! [Ma: fun(B,option(C)),M3: fun(B,option(C)),N: fun(B,option(C)),N3: fun(B,option(C))] :
          ( aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),Ma),M3)
         => ( aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),N),N3)
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,C,M3)),dom(B,C,N3)) = bot_bot(set(B)) )
             => aa(fun(B,option(C)),$o,aa(fun(B,option(C)),fun(fun(B,option(C)),$o),ord_less_eq(fun(B,option(C))),map_add(B,C,Ma,N)),map_add(B,C,M3,N3)) ) ) ) ) ).

% map_add_distinct_le
tff(fact_7590_dom__eq__singleton__conv,axiom,
    ! [B: $tType,C: $tType,F: fun(B,option(C)),X: B] :
      ( ( dom(B,C,F) = aa(set(B),set(B),insert(B,X),bot_bot(set(B))) )
    <=> ? [V5: C] : F = fun_upd(B,option(C),aTP_Lamp_jv(B,option(C)),X,aa(C,option(C),some(C),V5)) ) ).

% dom_eq_singleton_conv
tff(fact_7591_gfp__Kleene__iter,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F: fun(B,B),K: nat] :
          ( aa(fun(B,B),$o,order_mono(B,B),F)
         => ( ( aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(nat,nat,suc,K)),F),top_top(B)) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),K),F),top_top(B)) )
           => ( complete_lattice_gfp(B,F) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),K),F),top_top(B)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_7592_map__of__map__keys,axiom,
    ! [C: $tType,B: $tType,Xs: list(B),Ma: fun(B,option(C))] :
      ( ( aa(list(B),set(B),set2(B),Xs) = dom(B,C,Ma) )
     => ( map_of(B,C,aa(list(B),list(product_prod(B,C)),map(B,product_prod(B,C),aTP_Lamp_ans(fun(B,option(C)),fun(B,product_prod(B,C)),Ma)),Xs)) = Ma ) ) ).

% map_of_map_keys
tff(fact_7593_coinduct3,axiom,
    ! [B: $tType,F: fun(set(B),set(B)),A2: B,X5: set(B)] :
      ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
     => ( aa(set(B),$o,member(B,A2),X5)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),aa(set(B),set(B),F,complete_lattice_lfp(set(B),aa(set(B),fun(set(B),set(B)),aTP_Lamp_ant(fun(set(B),set(B)),fun(set(B),fun(set(B),set(B))),F),X5))))
         => aa(set(B),$o,member(B,A2),complete_lattice_gfp(set(B),F)) ) ) ) ).

% coinduct3
tff(fact_7594_def__coinduct3,axiom,
    ! [B: $tType,A4: set(B),F: fun(set(B),set(B)),A2: B,X5: set(B)] :
      ( ( A4 = complete_lattice_gfp(set(B),F) )
     => ( aa(fun(set(B),set(B)),$o,order_mono(set(B),set(B)),F)
       => ( aa(set(B),$o,member(B,A2),X5)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X5),aa(set(B),set(B),F,complete_lattice_lfp(set(B),aa(set(B),fun(set(B),set(B)),aa(fun(set(B),set(B)),fun(set(B),fun(set(B),set(B))),aTP_Lamp_anu(set(B),fun(fun(set(B),set(B)),fun(set(B),fun(set(B),set(B)))),A4),F),X5))))
           => aa(set(B),$o,member(B,A2),A4) ) ) ) ) ).

% def_coinduct3
tff(fact_7595_dom__override__on,axiom,
    ! [C: $tType,B: $tType,F: fun(B,option(C)),G: fun(B,option(C)),A4: set(B)] : dom(B,C,override_on(B,option(C),F,G,A4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,C,F)),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_anv(fun(B,option(C)),fun(set(B),fun(B,$o)),G),A4)))),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_anw(fun(B,option(C)),fun(set(B),fun(B,$o)),G),A4))) ).

% dom_override_on
tff(fact_7596_eq__f__restr__conv,axiom,
    ! [C: $tType,B: $tType,S2: set(B),F: fun(fun(B,option(C)),fun(B,option(C))),A4: fun(B,option(C))] :
      ( ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S2),dom(B,C,aa(fun(B,option(C)),fun(B,option(C)),F,A4)))
        & ( A4 = restrict_map(B,C,aa(fun(B,option(C)),fun(B,option(C)),F,A4),aa(set(B),set(B),uminus_uminus(set(B)),S2)) ) )
    <=> ( map_le(B,C,A4,aa(fun(B,option(C)),fun(B,option(C)),F,A4))
        & ( S2 = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,C,aa(fun(B,option(C)),fun(B,option(C)),F,A4))),dom(B,C,A4)) ) ) ) ).

% eq_f_restr_conv
tff(fact_7597_map__le__empty,axiom,
    ! [C: $tType,B: $tType,G: fun(B,option(C))] : map_le(B,C,aTP_Lamp_jv(B,option(C)),G) ).

% map_le_empty
tff(fact_7598_map__leD,axiom,
    ! [B: $tType,C: $tType,M1: fun(B,option(C)),M22: fun(B,option(C)),K: B,V2: C] :
      ( map_le(B,C,M1,M22)
     => ( ( aa(B,option(C),M1,K) = aa(C,option(C),some(C),V2) )
       => ( aa(B,option(C),M22,K) = aa(C,option(C),some(C),V2) ) ) ) ).

% map_leD
tff(fact_7599_map__leI,axiom,
    ! [C: $tType,B: $tType,M1: fun(B,option(C)),M22: fun(B,option(C))] :
      ( ! [X2: B,V4: C] :
          ( ( aa(B,option(C),M1,X2) = aa(C,option(C),some(C),V4) )
         => ( aa(B,option(C),M22,X2) = aa(C,option(C),some(C),V4) ) )
     => map_le(B,C,M1,M22) ) ).

% map_leI
tff(fact_7600_eq__f__restr__ss__eq,axiom,
    ! [C: $tType,B: $tType,S2: set(B),F: fun(fun(B,option(C)),fun(B,option(C))),A4: fun(B,option(C))] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),S2),dom(B,C,aa(fun(B,option(C)),fun(B,option(C)),F,A4)))
     => ( ( A4 = restrict_map(B,C,aa(fun(B,option(C)),fun(B,option(C)),F,A4),aa(set(B),set(B),uminus_uminus(set(B)),S2)) )
      <=> ( map_le(B,C,A4,aa(fun(B,option(C)),fun(B,option(C)),F,A4))
          & ( S2 = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,C,aa(fun(B,option(C)),fun(B,option(C)),F,A4))),dom(B,C,A4)) ) ) ) ) ).

% eq_f_restr_ss_eq
tff(fact_7601_le__map__mmupd__not__dom,axiom,
    ! [B: $tType,C: $tType,Ma: fun(B,option(C)),K3: set(B),V2: C] : map_le(B,C,Ma,map_mmupd(B,C,Ma,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),K3),dom(B,C,Ma)),V2)) ).

% le_map_mmupd_not_dom
tff(fact_7602_map__mmupd__update__less,axiom,
    ! [B: $tType,C: $tType,K3: set(B),K7: set(B),Ma: fun(B,option(C)),V2: C] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),K3),K7)
     => map_le(B,C,map_mmupd(B,C,Ma,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),K3),dom(B,C,Ma)),V2),map_mmupd(B,C,Ma,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),K7),dom(B,C,Ma)),V2)) ) ).

% map_mmupd_update_less
tff(fact_7603_mmupd__notin__upd,axiom,
    ! [C: $tType,B: $tType,K: B,K3: set(B),Ma: fun(B,option(C)),V2: C] :
      ( ~ aa(set(B),$o,member(B,K),K3)
     => ( aa(B,option(C),map_mmupd(B,C,Ma,K3,V2),K) = aa(B,option(C),Ma,K) ) ) ).

% mmupd_notin_upd
tff(fact_7604_map__mmupd__empty,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),V2: C] : map_mmupd(B,C,Ma,bot_bot(set(B)),V2) = Ma ).

% map_mmupd_empty
tff(fact_7605_mmupd__in__upd,axiom,
    ! [B: $tType,C: $tType,K: B,K3: set(B),Ma: fun(B,option(C)),V2: C] :
      ( aa(set(B),$o,member(B,K),K3)
     => ( aa(B,option(C),map_mmupd(B,C,Ma,K3,V2),K) = aa(C,option(C),some(C),V2) ) ) ).

% mmupd_in_upd
tff(fact_7606_dom__mmupd,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),K3: set(B),V2: C] : dom(B,C,map_mmupd(B,C,Ma,K3,V2)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),dom(B,C,Ma)),K3) ).

% dom_mmupd
tff(fact_7607_map__mmupdE,axiom,
    ! [C: $tType,B: $tType,Ma: fun(C,option(B)),K3: set(C),V2: B,K: C,X: B] :
      ( ( aa(C,option(B),map_mmupd(C,B,Ma,K3,V2),K) = aa(B,option(B),some(B),X) )
     => ( ( ~ aa(set(C),$o,member(C,K),K3)
         => ( aa(C,option(B),Ma,K) != aa(B,option(B),some(B),X) ) )
       => ~ ( aa(set(C),$o,member(C,K),K3)
           => ( X != V2 ) ) ) ) ).

% map_mmupdE
tff(fact_7608_map__mmupd__def,axiom,
    ! [B: $tType,C: $tType,Ma: fun(C,option(B)),K3: set(C),V2: B,K: C] :
      aa(C,option(B),map_mmupd(C,B,Ma,K3,V2),K) = $ite(aa(set(C),$o,member(C,K),K3),aa(B,option(B),some(B),V2),aa(C,option(B),Ma,K)) ).

% map_mmupd_def
tff(fact_7609_graph__eq__to__snd__dom,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : graph(B,C,Ma) = aa(set(B),set(product_prod(B,C)),image2(B,product_prod(B,C),aTP_Lamp_ans(fun(B,option(C)),fun(B,product_prod(B,C)),Ma)),dom(B,C,Ma)) ).

% graph_eq_to_snd_dom
tff(fact_7610_cSUP__UNION,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( condit1219197933456340205attice(D)
     => ! [A4: set(B),B4: fun(B,set(C)),F: fun(C,D)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => ( aa(B,set(C),B4,X2) != bot_bot(set(C)) ) )
           => ( condit941137186595557371_above(D,aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),aa(fun(C,D),fun(B,set(D)),aTP_Lamp_anx(fun(B,set(C)),fun(fun(C,D),fun(B,set(D))),B4),F)),A4)))
             => ( aa(set(D),D,complete_Sup_Sup(D),aa(set(C),set(D),image2(C,D,F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4)))) = aa(set(D),D,complete_Sup_Sup(D),aa(set(B),set(D),image2(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_any(fun(B,set(C)),fun(fun(C,D),fun(B,D)),B4),F)),A4)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_7611_bdd__above_OI,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B),M: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),M) )
         => condit941137186595557371_above(B,A4) ) ) ).

% bdd_above.I
tff(fact_7612_bdd__above__image__sup,axiom,
    ! [B: $tType,C: $tType] :
      ( lattice(B)
     => ! [F: fun(C,B),G: fun(C,B),A4: set(C)] :
          ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_anz(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),A4))
        <=> ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,F),A4))
            & condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G),A4)) ) ) ) ).

% bdd_above_image_sup
tff(fact_7613_graph__empty,axiom,
    ! [C: $tType,B: $tType] : graph(B,C,aTP_Lamp_jv(B,option(C))) = bot_bot(set(product_prod(B,C))) ).

% graph_empty
tff(fact_7614_bdd__above__UN,axiom,
    ! [C: $tType,B: $tType] :
      ( lattice(C)
     => ! [I5: set(B),A4: fun(B,set(C))] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)))
          <=> ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),I5)
               => condit941137186595557371_above(C,aa(B,set(C),A4,X3)) ) ) ) ) ).

% bdd_above_UN
tff(fact_7615_cSUP__upper2,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A4: set(C),X: C,U: B] :
          ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,F),A4))
         => ( aa(set(C),$o,member(C,X),A4)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(C,B,F,X))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))) ) ) ) ) ).

% cSUP_upper2
tff(fact_7616_cSUP__upper,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [X: B,A4: set(B),F: fun(B,C)] :
          ( aa(set(B),$o,member(B,X),A4)
         => ( condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,F),A4))
           => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X)),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ) ).

% cSUP_upper
tff(fact_7617_cSup__le__iff,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [S: set(B),A2: B] :
          ( ( S != bot_bot(set(B)) )
         => ( condit941137186595557371_above(B,S)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),S)),A2)
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),S)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),A2) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_7618_cSup__mono,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B4: set(B),A4: set(B)] :
          ( ( B4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(B,A4)
           => ( ! [B3: B] :
                  ( aa(set(B),$o,member(B,B3),B4)
                 => ? [X4: B] :
                      ( aa(set(B),$o,member(B,X4),A4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),X4) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),B4)),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ) ) ).

% cSup_mono
tff(fact_7619_bdd__above_OE,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B)] :
          ( condit941137186595557371_above(B,A4)
         => ~ ! [M8: B] :
                ~ ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),M8) ) ) ) ).

% bdd_above.E
tff(fact_7620_bdd__above_Ounfold,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B)] :
          ( condit941137186595557371_above(B,A4)
        <=> ? [M10: B] :
            ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),M10) ) ) ) ).

% bdd_above.unfold
tff(fact_7621_cSup__upper2,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X: B,X5: set(B),Y: B] :
          ( aa(set(B),$o,member(B,X),X5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X)
           => ( condit941137186595557371_above(B,X5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),aa(set(B),B,complete_Sup_Sup(B),X5)) ) ) ) ) ).

% cSup_upper2
tff(fact_7622_cSup__upper,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X: B,X5: set(B)] :
          ( aa(set(B),$o,member(B,X),X5)
         => ( condit941137186595557371_above(B,X5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),aa(set(B),B,complete_Sup_Sup(B),X5)) ) ) ) ).

% cSup_upper
tff(fact_7623_bdd__above_OI2,axiom,
    ! [C: $tType,B: $tType] :
      ( preorder(C)
     => ! [A4: set(B),F: fun(B,C),M: C] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),M) )
         => condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,F),A4)) ) ) ).

% bdd_above.I2
tff(fact_7624_cSUP__lessD,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A4: set(C),Y: B,I: C] :
          ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,F),A4))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4))),Y)
           => ( aa(set(C),$o,member(C,I),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(C,B,F,I)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_7625_cSUP__le__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),F: fun(B,C),U: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,F),A4))
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),U)
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X3)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_7626_cSUP__mono,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),G: fun(D,C),B4: set(D),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(C,aa(set(D),set(C),image2(D,C,G),B4))
           => ( ! [N2: B] :
                  ( aa(set(B),$o,member(B,N2),A4)
                 => ? [X4: D] :
                      ( aa(set(D),$o,member(D,X4),B4)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,N2)),aa(D,C,G,X4)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(D),set(C),image2(D,C,G),B4))) ) ) ) ) ).

% cSUP_mono
tff(fact_7627_cSup__subset__mono,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(B,B4)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),A4)),aa(set(B),B,complete_Sup_Sup(B),B4)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_7628_graph__def,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C))] : graph(B,C,Ma) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aTP_Lamp_aoa(fun(B,option(C)),fun(product_prod(B,C),$o),Ma)) ).

% graph_def
tff(fact_7629_less__cSUP__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( condit6923001295902523014norder(C)
     => ! [A4: set(B),F: fun(B,C),A2: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,F),A4))
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),A2),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4)))
            <=> ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),A2),aa(B,C,F,X3)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_7630_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,F),A4))
           => ( condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,G),A4))
             => ( aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),A4))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aob(fun(B,C),fun(fun(B,C),fun(B,C)),F),G)),A4)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_7631_graph__fun__upd__None,axiom,
    ! [C: $tType,B: $tType,Ma: fun(B,option(C)),K: B] : graph(B,C,fun_upd(B,option(C),Ma,K,none(C))) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(B,fun(product_prod(B,C),$o),aTP_Lamp_aoc(fun(B,option(C)),fun(B,fun(product_prod(B,C),$o)),Ma),K)) ).

% graph_fun_upd_None
tff(fact_7632_cSUP__subset__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),G: fun(B,C),B4: set(B),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(C,aa(set(B),set(C),image2(B,C,G),B4))
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),A4)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2)) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B4))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_7633_cSup__inter__less__eq,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( condit941137186595557371_above(B,A4)
         => ( condit941137186595557371_above(B,B4)
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) != bot_bot(set(B)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),A4)),aa(set(B),B,complete_Sup_Sup(B),B4))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_7634_bdd__above__multiset__imp__bdd__above__count,axiom,
    ! [B: $tType,A4: set(multiset(B)),X: B] :
      ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),A4)
     => condit941137186595557371_above(nat,aa(set(multiset(B)),set(nat),image2(multiset(B),nat,aTP_Lamp_ahp(B,fun(multiset(B),nat),X)),A4)) ) ).

% bdd_above_multiset_imp_bdd_above_count
tff(fact_7635_cSup__cInf,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [S: set(B)] :
          ( ( S != bot_bot(set(B)) )
         => ( condit941137186595557371_above(B,S)
           => ( aa(set(B),B,complete_Sup_Sup(B),S) = aa(set(B),B,complete_Inf_Inf(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_aod(set(B),fun(B,$o),S))) ) ) ) ) ).

% cSup_cInf
tff(fact_7636_mono__cSup,axiom,
    ! [C: $tType,B: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( condit941137186595557371_above(B,A4)
           => ( ( A4 != bot_bot(set(B)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(B,C,F,aa(set(B),B,complete_Sup_Sup(B),A4))) ) ) ) ) ).

% mono_cSup
tff(fact_7637_mono__cSUP,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(C) )
     => ! [F: fun(B,C),A4: fun(D,B),I5: set(D)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( condit941137186595557371_above(B,aa(set(D),set(B),image2(D,B,A4),I5))
           => ( ( I5 != bot_bot(set(D)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(D),set(C),image2(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aoe(fun(B,C),fun(fun(D,B),fun(D,C)),F),A4)),I5))),aa(B,C,F,aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,A4),I5)))) ) ) ) ) ).

% mono_cSUP
tff(fact_7638_trans__wf__iff,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( trans(B,R)
     => ( wf(B,R)
      <=> ! [A6: B] : wf(B,aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),set(product_prod(B,B))),inf_inf(set(product_prod(B,B))),R),product_Sigma(B,B,aa(set(B),set(B),image(B,B,converse(B,B,R)),aa(set(B),set(B),insert(B,A6),bot_bot(set(B)))),aa(B,fun(B,set(B)),aTP_Lamp_aof(set(product_prod(B,B)),fun(B,fun(B,set(B))),R),A6)))) ) ) ).

% trans_wf_iff
tff(fact_7639_cINF__UNION,axiom,
    ! [C: $tType,D: $tType,B: $tType] :
      ( condit1219197933456340205attice(D)
     => ! [A4: set(B),B4: fun(B,set(C)),F: fun(C,D)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),A4)
               => ( aa(B,set(C),B4,X2) != bot_bot(set(C)) ) )
           => ( condit1013018076250108175_below(D,aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),aa(fun(C,D),fun(B,set(D)),aTP_Lamp_anx(fun(B,set(C)),fun(fun(C,D),fun(B,set(D))),B4),F)),A4)))
             => ( aa(set(D),D,complete_Inf_Inf(D),aa(set(C),set(D),image2(C,D,F),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),B4),A4)))) = aa(set(D),D,complete_Inf_Inf(D),aa(set(B),set(D),image2(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_aog(fun(B,set(C)),fun(fun(C,D),fun(B,D)),B4),F)),A4)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_7640_bdd__belowI,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B),Ma: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ma),X2) )
         => condit1013018076250108175_below(B,A4) ) ) ).

% bdd_belowI
tff(fact_7641_bdd__below_OI,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B),M: B] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),X2) )
         => condit1013018076250108175_below(B,A4) ) ) ).

% bdd_below.I
tff(fact_7642_brk__rel__wf,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( wf(B,R2)
     => wf(product_prod($o,B),brk_rel(B,B,R2)) ) ).

% brk_rel_wf
tff(fact_7643_bdd__below__image__inf,axiom,
    ! [B: $tType,C: $tType] :
      ( lattice(B)
     => ! [F: fun(C,B),G: fun(C,B),A4: set(C)] :
          ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_aoh(fun(C,B),fun(fun(C,B),fun(C,B)),F),G)),A4))
        <=> ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F),A4))
            & condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,G),A4)) ) ) ) ).

% bdd_below_image_inf
tff(fact_7644_bdd__below__UN,axiom,
    ! [C: $tType,B: $tType] :
      ( lattice(C)
     => ! [I5: set(B),A4: fun(B,set(C))] :
          ( aa(set(B),$o,finite_finite2(B),I5)
         => ( condit1013018076250108175_below(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)))
          <=> ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),I5)
               => condit1013018076250108175_below(C,aa(B,set(C),A4,X3)) ) ) ) ) ).

% bdd_below_UN
tff(fact_7645_finite__wf__eq__wf__converse,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( aa(set(product_prod(B,B)),$o,finite_finite2(product_prod(B,B)),R2)
     => ( wf(B,converse(B,B,R2))
      <=> wf(B,R2) ) ) ).

% finite_wf_eq_wf_converse
tff(fact_7646_wf__finite__segments,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( irrefl(B,R)
     => ( trans(B,R)
       => ( ! [X2: B] : aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_aoi(set(product_prod(B,B)),fun(B,fun(B,$o)),R),X2)))
         => wf(B,R) ) ) ) ).

% wf_finite_segments
tff(fact_7647_wf__linord__ex__has__least,axiom,
    ! [C: $tType,B: $tType,R: set(product_prod(B,B)),Pa: fun(C,$o),K: C,Ma: fun(C,B)] :
      ( wf(B,R)
     => ( ! [X2: B,Y2: B] :
            ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X2),Y2)),transitive_trancl(B,R))
          <=> ~ aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),X2)),transitive_rtrancl(B,R)) )
       => ( aa(C,$o,Pa,K)
         => ? [X2: C] :
              ( aa(C,$o,Pa,X2)
              & ! [Y3: C] :
                  ( aa(C,$o,Pa,Y3)
                 => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(C,B,Ma,X2)),aa(C,B,Ma,Y3))),transitive_rtrancl(B,R)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_7648_wf__no__infinite__down__chainE,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),F: fun(nat,B)] :
      ( wf(B,R)
     => ~ ! [K4: nat] : aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,F,aa(nat,nat,suc,K4))),aa(nat,B,F,K4))),R) ) ).

% wf_no_infinite_down_chainE
tff(fact_7649_wf__iff__no__infinite__down__chain,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( wf(B,R)
    <=> ~ ? [F5: fun(nat,B)] :
          ! [I3: nat] : aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(nat,B,F5,aa(nat,nat,suc,I3))),aa(nat,B,F5,I3))),R) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_7650_wf__bounded__measure,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),Ub: fun(B,nat),F: fun(B,nat)] :
      ( ! [A3: B,B3: B] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B3),A3)),R)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,Ub,B3)),aa(B,nat,Ub,A3))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,F,B3)),aa(B,nat,Ub,A3))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,A3)),aa(B,nat,F,B3)) ) )
     => wf(B,R) ) ).

% wf_bounded_measure
tff(fact_7651_cINF__lower,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A4: set(C),X: C] :
          ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F),A4))
         => ( aa(set(C),$o,member(C,X),A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),aa(C,B,F,X)) ) ) ) ).

% cINF_lower
tff(fact_7652_cINF__lower2,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A4: set(C),X: C,U: B] :
          ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F),A4))
         => ( aa(set(C),$o,member(C,X),A4)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X)),U)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4))),U) ) ) ) ) ).

% cINF_lower2
tff(fact_7653_le__cInf__iff,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [S: set(B),A2: B] :
          ( ( S != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(B,S)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),aa(set(B),B,complete_Inf_Inf(B),S))
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),S)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_7654_cInf__mono,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B4: set(B),A4: set(B)] :
          ( ( B4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(B,A4)
           => ( ! [B3: B] :
                  ( aa(set(B),$o,member(B,B3),B4)
                 => ? [X4: B] :
                      ( aa(set(B),$o,member(B,X4),A4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B3) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Inf_Inf(B),B4)) ) ) ) ) ).

% cInf_mono
tff(fact_7655_bdd__belowI2,axiom,
    ! [C: $tType,B: $tType] :
      ( preorder(C)
     => ! [A4: set(B),Ma: C,F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Ma),aa(B,C,F,X2)) )
         => condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,F),A4)) ) ) ).

% bdd_belowI2
tff(fact_7656_bdd__below_OI2,axiom,
    ! [C: $tType,B: $tType] :
      ( preorder(C)
     => ! [A4: set(B),M: C,F: fun(B,C)] :
          ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),M),aa(B,C,F,X2)) )
         => condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,F),A4)) ) ) ).

% bdd_below.I2
tff(fact_7657_bdd__below_Ounfold,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B)] :
          ( condit1013018076250108175_below(B,A4)
        <=> ? [M10: B] :
            ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M10),X3) ) ) ) ).

% bdd_below.unfold
tff(fact_7658_bdd__below_OE,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [A4: set(B)] :
          ( condit1013018076250108175_below(B,A4)
         => ~ ! [M8: B] :
                ~ ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M8),X4) ) ) ) ).

% bdd_below.E
tff(fact_7659_cInf__lower,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X: B,X5: set(B)] :
          ( aa(set(B),$o,member(B,X),X5)
         => ( condit1013018076250108175_below(B,X5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),X5)),X) ) ) ) ).

% cInf_lower
tff(fact_7660_cInf__lower2,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [X: B,X5: set(B),Y: B] :
          ( aa(set(B),$o,member(B,X),X5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y)
           => ( condit1013018076250108175_below(B,X5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),X5)),Y) ) ) ) ) ).

% cInf_lower2
tff(fact_7661_wf__int__ge__less__than2,axiom,
    ! [D2: int] : wf(int,int_ge_less_than2(D2)) ).

% wf_int_ge_less_than2
tff(fact_7662_wf__int__ge__less__than,axiom,
    ! [D2: int] : wf(int,int_ge_less_than(D2)) ).

% wf_int_ge_less_than
tff(fact_7663_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat)))) ).

% wf_less
tff(fact_7664_wf,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => wf(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),ord_less(B)))) ) ).

% wf
tff(fact_7665_wf__if__measure,axiom,
    ! [B: $tType,Pa: fun(B,$o),F: fun(B,nat),G: fun(B,B)] :
      ( ! [X2: B] :
          ( aa(B,$o,Pa,X2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,F,aa(B,B,G,X2))),aa(B,nat,F,X2)) )
     => wf(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(fun(B,B),fun(B,fun(B,$o)),aTP_Lamp_aoj(fun(B,$o),fun(fun(B,B),fun(B,fun(B,$o))),Pa),G)))) ) ).

% wf_if_measure
tff(fact_7666_wfI,axiom,
    ! [B: $tType,R: set(product_prod(B,B)),A4: set(B),B4: set(B)] :
      ( aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),R),product_Sigma(B,B,A4,aTP_Lamp_zv(set(B),fun(B,set(B)),B4)))
     => ( ! [X2: B,P5: fun(B,$o)] :
            ( ! [Xa: B] :
                ( ! [Y2: B] :
                    ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Xa)),R)
                   => aa(B,$o,P5,Y2) )
               => aa(B,$o,P5,Xa) )
           => ( aa(set(B),$o,member(B,X2),A4)
             => ( aa(set(B),$o,member(B,X2),B4)
               => aa(B,$o,P5,X2) ) ) )
       => wf(B,R) ) ) ).

% wfI
tff(fact_7667_less__cINF__D,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A4: set(C),Y: B,I: C] :
          ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F),A4))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4)))
           => ( aa(set(C),$o,member(C,I),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),aa(C,B,F,I)) ) ) ) ) ).

% less_cINF_D
tff(fact_7668_cINF__mono,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [B4: set(B),F: fun(D,C),A4: set(D),G: fun(B,C)] :
          ( ( B4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(C,aa(set(D),set(C),image2(D,C,F),A4))
           => ( ! [M2: B] :
                  ( aa(set(B),$o,member(B,M2),B4)
                 => ? [X4: D] :
                      ( aa(set(D),$o,member(D,X4),A4)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(D,C,F,X4)),aa(B,C,G,M2)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(D),set(C),image2(D,C,F),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B4))) ) ) ) ) ).

% cINF_mono
tff(fact_7669_le__cINF__iff,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),F: fun(B,C),U: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,F),A4))
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),U),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4)))
            <=> ! [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),U),aa(B,C,F,X3)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_7670_cInf__superset__mono,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(B,B4)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),B4)),aa(set(B),B,complete_Inf_Inf(B),A4)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_7671_cINF__less__iff,axiom,
    ! [B: $tType,C: $tType] :
      ( condit6923001295902523014norder(C)
     => ! [A4: set(B),F: fun(B,C),A2: C] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,F),A4))
           => ( aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),A2)
            <=> ? [X3: B] :
                  ( aa(set(B),$o,member(B,X3),A4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,F,X3)),A2) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_7672_cINF__inf__distrib,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),F: fun(B,C),G: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,F),A4))
           => ( condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,G),A4))
             => ( aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),A4))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aok(fun(B,C),fun(fun(B,C),fun(B,C)),F),G)),A4)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_7673_wf__bounded__supset,axiom,
    ! [B: $tType,S: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),S)
     => wf(set(B),aa(fun(product_prod(set(B),set(B)),$o),set(product_prod(set(B),set(B))),collect(product_prod(set(B),set(B))),aa(fun(set(B),fun(set(B),$o)),fun(product_prod(set(B),set(B)),$o),product_case_prod(set(B),set(B),$o),aTP_Lamp_aol(set(B),fun(set(B),fun(set(B),$o)),S)))) ) ).

% wf_bounded_supset
tff(fact_7674_finite__subset__wf,axiom,
    ! [B: $tType,A4: set(B)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => wf(set(B),aa(fun(product_prod(set(B),set(B)),$o),set(product_prod(set(B),set(B))),collect(product_prod(set(B),set(B))),aa(fun(set(B),fun(set(B),$o)),fun(product_prod(set(B),set(B)),$o),product_case_prod(set(B),set(B),$o),aTP_Lamp_aom(set(B),fun(set(B),fun(set(B),$o)),A4)))) ) ).

% finite_subset_wf
tff(fact_7675_cSUP__eq__cINF__D,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A4: set(C),A2: C] :
          ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A4)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,F),A4))
           => ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F),A4))
             => ( aa(set(C),$o,member(C,A2),A4)
               => ( aa(C,B,F,A2) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A4)) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_7676_cINF__superset__mono,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(B),G: fun(B,C),B4: set(B),F: fun(B,C)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(C,aa(set(B),set(C),image2(B,C,G),B4))
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B4)
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),B4)
                   => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X2)),aa(B,C,F,X2)) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_7677_less__eq__cInf__inter,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(B),B4: set(B)] :
          ( condit1013018076250108175_below(B,A4)
         => ( condit1013018076250108175_below(B,B4)
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4) != bot_bot(set(B)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Inf_Inf(B),B4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B4))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_7678_cInf__le__cSup,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(B)] :
          ( ( A4 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(B,A4)
           => ( condit1013018076250108175_below(B,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),A4)),aa(set(B),B,complete_Sup_Sup(B),A4)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_7679_cInf__cSup,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [S: set(B)] :
          ( ( S != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(B,S)
           => ( aa(set(B),B,complete_Inf_Inf(B),S) = aa(set(B),B,complete_Sup_Sup(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_aon(set(B),fun(B,$o),S))) ) ) ) ) ).

% cInf_cSup
tff(fact_7680_mono__cInf,axiom,
    ! [C: $tType,B: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(C) )
     => ! [F: fun(B,C),A4: set(B)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( condit1013018076250108175_below(B,A4)
           => ( ( A4 != bot_bot(set(B)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,aa(set(B),B,complete_Inf_Inf(B),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),A4))) ) ) ) ) ).

% mono_cInf
tff(fact_7681_mono__cINF,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( ( condit1219197933456340205attice(C)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(B,C),A4: fun(D,B),I5: set(D)] :
          ( aa(fun(B,C),$o,order_mono(B,C),F)
         => ( condit1013018076250108175_below(B,aa(set(D),set(B),image2(D,B,A4),I5))
           => ( ( I5 != bot_bot(set(D)) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F,aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,A4),I5)))),aa(set(C),C,complete_Inf_Inf(C),aa(set(D),set(C),image2(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aoe(fun(B,C),fun(fun(D,B),fun(D,C)),F),A4)),I5))) ) ) ) ) ).

% mono_cINF
tff(fact_7682_relation__of__def,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),A4: set(B)] : order_relation_of(B,Pa,A4) = aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(set(B),fun(B,fun(B,$o)),aTP_Lamp_aoo(fun(B,fun(B,$o)),fun(set(B),fun(B,fun(B,$o))),Pa),A4))) ).

% relation_of_def
tff(fact_7683_filterlim__INF__INF,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,J4: set(B),I5: set(C),F: fun(E,D),F3: fun(C,filter(E)),G5: fun(B,filter(D))] :
      ( ! [M2: B] :
          ( aa(set(B),$o,member(B,M2),J4)
         => ? [X4: C] :
              ( aa(set(C),$o,member(C,X4),I5)
              & aa(filter(D),$o,aa(filter(D),fun(filter(D),$o),ord_less_eq(filter(D)),filtermap(E,D,F,aa(C,filter(E),F3,X4))),aa(B,filter(D),G5,M2)) ) )
     => filterlim(E,D,F,aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),G5),J4)),aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(C),set(filter(E)),image2(C,filter(E),F3),I5))) ) ).

% filterlim_INF_INF
tff(fact_7684_filtermap__id_H,axiom,
    ! [B: $tType,X4: filter(B)] : filtermap(B,B,aTP_Lamp_il(B,B),X4) = X4 ).

% filtermap_id'
tff(fact_7685_filtermap__ident,axiom,
    ! [B: $tType,F3: filter(B)] : filtermap(B,B,aTP_Lamp_il(B,B),F3) = F3 ).

% filtermap_ident
tff(fact_7686_filtermap__filtermap,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,B),G: fun(D,C),F3: filter(D)] : filtermap(C,B,F,filtermap(D,C,G,F3)) = filtermap(D,B,aa(fun(D,C),fun(D,B),aTP_Lamp_ip(fun(C,B),fun(fun(D,C),fun(D,B)),F),G),F3) ).

% filtermap_filtermap
tff(fact_7687_filterlim__filtermap,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(B,C),F12: filter(C),G: fun(D,B),F23: filter(D)] :
      ( filterlim(B,C,F,F12,filtermap(D,B,G,F23))
    <=> filterlim(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aji(fun(B,C),fun(fun(D,B),fun(D,C)),F),G),F12,F23) ) ).

% filterlim_filtermap
tff(fact_7688_eventually__filtermap,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),F: fun(C,B),F3: filter(C)] :
      ( eventually(B,Pa,filtermap(C,B,F,F3))
    <=> eventually(C,aa(fun(C,B),fun(C,$o),aTP_Lamp_als(fun(B,$o),fun(fun(C,B),fun(C,$o)),Pa),F),F3) ) ).

% eventually_filtermap
tff(fact_7689_filtermap__fun__inverse,axiom,
    ! [C: $tType,B: $tType,G: fun(B,C),F3: filter(C),G5: filter(B),F: fun(C,B)] :
      ( filterlim(B,C,G,F3,G5)
     => ( filterlim(C,B,F,G5,F3)
       => ( eventually(B,aa(fun(C,B),fun(B,$o),aTP_Lamp_aop(fun(B,C),fun(fun(C,B),fun(B,$o)),G),F),G5)
         => ( filtermap(C,B,F,F3) = G5 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_7690_filtermap__SUP,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,B),F3: fun(D,filter(C)),B4: set(D)] : filtermap(C,B,F,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),F3),B4))) = aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(D),set(filter(B)),image2(D,filter(B),aa(fun(D,filter(C)),fun(D,filter(B)),aTP_Lamp_aoq(fun(C,B),fun(fun(D,filter(C)),fun(D,filter(B))),F),F3)),B4)) ).

% filtermap_SUP
tff(fact_7691_filtermap__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),F3: filter(C)] : filtermap(C,B,F,F3) = abs_filter(B,aa(filter(C),fun(fun(B,$o),$o),aTP_Lamp_aos(fun(C,B),fun(filter(C),fun(fun(B,$o),$o)),F),F3)) ).

% filtermap_def
tff(fact_7692_filtermap__INF,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,B),F3: fun(D,filter(C)),B4: set(D)] : aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),filtermap(C,B,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),F3),B4)))),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(D),set(filter(B)),image2(D,filter(B),aa(fun(D,filter(C)),fun(D,filter(B)),aTP_Lamp_aoq(fun(C,B),fun(fun(D,filter(C)),fun(D,filter(B))),F),F3)),B4))) ).

% filtermap_INF
tff(fact_7693_prod__filter__principal__singleton2,axiom,
    ! [C: $tType,B: $tType,F3: filter(B),X: C] : prod_filter(B,C,F3,principal(C,aa(set(C),set(C),insert(C,X),bot_bot(set(C))))) = filtermap(B,product_prod(B,C),aa(C,fun(B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))),X),F3) ).

% prod_filter_principal_singleton2
tff(fact_7694_num__of__integer_Otransfer,axiom,
    aa(fun(code_integer,num),$o,aa(fun(int,num),fun(fun(code_integer,num),$o),bNF_rel_fun(int,code_integer,num,num,code_pcr_integer,fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_7695_filtermap__Pair,axiom,
    ! [B: $tType,C: $tType,D: $tType,F: fun(D,B),G: fun(D,C),F3: filter(D)] : aa(filter(product_prod(B,C)),$o,aa(filter(product_prod(B,C)),fun(filter(product_prod(B,C)),$o),ord_less_eq(filter(product_prod(B,C))),filtermap(D,product_prod(B,C),aa(fun(D,C),fun(D,product_prod(B,C)),aTP_Lamp_adn(fun(D,B),fun(fun(D,C),fun(D,product_prod(B,C))),F),G),F3)),prod_filter(B,C,filtermap(D,B,F,F3),filtermap(D,C,G,F3))) ).

% filtermap_Pair
tff(fact_7696_prod__filter__assoc,axiom,
    ! [B: $tType,C: $tType,D: $tType,F3: filter(B),G5: filter(C),H7: filter(D)] : prod_filter(product_prod(B,C),D,prod_filter(B,C,F3,G5),H7) = filtermap(product_prod(B,product_prod(C,D)),product_prod(product_prod(B,C),D),aa(fun(B,fun(product_prod(C,D),product_prod(product_prod(B,C),D))),fun(product_prod(B,product_prod(C,D)),product_prod(product_prod(B,C),D)),product_case_prod(B,product_prod(C,D),product_prod(product_prod(B,C),D)),aTP_Lamp_aou(B,fun(product_prod(C,D),product_prod(product_prod(B,C),D)))),prod_filter(B,product_prod(C,D),F3,prod_filter(C,D,G5,H7))) ).

% prod_filter_assoc
tff(fact_7697_euclidean__size__integer_Otransfer,axiom,
    aa(fun(code_integer,nat),$o,aa(fun(int,nat),fun(fun(code_integer,nat),$o),bNF_rel_fun(int,code_integer,nat,nat,code_pcr_integer,fequal(nat)),euclid6346220572633701492n_size(int)),euclid6346220572633701492n_size(code_integer)) ).

% euclidean_size_integer.transfer
tff(fact_7698_filterlim__Pair,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,C),G5: filter(C),F3: filter(B),G: fun(B,D),H7: filter(D)] :
      ( filterlim(B,C,F,G5,F3)
     => ( filterlim(B,D,G,H7,F3)
       => filterlim(B,product_prod(C,D),aa(fun(B,D),fun(B,product_prod(C,D)),aTP_Lamp_aov(fun(B,C),fun(fun(B,D),fun(B,product_prod(C,D))),F),G),prod_filter(C,D,G5,H7),F3) ) ) ).

% filterlim_Pair
tff(fact_7699_integer_Oleft__unique,axiom,
    left_unique(int,code_integer,code_pcr_integer) ).

% integer.left_unique
tff(fact_7700_zero__integer_Otransfer,axiom,
    aa(code_integer,$o,aa(int,fun(code_integer,$o),code_pcr_integer,zero_zero(int)),zero_zero(code_integer)) ).

% zero_integer.transfer
tff(fact_7701_one__integer_Otransfer,axiom,
    aa(code_integer,$o,aa(int,fun(code_integer,$o),code_pcr_integer,one_one(int)),one_one(code_integer)) ).

% one_integer.transfer
tff(fact_7702_integer_Obi__unique,axiom,
    bi_unique(int,code_integer,code_pcr_integer) ).

% integer.bi_unique
tff(fact_7703_integer_Odomain,axiom,
    ! [Pa: fun(int,$o)] :
      ( ( aa(fun(int,fun(int,$o)),fun(int,$o),domainp(int,int),fequal(int)) = Pa )
     => ( aa(fun(int,fun(code_integer,$o)),fun(int,$o),domainp(int,code_integer),code_pcr_integer) = Pa ) ) ).

% integer.domain
tff(fact_7704_integer_Obi__total,axiom,
    bi_total(int,code_integer,code_pcr_integer) ).

% integer.bi_total
tff(fact_7705_flip__bit__integer_Otransfer,axiom,
    aa(fun(nat,fun(code_integer,code_integer)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(code_integer)) ).

% flip_bit_integer.transfer
tff(fact_7706_push__bit__integer_Otransfer,axiom,
    aa(fun(nat,fun(code_integer,code_integer)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se4730199178511100633sh_bit(int)),bit_se4730199178511100633sh_bit(code_integer)) ).

% push_bit_integer.transfer
tff(fact_7707_drop__bit__integer_Otransfer,axiom,
    aa(fun(nat,fun(code_integer,code_integer)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se4197421643247451524op_bit(int)),bit_se4197421643247451524op_bit(code_integer)) ).

% drop_bit_integer.transfer
tff(fact_7708_xor__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se5824344971392196577ns_xor(int)),bit_se5824344971392196577ns_xor(code_integer)) ).

% xor_integer.transfer
tff(fact_7709_bit__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(code_integer,fun(nat,$o)),$o),bNF_rel_fun(int,code_integer,fun(nat,$o),fun(nat,$o),code_pcr_integer,fequal(fun(nat,$o))),bit_se5641148757651400278ts_bit(int)),bit_se5641148757651400278ts_bit(code_integer)) ).

% bit_integer.transfer
tff(fact_7710_or__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se1065995026697491101ons_or(int)),bit_se1065995026697491101ons_or(code_integer)) ).

% or_integer.transfer
tff(fact_7711_take__bit__integer_Otransfer,axiom,
    aa(fun(nat,fun(code_integer,code_integer)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se2584673776208193580ke_bit(int)),bit_se2584673776208193580ke_bit(code_integer)) ).

% take_bit_integer.transfer
tff(fact_7712_not__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),bit_ri4277139882892585799ns_not(int)),bit_ri4277139882892585799ns_not(code_integer)) ).

% not_integer.transfer
tff(fact_7713_and__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se5824344872417868541ns_and(int)),bit_se5824344872417868541ns_and(code_integer)) ).

% and_integer.transfer
tff(fact_7714_dup_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),aTP_Lamp_ty(int,int)),code_dup) ).

% dup.transfer
tff(fact_7715_integer__of__nat_Otransfer,axiom,
    aa(fun(nat,code_integer),$o,aa(fun(nat,int),fun(fun(nat,code_integer),$o),bNF_rel_fun(nat,nat,int,code_integer,fequal(nat),code_pcr_integer),semiring_1_of_nat(int)),code_integer_of_nat) ).

% integer_of_nat.transfer
tff(fact_7716_less__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(code_integer,fun(code_integer,$o)),$o),bNF_rel_fun(int,code_integer,fun(int,$o),fun(code_integer,$o),code_pcr_integer,bNF_rel_fun(int,code_integer,$o,$o,code_pcr_integer,fequal($o))),ord_less(int)),ord_less(code_integer)) ).

% less_integer.transfer
tff(fact_7717_times__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),times_times(int)),times_times(code_integer)) ).

% times_integer.transfer
tff(fact_7718_divide__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),divide_divide(int)),divide_divide(code_integer)) ).

% divide_integer.transfer
tff(fact_7719_abs__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),abs_abs(int)),abs_abs(code_integer)) ).

% abs_integer.transfer
tff(fact_7720_nat__of__integer_Otransfer,axiom,
    aa(fun(code_integer,nat),$o,aa(fun(int,nat),fun(fun(code_integer,nat),$o),bNF_rel_fun(int,code_integer,nat,nat,code_pcr_integer,fequal(nat)),nat2),code_nat_of_integer) ).

% nat_of_integer.transfer
tff(fact_7721_modulo__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),modulo_modulo(int)),modulo_modulo(code_integer)) ).

% modulo_integer.transfer
tff(fact_7722_plus__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),plus_plus(int)),plus_plus(code_integer)) ).

% plus_integer.transfer
tff(fact_7723_sgn__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),sgn_sgn(int)),sgn_sgn(code_integer)) ).

% sgn_integer.transfer
tff(fact_7724_integer_Orep__transfer,axiom,
    aa(fun(code_integer,int),$o,aa(fun(int,int),fun(fun(code_integer,int),$o),bNF_rel_fun(int,code_integer,int,int,code_pcr_integer,fequal(int)),aTP_Lamp_dk(int,int)),code_int_of_integer) ).

% integer.rep_transfer
tff(fact_7725_integer_Oid__abs__transfer,axiom,
    aa(fun(int,code_integer),$o,aa(fun(int,int),fun(fun(int,code_integer),$o),bNF_rel_fun(int,int,int,code_integer,fequal(int),code_pcr_integer),aTP_Lamp_dk(int,int)),code_integer_of_int) ).

% integer.id_abs_transfer
tff(fact_7726_gcd__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),gcd_gcd(int)),gcd_gcd(code_integer)) ).

% gcd_integer.transfer
tff(fact_7727_minus__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,code_integer)),$o,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),$o),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),minus_minus(int)),minus_minus(code_integer)) ).

% minus_integer.transfer
tff(fact_7728_uminus__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),uminus_uminus(int)),uminus_uminus(code_integer)) ).

% uminus_integer.transfer
tff(fact_7729_mask__integer_Otransfer,axiom,
    aa(fun(nat,code_integer),$o,aa(fun(nat,int),fun(fun(nat,code_integer),$o),bNF_rel_fun(nat,nat,int,code_integer,fequal(nat),code_pcr_integer),bit_se2239418461657761734s_mask(int)),bit_se2239418461657761734s_mask(code_integer)) ).

% mask_integer.transfer
tff(fact_7730_unset__bit__integer_Otransfer,axiom,
    aa(fun(nat,fun(code_integer,code_integer)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(code_integer)) ).

% unset_bit_integer.transfer
tff(fact_7731_set__bit__integer_Otransfer,axiom,
    aa(fun(nat,fun(code_integer,code_integer)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(code_integer)) ).

% set_bit_integer.transfer
tff(fact_7732_integer_Oright__total,axiom,
    right_total(int,code_integer,code_pcr_integer) ).

% integer.right_total
tff(fact_7733_less__eq__integer_Otransfer,axiom,
    aa(fun(code_integer,fun(code_integer,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(code_integer,fun(code_integer,$o)),$o),bNF_rel_fun(int,code_integer,fun(int,$o),fun(code_integer,$o),code_pcr_integer,bNF_rel_fun(int,code_integer,$o,$o,code_pcr_integer,fequal($o))),ord_less_eq(int)),ord_less_eq(code_integer)) ).

% less_eq_integer.transfer
tff(fact_7734_division__segment__integer_Otransfer,axiom,
    aa(fun(code_integer,code_integer),$o,aa(fun(int,int),fun(fun(code_integer,code_integer),$o),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),euclid7384307370059645450egment(int)),euclid7384307370059645450egment(code_integer)) ).

% division_segment_integer.transfer
tff(fact_7735_principal__prod__principal,axiom,
    ! [B: $tType,C: $tType,A4: set(B),B4: set(C)] : prod_filter(B,C,principal(B,A4),principal(C,B4)) = principal(product_prod(B,C),product_Sigma(B,C,A4,aTP_Lamp_vu(set(C),fun(B,set(C)),B4))) ).

% principal_prod_principal
tff(fact_7736_prod__filter__def,axiom,
    ! [B: $tType,C: $tType,F3: filter(B),G5: filter(C)] : prod_filter(B,C,F3,G5) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(product_prod(fun(B,$o),fun(C,$o))),set(filter(product_prod(B,C))),image2(product_prod(fun(B,$o),fun(C,$o)),filter(product_prod(B,C)),aa(fun(fun(B,$o),fun(fun(C,$o),filter(product_prod(B,C)))),fun(product_prod(fun(B,$o),fun(C,$o)),filter(product_prod(B,C))),product_case_prod(fun(B,$o),fun(C,$o),filter(product_prod(B,C))),aTP_Lamp_aow(fun(B,$o),fun(fun(C,$o),filter(product_prod(B,C)))))),aa(fun(product_prod(fun(B,$o),fun(C,$o)),$o),set(product_prod(fun(B,$o),fun(C,$o))),collect(product_prod(fun(B,$o),fun(C,$o))),aa(fun(fun(B,$o),fun(fun(C,$o),$o)),fun(product_prod(fun(B,$o),fun(C,$o)),$o),product_case_prod(fun(B,$o),fun(C,$o),$o),aa(filter(C),fun(fun(B,$o),fun(fun(C,$o),$o)),aTP_Lamp_aox(filter(B),fun(filter(C),fun(fun(B,$o),fun(fun(C,$o),$o))),F3),G5))))) ).

% prod_filter_def
tff(fact_7737_sub_Otransfer,axiom,
    aa(fun(num,fun(num,code_integer)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,code_integer)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,code_integer),fequal(num),bNF_rel_fun(num,num,int,code_integer,fequal(num),code_pcr_integer)),aTP_Lamp_uc(num,fun(num,int))),code_sub) ).

% sub.transfer
tff(fact_7738_eventually__prod__sequentially,axiom,
    ! [Pa: fun(product_prod(nat,nat),$o)] :
      ( eventually(product_prod(nat,nat),Pa,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N11: nat] :
        ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),M4)
         => ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N11),N4)
             => aa(product_prod(nat,nat),$o,Pa,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N4),M4)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_7739_eventually__prodI,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),F3: filter(B),Q: fun(C,$o),G5: filter(C)] :
      ( eventually(B,Pa,F3)
     => ( eventually(C,Q,G5)
       => eventually(product_prod(B,C),aa(fun(C,$o),fun(product_prod(B,C),$o),aTP_Lamp_aoy(fun(B,$o),fun(fun(C,$o),fun(product_prod(B,C),$o)),Pa),Q),prod_filter(B,C,F3,G5)) ) ) ).

% eventually_prodI
tff(fact_7740_eventually__prod2,axiom,
    ! [B: $tType,C: $tType,A4: filter(B),Pa: fun(C,$o),B4: filter(C)] :
      ( ( A4 != bot_bot(filter(B)) )
     => ( eventually(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_aoz(fun(C,$o),fun(B,fun(C,$o)),Pa)),prod_filter(B,C,A4,B4))
      <=> eventually(C,Pa,B4) ) ) ).

% eventually_prod2
tff(fact_7741_eventually__prod1,axiom,
    ! [B: $tType,C: $tType,B4: filter(B),Pa: fun(C,$o),A4: filter(C)] :
      ( ( B4 != bot_bot(filter(B)) )
     => ( eventually(product_prod(C,B),aa(fun(C,fun(B,$o)),fun(product_prod(C,B),$o),product_case_prod(C,B,$o),aTP_Lamp_apa(fun(C,$o),fun(C,fun(B,$o)),Pa)),prod_filter(C,B,A4,B4))
      <=> eventually(C,Pa,A4) ) ) ).

% eventually_prod1
tff(fact_7742_prod__filter__INF,axiom,
    ! [C: $tType,E: $tType,D: $tType,B: $tType,I5: set(B),J4: set(C),A4: fun(B,filter(D)),B4: fun(C,filter(E))] :
      ( ( I5 != bot_bot(set(B)) )
     => ( ( J4 != bot_bot(set(C)) )
       => ( prod_filter(D,E,aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),A4),I5)),aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(C),set(filter(E)),image2(C,filter(E),B4),J4))) = aa(set(filter(product_prod(D,E))),filter(product_prod(D,E)),complete_Inf_Inf(filter(product_prod(D,E))),aa(set(B),set(filter(product_prod(D,E))),image2(B,filter(product_prod(D,E)),aa(fun(C,filter(E)),fun(B,filter(product_prod(D,E))),aa(fun(B,filter(D)),fun(fun(C,filter(E)),fun(B,filter(product_prod(D,E)))),aTP_Lamp_apc(set(C),fun(fun(B,filter(D)),fun(fun(C,filter(E)),fun(B,filter(product_prod(D,E))))),J4),A4),B4)),I5)) ) ) ) ).

% prod_filter_INF
tff(fact_7743_prod__filter__INF1,axiom,
    ! [D: $tType,C: $tType,B: $tType,I5: set(B),A4: fun(B,filter(C)),B4: filter(D)] :
      ( ( I5 != bot_bot(set(B)) )
     => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),A4),I5)),B4) = 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(filter(D),fun(B,filter(product_prod(C,D))),aTP_Lamp_apd(fun(B,filter(C)),fun(filter(D),fun(B,filter(product_prod(C,D)))),A4),B4)),I5)) ) ) ).

% prod_filter_INF1
tff(fact_7744_prod__filter__INF2,axiom,
    ! [D: $tType,C: $tType,B: $tType,J4: set(B),A4: filter(C),B4: fun(B,filter(D))] :
      ( ( J4 != bot_bot(set(B)) )
     => ( prod_filter(C,D,A4,aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B4),J4))) = 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(fun(B,filter(D)),fun(B,filter(product_prod(C,D))),aTP_Lamp_ape(filter(C),fun(fun(B,filter(D)),fun(B,filter(product_prod(C,D)))),A4),B4)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_7745_char__of__take__bit__eq,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat,Ma: B] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),N)
         => ( aa(B,char,unique5772411509450598832har_of(B),aa(B,B,aa(nat,fun(B,B),bit_se2584673776208193580ke_bit(B),N),Ma)) = aa(B,char,unique5772411509450598832har_of(B),Ma) ) ) ) ).

% char_of_take_bit_eq
tff(fact_7746_Max_Osemilattice__order__set__axioms,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => lattic4895041142388067077er_set(B,ord_max(B),aTP_Lamp_kh(B,fun(B,$o)),aTP_Lamp_apf(B,fun(B,$o))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_7747_char__of__nat,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: nat] : aa(B,char,unique5772411509450598832har_of(B),aa(nat,B,semiring_1_of_nat(B),N)) = aa(nat,char,unique5772411509450598832har_of(nat),N) ) ).

% char_of_nat
tff(fact_7748_Inf__fin_Osemilattice__order__set__axioms,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => lattic4895041142388067077er_set(B,inf_inf(B),ord_less_eq(B),ord_less(B)) ) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_7749_Min_Osemilattice__order__set__axioms,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => lattic4895041142388067077er_set(B,ord_min(B),ord_less_eq(B),ord_less(B)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_7750_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image2(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_7751_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_7752_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => lattic4895041142388067077er_set(B,sup_sup(B),aTP_Lamp_apg(B,fun(B,$o)),aTP_Lamp_aph(B,fun(B,$o))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_7753_char__of__def,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [N: B] : aa(B,char,unique5772411509450598832har_of(B),N) = aa($o,char,char2(~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),N),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),one_one(nat)),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2))))),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_7754_range__nat__of__char,axiom,
    aa(set(char),set(nat),image2(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% range_nat_of_char
tff(fact_7755_char_Osize_I2_J,axiom,
    ! [X12: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : aa(char,nat,size_size(char),aa($o,char,char2((X12),(X22),(X32),(X42),(X52),(X62),(X72)),(X82))) = zero_zero(nat) ).

% char.size(2)
tff(fact_7756_assert__def,axiom,
    ! [B: $tType,Pa: fun(B,$o),X: B] :
      heap_Time_assert(B,Pa,X) = $ite(aa(B,$o,Pa,X),aa(B,heap_Time_Heap(B),heap_Time_return(B),X),heap_Time_raise(B,aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$false,$false,$false,$false,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$true,$false,$false,$true,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$true,$false,$false,$true,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$false,$true,$false,$false,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($false,$true,$false,$false,$true,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($false,$false,$true,$false,$true,$true,$true),$false)),nil(char))))))))) ).

% assert_def
tff(fact_7757_integer__of__char__code,axiom,
    ! [B0: $o,B13: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa($o,char,char2((B0),(B13),(B23),(B32),(B42),(B52),(B62)),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B72))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B62)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B52)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B42)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B32)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B23)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B13)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B0))) ).

% integer_of_char_code
tff(fact_7758_char_Osize__gen,axiom,
    ! [X12: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : size_char(aa($o,char,char2((X12),(X22),(X32),(X42),(X52),(X62),(X72)),(X82))) = zero_zero(nat) ).

% char.size_gen
tff(fact_7759_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_app(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_7760_assert_H__def,axiom,
    ! [Pa: $o] :
      refine_Imp_assert((Pa)) = $ite((Pa),aa(product_unit,heap_Time_Heap(product_unit),heap_Time_ureturn(product_unit),product_Unity),heap_Time_raise(product_unit,aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$false,$false,$false,$false,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$true,$false,$false,$true,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$true,$false,$false,$true,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($true,$false,$true,$false,$false,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($false,$true,$false,$false,$true,$true,$true),$false)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),aa($o,char,char2($false,$false,$true,$false,$true,$true,$true),$false)),nil(char))))))))) ).

% assert'_def
tff(fact_7761_unit__abs__eta__conv,axiom,
    ! [B: $tType,F: fun(product_unit,B)] : aTP_Lamp_apq(fun(product_unit,B),fun(product_unit,B),F) = F ).

% unit_abs_eta_conv
tff(fact_7762_wait__def,axiom,
    ! [N: nat] : heap_Time_wait(N) = heap_Time_Heap2(product_unit,aTP_Lamp_apr(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ).

% wait_def
tff(fact_7763_unit__all__impI,axiom,
    ! [Pa: fun(product_unit,$o),Q: fun(product_unit,$o)] :
      ( ( aa(product_unit,$o,Pa,product_Unity)
       => aa(product_unit,$o,Q,product_Unity) )
     => ! [X4: product_unit] :
          ( aa(product_unit,$o,Pa,X4)
         => aa(product_unit,$o,Q,X4) ) ) ).

% unit_all_impI
tff(fact_7764_eq__sym__Unity__conv,axiom,
    ! [X: $o] :
      ( ( (X)
      <=> ( product_Unity = product_Unity ) )
    <=> (X) ) ).

% eq_sym_Unity_conv
tff(fact_7765_refines__assert_H,axiom,
    ! [Phi: $o] : refine_Imp_refines(product_unit,aa(product_unit,heap_Time_Heap(product_unit),heap_Time_ureturn(product_unit),product_Unity),refine_Imp_assert((Phi))) ).

% refines_assert'
tff(fact_7766_execute__assert_H,axiom,
    ! [Pa: $o,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,refine_Imp_assert((Pa))),H) = $ite((Pa),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),H),zero_zero(nat)))),none(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))) ).

% execute_assert'
tff(fact_7767_upd_H__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [A2: array(B),I: code_integer,X: B] : array_upd2(B,A2,I,X) = heap_Time_bind(array(B),product_unit,array_upd(B,aa(code_integer,nat,code_nat_of_integer,I),X,A2),aTP_Lamp_aps(array(B),heap_Time_Heap(product_unit))) ) ).

% upd'_def
tff(fact_7768_CODE__ABORT__def,axiom,
    ! [B: $tType,F: fun(product_unit,B)] : cODE_ABORT(B,F) = aa(product_unit,B,F,product_Unity) ).

% CODE_ABORT_def
tff(fact_7769_natural__zero__minus__one,axiom,
    aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),zero_zero(code_natural)),one_one(code_natural)) = zero_zero(code_natural) ).

% natural_zero_minus_one
tff(fact_7770_execute__update,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [R: ref(B),V2: B,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(B,R,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(B,R,V2,H)),one_one(nat)))) ) ).

% execute_update
tff(fact_7771_minus__shift__def,axiom,
    ! [R: code_natural,K: code_natural,L: code_natural] :
      minus_shift(R,K,L) = $ite(aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),K),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),R),K)),L),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),L)) ).

% minus_shift_def
tff(fact_7772_effect__updateE,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [R3: ref(B),V2: B,H: heap_ext(product_unit),H4: heap_ext(product_unit),R: product_unit,N: nat] :
          ( heap_Time_effect(product_unit,ref_update(B,R3,V2),H,H4,R,N)
         => ~ ( ( H4 = ref_set(B,R3,V2,H) )
             => ( N != one_one(nat) ) ) ) ) ).

% effect_updateE
tff(fact_7773_effect__updateI,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [H4: heap_ext(product_unit),R: ref(B),V2: B,H: heap_ext(product_unit),N: nat,X: product_unit] :
          ( ( H4 = ref_set(B,R,V2,H) )
         => ( ( N = one_one(nat) )
           => heap_Time_effect(product_unit,ref_update(B,R,V2),H,H4,X,N) ) ) ) ).

% effect_updateI
tff(fact_7774_Ref__Time_Oupdate__def,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [R: ref(B),V2: B] : ref_update(B,R,V2) = heap_Time_heap(product_unit,aa(B,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_apt(ref(B),fun(B,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R),V2)) ) ).

% Ref_Time.update_def
tff(fact_7775_next_Osimps,axiom,
    ! [V2: code_natural,W: 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),W)) = $let(
        v: code_natural,
        v:= 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,one2))))))))))))))))))))))))))))))),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),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,one2)))))))))))))))))),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,one2))))))))))))))))),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,one2)))))))))))))))))),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,one2)))))))))))))))),
        $let(
          w2: code_natural,
          w2:= 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,one2))))))))))))))))))))))))))))))),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),modulo_modulo(code_natural),W),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,one2)))))))))))))))))),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,one2))))))))))))))))),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),W),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,one2)))))))))))))))))),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,one2)))))))))))))),
          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,one2))))))))))))))))))))))))))))))),v,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),w2),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),v),w2)) ) ) ).

% next.simps
tff(fact_7776_log_Oelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( Y = $ite(
            ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),X),one_one(code_natural))
            | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xa2),X) ),
            one_one(code_natural),
            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_7777_log_Osimps,axiom,
    ! [B2: code_natural,I: code_natural] :
      log(B2,I) = $ite(
        ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),B2),one_one(code_natural))
        | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),I),B2) ),
        one_one(code_natural),
        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),I),B2))) ) ).

% log.simps
tff(fact_7778_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_apv(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_7779_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = 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,one2))))))))))))))))))))))))))))))),K),aTP_Lamp_apx(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_apy(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ).

% Random.range_def
tff(fact_7780_scomp__scomp,axiom,
    ! [B: $tType,F6: $tType,G4: $tType,C: $tType,E: $tType,D: $tType,F: fun(B,product_prod(F6,G4)),G: fun(F6,fun(G4,product_prod(D,E))),H: fun(D,fun(E,C))] : product_scomp(B,D,E,C,product_scomp(B,F6,G4,product_prod(D,E),F,G),H) = product_scomp(B,F6,G4,C,F,aa(fun(D,fun(E,C)),fun(F6,fun(G4,C)),aTP_Lamp_apz(fun(F6,fun(G4,product_prod(D,E))),fun(fun(D,fun(E,C)),fun(F6,fun(G4,C))),G),H)) ).

% scomp_scomp
tff(fact_7781_iterate_Oelims,axiom,
    ! [B: $tType,C: $tType,X: code_natural,Xa2: fun(C,fun(B,product_prod(C,B))),Xb: C,Y: fun(B,product_prod(C,B))] :
      ( ( aa(C,fun(B,product_prod(C,B)),iterate(C,B,X,Xa2),Xb) = Y )
     => ( Y = $ite(X = zero_zero(code_natural),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Xb),product_scomp(B,C,B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),Xa2,Xb),iterate(C,B,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_7782_iterate_Osimps,axiom,
    ! [B: $tType,C: $tType,K: code_natural,F: fun(C,fun(B,product_prod(C,B))),X: C] :
      aa(C,fun(B,product_prod(C,B)),iterate(C,B,K,F),X) = $ite(K = zero_zero(code_natural),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),product_scomp(B,C,B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),F,X),iterate(C,B,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F))) ).

% iterate.simps
tff(fact_7783_scomp__unfold,axiom,
    ! [E: $tType,C: $tType,D: $tType,B: $tType,X4: fun(B,product_prod(C,D)),Xa: fun(C,fun(D,E)),Xb2: B] : aa(B,E,product_scomp(B,C,D,E,X4,Xa),Xb2) = aa(D,E,aa(C,fun(D,E),Xa,aa(product_prod(C,D),C,product_fst(C,D),aa(B,product_prod(C,D),X4,Xb2))),aa(product_prod(C,D),D,product_snd(C,D),aa(B,product_prod(C,D),X4,Xb2))) ).

% scomp_unfold
tff(fact_7784_inc__shift__def,axiom,
    ! [V2: code_natural,K: code_natural] :
      inc_shift(V2,K) = $ite(V2 = K,one_one(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K),one_one(code_natural))) ).

% inc_shift_def
tff(fact_7785_natural_Osimps_I4_J,axiom,
    ! [B: $tType,F1: B,F22: fun(code_natural,B)] : aa(code_natural,B,code_case_natural(B,F1,F22),zero_zero(code_natural)) = F1 ).

% natural.simps(4)
tff(fact_7786_log_Opelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( 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))
       => ~ ( ( Y = $ite(
                  ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),X),one_one(code_natural))
                  | aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xa2),X) ),
                  one_one(code_natural),
                  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))) ) )
           => ~ 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_7787_Predicate_Oiterate__upto_Opinduct,axiom,
    ! [B: $tType,A0: fun(code_natural,B),A12: code_natural,A23: code_natural,Pa: fun(fun(code_natural,B),fun(code_natural,fun(code_natural,$o)))] :
      ( accp(product_prod(fun(code_natural,B),product_prod(code_natural,code_natural)),iterate_upto_rel(B),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,B),product_prod(code_natural,code_natural)),aa(fun(code_natural,B),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,B),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,B),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),A12),A23)))
     => ( ! [F4: fun(code_natural,B),N2: code_natural,M2: code_natural] :
            ( accp(product_prod(fun(code_natural,B),product_prod(code_natural,code_natural)),iterate_upto_rel(B),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,B),product_prod(code_natural,code_natural)),aa(fun(code_natural,B),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,B),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,B),product_prod(code_natural,code_natural)),F4),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),N2),M2)))
           => ( ! [X4: product_unit] :
                  ( ~ aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),M2),N2)
                 => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,B),fun(code_natural,fun(code_natural,$o)),Pa,F4),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N2),one_one(code_natural))),M2) )
             => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,B),fun(code_natural,fun(code_natural,$o)),Pa,F4),N2),M2) ) )
       => aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,B),fun(code_natural,fun(code_natural,$o)),Pa,A0),A12),A23) ) ) ).

% Predicate.iterate_upto.pinduct
tff(fact_7788_case__natural__def,axiom,
    ! [B: $tType,X4: B,Xa: fun(code_natural,B)] : code_case_natural(B,X4,Xa) = code_rec_natural(B,X4,aTP_Lamp_aqa(fun(code_natural,B),fun(code_natural,fun(B,B)),Xa)) ).

% case_natural_def
tff(fact_7789_natural_Osimps_I6_J,axiom,
    ! [B: $tType,F1: B,F22: fun(code_natural,fun(B,B))] : aa(code_natural,B,code_rec_natural(B,F1,F22),zero_zero(code_natural)) = F1 ).

% natural.simps(6)
tff(fact_7790_rec__natural__def,axiom,
    ! [B: $tType,X4: B,Xa: fun(code_natural,fun(B,B)),Xb2: code_natural] : aa(code_natural,B,code_rec_natural(B,X4,Xa),Xb2) = the(B,code_rec_set_natural(B,X4,Xa,Xb2)) ).

% rec_natural_def
tff(fact_7791_iterate_Opelims,axiom,
    ! [B: $tType,C: $tType,X: code_natural,Xa2: fun(C,fun(B,product_prod(C,B))),Xb: C,Y: fun(B,product_prod(C,B))] :
      ( ( aa(C,fun(B,product_prod(C,B)),iterate(C,B,X,Xa2),Xb) = Y )
     => ( accp(product_prod(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C)),iterate_rel(C,B),aa(product_prod(fun(C,fun(B,product_prod(C,B))),C),product_prod(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C)),aa(code_natural,fun(product_prod(fun(C,fun(B,product_prod(C,B))),C),product_prod(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C))),product_Pair(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C)),X),aa(C,product_prod(fun(C,fun(B,product_prod(C,B))),C),aa(fun(C,fun(B,product_prod(C,B))),fun(C,product_prod(fun(C,fun(B,product_prod(C,B))),C)),product_Pair(fun(C,fun(B,product_prod(C,B))),C),Xa2),Xb)))
       => ~ ( ( Y = $ite(X = zero_zero(code_natural),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Xb),product_scomp(B,C,B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),Xa2,Xb),iterate(C,B,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2))) )
           => ~ accp(product_prod(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C)),iterate_rel(C,B),aa(product_prod(fun(C,fun(B,product_prod(C,B))),C),product_prod(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C)),aa(code_natural,fun(product_prod(fun(C,fun(B,product_prod(C,B))),C),product_prod(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C))),product_Pair(code_natural,product_prod(fun(C,fun(B,product_prod(C,B))),C)),X),aa(C,product_prod(fun(C,fun(B,product_prod(C,B))),C),aa(fun(C,fun(B,product_prod(C,B))),fun(C,product_prod(fun(C,fun(B,product_prod(C,B))),C)),product_Pair(fun(C,fun(B,product_prod(C,B))),C),Xa2),Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7792_pick__drop__zero,axiom,
    ! [B: $tType,Xs: list(product_prod(code_natural,B))] : pick(B,filter2(product_prod(code_natural,B),aa(fun(code_natural,fun(B,$o)),fun(product_prod(code_natural,B),$o),product_case_prod(code_natural,B,$o),aTP_Lamp_aqb(code_natural,fun(B,$o))),Xs)) = pick(B,Xs) ).

% pick_drop_zero
tff(fact_7793_select__weight__drop__zero,axiom,
    ! [B: $tType,Xs: list(product_prod(code_natural,B))] : select_weight(B,filter2(product_prod(code_natural,B),aa(fun(code_natural,fun(B,$o)),fun(product_prod(code_natural,B),$o),product_case_prod(code_natural,B,$o),aTP_Lamp_aqb(code_natural,fun(B,$o))),Xs)) = select_weight(B,Xs) ).

% select_weight_drop_zero
tff(fact_7794_select__weight__def,axiom,
    ! [B: $tType,Xs: list(product_prod(code_natural,B))] : select_weight(B,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural)),range(aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,B)),list(code_natural),map(product_prod(code_natural,B),code_natural,product_fst(code_natural,B)),Xs))),aTP_Lamp_aqc(list(product_prod(code_natural,B)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural)))),Xs)) ).

% select_weight_def
tff(fact_7795_select__weight__select,axiom,
    ! [B: $tType,Xs: list(B)] :
      ( ( Xs != nil(B) )
     => ( select_weight(B,aa(list(B),list(product_prod(code_natural,B)),map(B,product_prod(code_natural,B),aa(code_natural,fun(B,product_prod(code_natural,B)),product_Pair(code_natural,B),one_one(code_natural))),Xs)) = select(B,Xs) ) ) ).

% select_weight_select
tff(fact_7796_pick__same,axiom,
    ! [B: $tType,L: nat,Xs: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),L),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(code_natural,B,pick(B,aa(list(B),list(product_prod(code_natural,B)),map(B,product_prod(code_natural,B),aa(code_natural,fun(B,product_prod(code_natural,B)),product_Pair(code_natural,B),one_one(code_natural))),Xs)),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,B,nth(B,Xs),L) ) ) ).

% pick_same
tff(fact_7797_natural__of__nat__of__nat,axiom,
    code_natural_of_nat = semiring_1_of_nat(code_natural) ).

% natural_of_nat_of_nat
tff(fact_7798_divide__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(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),divide_divide(nat),Xa2),X)) ).

% divide_natural.abs_eq
tff(fact_7799_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa2),X) ) ).

% less_eq_natural.abs_eq
tff(fact_7800_minus__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(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),minus_minus(nat),Xa2),X)) ).

% minus_natural.abs_eq
tff(fact_7801_zero__natural__def,axiom,
    zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).

% zero_natural_def
tff(fact_7802_one__natural__def,axiom,
    one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).

% one_natural_def
tff(fact_7803_euclidean__size__natural_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,nat,euclid6346220572633701492n_size(code_natural),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,nat,euclid6346220572633701492n_size(nat),X) ).

% euclidean_size_natural.abs_eq
tff(fact_7804_mask__natural__def,axiom,
    bit_se2239418461657761734s_mask(code_natural) = aa(fun(nat,nat),fun(nat,code_natural),map_fun(nat,nat,nat,code_natural,id(nat),code_natural_of_nat),bit_se2239418461657761734s_mask(nat)) ).

% mask_natural_def
tff(fact_7805_drop__bit__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4197421643247451524op_bit(code_natural),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),bit_se4197421643247451524op_bit(nat),Xa2),X)) ).

% drop_bit_natural.abs_eq
tff(fact_7806_and__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se5824344872417868541ns_and(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),bit_se5824344872417868541ns_and(nat),Xa2),X)) ).

% and_natural.abs_eq
tff(fact_7807_bit__natural_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,fun(nat,$o),bit_se5641148757651400278ts_bit(code_natural),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),X) ).

% bit_natural.abs_eq
tff(fact_7808_take__bit__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2584673776208193580ke_bit(code_natural),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),bit_se2584673776208193580ke_bit(nat),Xa2),X)) ).

% take_bit_natural.abs_eq
tff(fact_7809_xor__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se5824344971392196577ns_xor(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),bit_se5824344971392196577ns_xor(nat),Xa2),X)) ).

% xor_natural.abs_eq
tff(fact_7810_or__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se1065995026697491101ons_or(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),bit_se1065995026697491101ons_or(nat),Xa2),X)) ).

% or_natural.abs_eq
tff(fact_7811_push__bit__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4730199178511100633sh_bit(code_natural),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),bit_se4730199178511100633sh_bit(nat),Xa2),X)) ).

% push_bit_natural.abs_eq
tff(fact_7812_mask__natural_Oabs__eq,axiom,
    ! [X: nat] : aa(nat,code_natural,bit_se2239418461657761734s_mask(code_natural),X) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,bit_se2239418461657761734s_mask(nat),X)) ).

% mask_natural.abs_eq
tff(fact_7813_flip__bit__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se8732182000553998342ip_bit(code_natural),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),bit_se8732182000553998342ip_bit(nat),Xa2),X)) ).

% flip_bit_natural.abs_eq
tff(fact_7814_unset__bit__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2638667681897837118et_bit(code_natural),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),bit_se2638667681897837118et_bit(nat),Xa2),X)) ).

% unset_bit_natural.abs_eq
tff(fact_7815_set__bit__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se5668285175392031749et_bit(code_natural),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),bit_se5668285175392031749et_bit(nat),Xa2),X)) ).

% set_bit_natural.abs_eq
tff(fact_7816_division__segment__natural_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,code_natural,euclid7384307370059645450egment(code_natural),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,euclid7384307370059645450egment(nat),X)) ).

% division_segment_natural.abs_eq
tff(fact_7817_natural__of__nat__cases,axiom,
    ! [X: code_natural] :
      ~ ! [Y2: nat] :
          ( ( X = aa(nat,code_natural,code_natural_of_nat,Y2) )
         => ~ aa(set(nat),$o,member(nat,Y2),top_top(set(nat))) ) ).

% natural_of_nat_cases
tff(fact_7818_natural__of__nat__induct,axiom,
    ! [Pa: fun(code_natural,$o),X: code_natural] :
      ( ! [Y2: nat] :
          ( aa(set(nat),$o,member(nat,Y2),top_top(set(nat)))
         => aa(code_natural,$o,Pa,aa(nat,code_natural,code_natural_of_nat,Y2)) )
     => aa(code_natural,$o,Pa,X) ) ).

% natural_of_nat_induct
tff(fact_7819_natural__of__nat__inject,axiom,
    ! [X: nat,Y: nat] :
      ( aa(set(nat),$o,member(nat,X),top_top(set(nat)))
     => ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
       => ( ( aa(nat,code_natural,code_natural_of_nat,X) = aa(nat,code_natural,code_natural_of_nat,Y) )
        <=> ( X = Y ) ) ) ) ).

% natural_of_nat_inject
tff(fact_7820_less__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa2),X) ) ).

% less_natural.abs_eq
tff(fact_7821_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_7822_times__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(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),times_times(nat),Xa2),X)) ).

% times_natural.abs_eq
tff(fact_7823_modulo__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(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),modulo_modulo(nat),Xa2),X)) ).

% modulo_natural.abs_eq
tff(fact_7824_select__def,axiom,
    ! [B: $tType,Xs: list(B)] : select(B,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural)),range(aa(nat,code_natural,code_natural_of_nat,aa(list(B),nat,size_size(list(B)),Xs))),aTP_Lamp_aqd(list(B),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural)))),Xs)) ).

% select_def
tff(fact_7825_Nat_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_natural,code_Nat,aa(int,code_integer,code_integer_of_int,X)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,X)) ).

% Nat.abs_eq
tff(fact_7826_nat__of__natural__inverse,axiom,
    ! [X: code_natural] : aa(nat,code_natural,code_natural_of_nat,aa(code_natural,nat,code_nat_of_natural,X)) = X ).

% nat_of_natural_inverse
tff(fact_7827_nat__of__natural__of__nat__inverse,axiom,
    ! [N: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,code_natural_of_nat,N)) = N ).

% nat_of_natural_of_nat_inverse
tff(fact_7828_natural__of__nat__of__natural__inverse,axiom,
    ! [N: code_natural] : aa(nat,code_natural,code_natural_of_nat,aa(code_natural,nat,code_nat_of_natural,N)) = N ).

% natural_of_nat_of_natural_inverse
tff(fact_7829_nat__of__natural__min,axiom,
    ! [K: code_natural,L: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),ord_min(code_natural),K),L)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(code_natural,nat,code_nat_of_natural,K)),aa(code_natural,nat,code_nat_of_natural,L)) ).

% nat_of_natural_min
tff(fact_7830_nat__of__natural__max,axiom,
    ! [K: code_natural,L: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),ord_max(code_natural),K),L)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(code_natural,nat,code_nat_of_natural,K)),aa(code_natural,nat,code_nat_of_natural,L)) ).

% nat_of_natural_max
tff(fact_7831_nat__of__natural__of__nat,axiom,
    ! [N: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,semiring_1_of_nat(code_natural),N)) = N ).

% nat_of_natural_of_nat
tff(fact_7832_of__nat__of__natural,axiom,
    ! [N: code_natural] : aa(nat,code_natural,semiring_1_of_nat(code_natural),aa(code_natural,nat,code_nat_of_natural,N)) = N ).

% of_nat_of_natural
tff(fact_7833_zero__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,zero_zero(code_natural)) = zero_zero(nat) ).

% zero_natural.rep_eq
tff(fact_7834_one__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,one_one(code_natural)) = one_one(nat) ).

% one_natural.rep_eq
tff(fact_7835_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_7836_minus__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),minus_minus(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% minus_natural.rep_eq
tff(fact_7837_nat__of__natural__numeral,axiom,
    ! [K: num] : aa(code_natural,nat,code_nat_of_natural,aa(num,code_natural,numeral_numeral(code_natural),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_natural_numeral
tff(fact_7838_times__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),times_times(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% times_natural.rep_eq
tff(fact_7839_divide__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),divide_divide(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% divide_natural.rep_eq
tff(fact_7840_modulo__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),modulo_modulo(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% modulo_natural.rep_eq
tff(fact_7841_division__segment__natural_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,euclid7384307370059645450egment(code_natural),X)) = aa(nat,nat,euclid7384307370059645450egment(nat),aa(code_natural,nat,code_nat_of_natural,X)) ).

% division_segment_natural.rep_eq
tff(fact_7842_euclidean__size__natural_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_natural,nat,euclid6346220572633701492n_size(code_natural),X) = aa(nat,nat,euclid6346220572633701492n_size(nat),aa(code_natural,nat,code_nat_of_natural,X)) ).

% euclidean_size_natural.rep_eq
tff(fact_7843_minus__natural__def,axiom,
    minus_minus(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)),minus_minus(nat)) ).

% minus_natural_def
tff(fact_7844_or__natural__def,axiom,
    bit_se1065995026697491101ons_or(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)),bit_se1065995026697491101ons_or(nat)) ).

% or_natural_def
tff(fact_7845_xor__natural__def,axiom,
    bit_se5824344971392196577ns_xor(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)),bit_se5824344971392196577ns_xor(nat)) ).

% xor_natural_def
tff(fact_7846_and__natural__def,axiom,
    bit_se5824344872417868541ns_and(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)),bit_se5824344872417868541ns_and(nat)) ).

% and_natural_def
tff(fact_7847_division__segment__natural__def,axiom,
    euclid7384307370059645450egment(code_natural) = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),euclid7384307370059645450egment(nat)) ).

% division_segment_natural_def
tff(fact_7848_natural__of__nat__inverse,axiom,
    ! [Y: nat] :
      ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
     => ( aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,code_natural_of_nat,Y)) = Y ) ) ).

% natural_of_nat_inverse
tff(fact_7849_type__definition__natural,axiom,
    type_definition(code_natural,nat,code_nat_of_natural,code_natural_of_nat,top_top(set(nat))) ).

% type_definition_natural
tff(fact_7850_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_7851_less__natural__def,axiom,
    ord_less(code_natural) = aa(fun(nat,fun(nat,$o)),fun(code_natural,fun(code_natural,$o)),map_fun(code_natural,nat,fun(nat,$o),fun(code_natural,$o),code_nat_of_natural,map_fun(code_natural,nat,$o,$o,code_nat_of_natural,id($o))),ord_less(nat)) ).

% less_natural_def
tff(fact_7852_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),X),Xa2)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),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_7853_nat__of__natural__induct,axiom,
    ! [Y: nat,Pa: fun(nat,$o)] :
      ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
     => ( ! [X2: code_natural] : aa(nat,$o,Pa,aa(code_natural,nat,code_nat_of_natural,X2))
       => aa(nat,$o,Pa,Y) ) ) ).

% nat_of_natural_induct
tff(fact_7854_nat__of__natural__cases,axiom,
    ! [Y: nat] :
      ( aa(set(nat),$o,member(nat,Y),top_top(set(nat)))
     => ~ ! [X2: code_natural] : Y != aa(code_natural,nat,code_nat_of_natural,X2) ) ).

% nat_of_natural_cases
tff(fact_7855_nat__of__natural,axiom,
    ! [X: code_natural] : aa(set(nat),$o,member(nat,aa(code_natural,nat,code_nat_of_natural,X)),top_top(set(nat))) ).

% nat_of_natural
tff(fact_7856_drop__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4197421643247451524op_bit(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% drop_bit_natural.rep_eq
tff(fact_7857_bit__natural_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_natural,fun(nat,$o),bit_se5641148757651400278ts_bit(code_natural),X) = aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(code_natural,nat,code_nat_of_natural,X)) ).

% bit_natural.rep_eq
tff(fact_7858_or__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),bit_se1065995026697491101ons_or(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% or_natural.rep_eq
tff(fact_7859_xor__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),bit_se5824344971392196577ns_xor(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% xor_natural.rep_eq
tff(fact_7860_natural__eq__iff,axiom,
    ! [Ma: code_natural,N: code_natural] :
      ( ( Ma = N )
    <=> ( aa(code_natural,nat,code_nat_of_natural,Ma) = aa(code_natural,nat,code_nat_of_natural,N) ) ) ).

% natural_eq_iff
tff(fact_7861_natural__eqI,axiom,
    ! [Ma: code_natural,N: code_natural] :
      ( ( aa(code_natural,nat,code_nat_of_natural,Ma) = aa(code_natural,nat,code_nat_of_natural,N) )
     => ( Ma = N ) ) ).

% natural_eqI
tff(fact_7862_nat__of__natural__inject,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( aa(code_natural,nat,code_nat_of_natural,X) = aa(code_natural,nat,code_nat_of_natural,Y) )
    <=> ( X = Y ) ) ).

% nat_of_natural_inject
tff(fact_7863_take__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2584673776208193580ke_bit(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% take_bit_natural.rep_eq
tff(fact_7864_and__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),bit_se5824344872417868541ns_and(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% and_natural.rep_eq
tff(fact_7865_mask__natural_Orep__eq,axiom,
    ! [X: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,bit_se2239418461657761734s_mask(code_natural),X)) = aa(nat,nat,bit_se2239418461657761734s_mask(nat),X) ).

% mask_natural.rep_eq
tff(fact_7866_push__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4730199178511100633sh_bit(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% push_bit_natural.rep_eq
tff(fact_7867_set__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se5668285175392031749et_bit(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% set_bit_natural.rep_eq
tff(fact_7868_unset__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2638667681897837118et_bit(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% unset_bit_natural.rep_eq
tff(fact_7869_flip__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se8732182000553998342ip_bit(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% flip_bit_natural.rep_eq
tff(fact_7870_Nat_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_Nat,X)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).

% Nat.rep_eq
tff(fact_7871_euclidean__size__natural__def,axiom,
    euclid6346220572633701492n_size(code_natural) = aa(fun(nat,nat),fun(code_natural,nat),map_fun(code_natural,nat,nat,nat,code_nat_of_natural,id(nat)),euclid6346220572633701492n_size(nat)) ).

% euclidean_size_natural_def
tff(fact_7872_bit__natural__def,axiom,
    bit_se5641148757651400278ts_bit(code_natural) = aa(fun(nat,fun(nat,$o)),fun(code_natural,fun(nat,$o)),map_fun(code_natural,nat,fun(nat,$o),fun(nat,$o),code_nat_of_natural,id(fun(nat,$o))),bit_se5641148757651400278ts_bit(nat)) ).

% bit_natural_def
tff(fact_7873_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),X),Xa2)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),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_7874_times__natural__def,axiom,
    times_times(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)),times_times(nat)) ).

% times_natural_def
tff(fact_7875_divide__natural__def,axiom,
    divide_divide(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)),divide_divide(nat)) ).

% divide_natural_def
tff(fact_7876_take__bit__natural__def,axiom,
    bit_se2584673776208193580ke_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se2584673776208193580ke_bit(nat)) ).

% take_bit_natural_def
tff(fact_7877_push__bit__natural__def,axiom,
    bit_se4730199178511100633sh_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se4730199178511100633sh_bit(nat)) ).

% push_bit_natural_def
tff(fact_7878_less__eq__natural__def,axiom,
    ord_less_eq(code_natural) = aa(fun(nat,fun(nat,$o)),fun(code_natural,fun(code_natural,$o)),map_fun(code_natural,nat,fun(nat,$o),fun(code_natural,$o),code_nat_of_natural,map_fun(code_natural,nat,$o,$o,code_nat_of_natural,id($o))),ord_less_eq(nat)) ).

% less_eq_natural_def
tff(fact_7879_modulo__natural__def,axiom,
    modulo_modulo(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)),modulo_modulo(nat)) ).

% modulo_natural_def
tff(fact_7880_drop__bit__natural__def,axiom,
    bit_se4197421643247451524op_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se4197421643247451524op_bit(nat)) ).

% drop_bit_natural_def
tff(fact_7881_flip__bit__natural__def,axiom,
    bit_se8732182000553998342ip_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se8732182000553998342ip_bit(nat)) ).

% flip_bit_natural_def
tff(fact_7882_set__bit__natural__def,axiom,
    bit_se5668285175392031749et_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se5668285175392031749et_bit(nat)) ).

% set_bit_natural_def
tff(fact_7883_unset__bit__natural__def,axiom,
    bit_se2638667681897837118et_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se2638667681897837118et_bit(nat)) ).

% unset_bit_natural_def
tff(fact_7884_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_zero(code_natural) )
     => aa(nat,$o,aa(nat,fun(nat,$o),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_7885_Nat__def,axiom,
    code_Nat = aa(fun(int,nat),fun(code_integer,code_natural),map_fun(code_integer,int,nat,code_natural,code_int_of_integer,code_natural_of_nat),nat2) ).

% Nat_def
tff(fact_7886_integer__of__natural_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,X)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,X)) ).

% integer_of_natural.rep_eq
tff(fact_7887_int__of__integer__of__natural,axiom,
    ! [N: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,N)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,N)) ).

% int_of_integer_of_natural
tff(fact_7888_integer__of__natural__of__nat,axiom,
    ! [N: nat] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,semiring_1_of_nat(code_natural),N)) = aa(nat,code_integer,semiring_1_of_nat(code_integer),N) ).

% integer_of_natural_of_nat
tff(fact_7889_integer__of__natural__def,axiom,
    code_i5400310926305786745atural = aa(fun(nat,int),fun(code_natural,code_integer),map_fun(code_natural,nat,int,code_integer,code_nat_of_natural,code_integer_of_int),semiring_1_of_nat(int)) ).

% integer_of_natural_def
tff(fact_7890_integer__of__natural_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,code_natural_of_nat,X)) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).

% integer_of_natural.abs_eq
tff(fact_7891_integer__of__natural__of__integer,axiom,
    ! [K: code_integer] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(code_integer,code_natural,code_n4118661773612635043nteger,K)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).

% integer_of_natural_of_integer
tff(fact_7892_natural__of__integer__def,axiom,
    code_n4118661773612635043nteger = aa(fun(int,nat),fun(code_integer,code_natural),map_fun(code_integer,int,nat,code_natural,code_int_of_integer,code_natural_of_nat),nat2) ).

% natural_of_integer_def
tff(fact_7893_natural__of__integer__of__natural,axiom,
    ! [N: code_natural] : aa(code_integer,code_natural,code_n4118661773612635043nteger,aa(code_natural,code_integer,code_i5400310926305786745atural,N)) = N ).

% natural_of_integer_of_natural
tff(fact_7894_natural__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_n4118661773612635043nteger,X)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).

% natural_of_integer.rep_eq
tff(fact_7895_natural__of__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_natural,code_n4118661773612635043nteger,aa(int,code_integer,code_integer_of_int,X)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,X)) ).

% natural_of_integer.abs_eq
tff(fact_7896_Code__Numeral_Osize__nat__def,axiom,
    code_size_nat = code_nat_of_natural ).

% Code_Numeral.size_nat_def
tff(fact_7897_Code__Numeral_OSuc__def,axiom,
    code_Suc = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),suc) ).

% Code_Numeral.Suc_def
tff(fact_7898_natural_Oinject,axiom,
    ! [Natural: code_natural,Natural2: code_natural] :
      ( ( aa(code_natural,code_natural,code_Suc,Natural) = aa(code_natural,code_natural,code_Suc,Natural2) )
    <=> ( Natural = Natural2 ) ) ).

% natural.inject
tff(fact_7899_natural_Osimps_I7_J,axiom,
    ! [B: $tType,F1: B,F22: fun(code_natural,fun(B,B)),Natural: code_natural] : aa(code_natural,B,code_rec_natural(B,F1,F22),aa(code_natural,code_natural,code_Suc,Natural)) = aa(B,B,aa(code_natural,fun(B,B),F22,Natural),aa(code_natural,B,code_rec_natural(B,F1,F22),Natural)) ).

% natural.simps(7)
tff(fact_7900_natural_Osimps_I5_J,axiom,
    ! [B: $tType,F1: B,F22: fun(code_natural,B),Natural: code_natural] : aa(code_natural,B,code_case_natural(B,F1,F22),aa(code_natural,code_natural,code_Suc,Natural)) = aa(code_natural,B,F22,Natural) ).

% natural.simps(5)
tff(fact_7901_Suc_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,code_Suc,X)) = aa(nat,nat,suc,aa(code_natural,nat,code_nat_of_natural,X)) ).

% Suc.rep_eq
tff(fact_7902_natural_Odistinct_I2_J,axiom,
    ! [Natural2: code_natural] : aa(code_natural,code_natural,code_Suc,Natural2) != zero_zero(code_natural) ).

% natural.distinct(2)
tff(fact_7903_natural_Odistinct_I1_J,axiom,
    ! [Natural2: code_natural] : zero_zero(code_natural) != aa(code_natural,code_natural,code_Suc,Natural2) ).

% natural.distinct(1)
tff(fact_7904_natural_Oexhaust,axiom,
    ! [Y: code_natural] :
      ( ( Y != zero_zero(code_natural) )
     => ~ ! [Natural3: code_natural] : Y != aa(code_natural,code_natural,code_Suc,Natural3) ) ).

% natural.exhaust
tff(fact_7905_Suc_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,code_natural,code_Suc,aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,suc,X)) ).

% Suc.abs_eq
tff(fact_7906_Suc__natural__minus__one,axiom,
    ! [N: code_natural] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,code_Suc,N)),one_one(code_natural)) = N ).

% Suc_natural_minus_one
tff(fact_7907_subset__singleton__iff__Uniq,axiom,
    ! [B: $tType,A4: set(B)] :
      ( ? [A6: B] : aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(B),set(B),insert(B,A6),bot_bot(set(B))))
    <=> uniq(B,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),A4)) ) ).

% subset_singleton_iff_Uniq
tff(fact_7908_prod__decode__triangle__add,axiom,
    ! [K: nat,Ma: nat] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Ma)) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),Ma) ).

% prod_decode_triangle_add
tff(fact_7909_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero(nat)) ).

% prod_decode_def
tff(fact_7910_inj__on__iff__Uniq,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),A4: set(B)] :
      ( inj_on(B,C,F,A4)
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),A4)
         => uniq(B,aa(B,fun(B,$o),aa(set(B),fun(B,fun(B,$o)),aTP_Lamp_aqe(fun(B,C),fun(set(B),fun(B,fun(B,$o))),F),A4),X3)) ) ) ).

% inj_on_iff_Uniq
tff(fact_7911_the1__equality_H,axiom,
    ! [B: $tType,Pa: fun(B,$o),A2: B] :
      ( uniq(B,Pa)
     => ( aa(B,$o,Pa,A2)
       => ( the(B,Pa) = A2 ) ) ) ).

% the1_equality'
tff(fact_7912_ex1__iff__ex__Uniq,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ? [X3: B] :
          ( aa(B,$o,Pa,X3)
          & ! [Y5: B] :
              ( aa(B,$o,Pa,Y5)
             => ( Y5 = X3 ) ) )
    <=> ( ? [X_12: B] : aa(B,$o,Pa,X_12)
        & uniq(B,Pa) ) ) ).

% ex1_iff_ex_Uniq
tff(fact_7913_alt__ex1E_H,axiom,
    ! [B: $tType,Pa: fun(B,$o)] :
      ( ? [X4: B] :
          ( aa(B,$o,Pa,X4)
          & ! [Y2: B] :
              ( aa(B,$o,Pa,Y2)
             => ( Y2 = X4 ) ) )
     => ~ ( ? [X_13: B] : aa(B,$o,Pa,X_13)
         => ~ uniq(B,Pa) ) ) ).

% alt_ex1E'
tff(fact_7914_left__unique__iff,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o))] :
      ( left_unique(B,C,R2)
    <=> ! [Z5: C] : uniq(B,aa(C,fun(B,$o),aTP_Lamp_yp(fun(B,fun(C,$o)),fun(C,fun(B,$o)),R2),Z5)) ) ).

% left_unique_iff
tff(fact_7915_bi__unique__iff,axiom,
    ! [C: $tType,B: $tType,R2: fun(B,fun(C,$o))] :
      ( bi_unique(B,C,R2)
    <=> ( ! [Z5: C] : uniq(B,aa(C,fun(B,$o),aTP_Lamp_yp(fun(B,fun(C,$o)),fun(C,fun(B,$o)),R2),Z5))
        & ! [Z5: B] : uniq(C,aa(B,fun(C,$o),R2,Z5)) ) ) ).

% bi_unique_iff
tff(fact_7916_strict__sorted__equal__Uniq,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [A4: set(B)] : uniq(list(B),aTP_Lamp_aqf(set(B),fun(list(B),$o),A4)) ) ).

% strict_sorted_equal_Uniq
tff(fact_7917_list__decode_Oelims,axiom,
    ! [X: nat,Y: list(nat)] :
      ( ( nat_list_decode(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != nil(nat) ) )
       => ~ ! [N2: nat] :
              ( ( X = aa(nat,nat,suc,N2) )
             => ( 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_aqg(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_7918_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_aqg(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_7919_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero(nat)) = nil(nat) ).

% list_decode.simps(1)
tff(fact_7920_list__decode_Opelims,axiom,
    ! [X: nat,Y: list(nat)] :
      ( ( nat_list_decode(X) = Y )
     => ( accp(nat,nat_list_decode_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = nil(nat) )
             => ~ accp(nat,nat_list_decode_rel,zero_zero(nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,N2) )
               => ( ( 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_aqg(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) )
                 => ~ accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N2)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_7921_Gcd__fin__def,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ( semiring_gcd_Gcd_fin(B) = bounde2362111253966948842tice_F(B,gcd_gcd(B),zero_zero(B),one_one(B)) ) ) ).

% Gcd_fin_def
tff(fact_7922_list__decode_Opsimps_I1_J,axiom,
    ( accp(nat,nat_list_decode_rel,zero_zero(nat))
   => ( nat_list_decode(zero_zero(nat)) = nil(nat) ) ) ).

% list_decode.psimps(1)
tff(fact_7923_bounded__quasi__semilattice__set_OF_Ocong,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B] : bounde2362111253966948842tice_F(B,F,Top,Bot) = bounde2362111253966948842tice_F(B,F,Top,Bot) ).

% bounded_quasi_semilattice_set.F.cong
tff(fact_7924_list__decode_Opinduct,axiom,
    ! [A0: nat,Pa: fun(nat,$o)] :
      ( accp(nat,nat_list_decode_rel,A0)
     => ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
         => aa(nat,$o,Pa,zero_zero(nat)) )
       => ( ! [N2: nat] :
              ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N2))
             => ( ! [X4: nat,Y3: nat] :
                    ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X4),Y3) = aa(nat,product_prod(nat,nat),nat_prod_decode,N2) )
                   => aa(nat,$o,Pa,Y3) )
               => aa(nat,$o,Pa,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,Pa,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_7925_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( 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_aqg(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_7926_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A2: B,A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),aa(set(B),set(B),insert(B,A2),A4)) = aa(B,B,aa(B,fun(B,B),F,A2),aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) ) ).

% bounded_quasi_semilattice_set.insert_remove
tff(fact_7927_bounded__quasi__semilattice__set_Oremove,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A2: B,A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),$o,member(B,A2),A4)
       => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4) = aa(B,B,aa(B,fun(B,B),F,A2),aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert(B,A2),bot_bot(set(B)))))) ) ) ) ).

% bounded_quasi_semilattice_set.remove
tff(fact_7928_bounded__quasi__semilattice__set_Onormalize,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(B,B,Normalize,aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4)) = aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4) ) ) ).

% bounded_quasi_semilattice_set.normalize
tff(fact_7929_bounded__quasi__semilattice__set_Oin__idem,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A2: B,A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),$o,member(B,A2),A4)
       => ( aa(B,B,aa(B,fun(B,B),F,A2),aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4)) = aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4) ) ) ) ).

% bounded_quasi_semilattice_set.in_idem
tff(fact_7930_bounded__quasi__semilattice__set_Oempty,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),bot_bot(set(B))) = Top ) ) ).

% bounded_quasi_semilattice_set.empty
tff(fact_7931_bounded__quasi__semilattice__set_Oinfinite,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4) = Bot ) ) ) ).

% bounded_quasi_semilattice_set.infinite
tff(fact_7932_bounded__quasi__semilattice__set_Osubset,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),B4: set(B),A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),A4)
       => ( aa(B,B,aa(B,fun(B,B),F,aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),B4)),aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4)) = aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4) ) ) ) ).

% bounded_quasi_semilattice_set.subset
tff(fact_7933_bounded__quasi__semilattice__set_Oinsert,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A2: B,A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),aa(set(B),set(B),insert(B,A2),A4)) = aa(B,B,aa(B,fun(B,B),F,A2),aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4)) ) ) ).

% bounded_quasi_semilattice_set.insert
tff(fact_7934_bounded__quasi__semilattice__set_Ounion,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A4: set(B),B4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B4)) = aa(B,B,aa(B,fun(B,B),F,aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4)),aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),B4)) ) ) ).

% bounded_quasi_semilattice_set.union
tff(fact_7935_bounded__quasi__semilattice__set_Oeq__fold,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),A4: set(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),A4) = $ite(aa(set(B),$o,finite_finite2(B),A4),finite_fold(B,B,F,Top,A4),Bot) ) ) ).

% bounded_quasi_semilattice_set.eq_fold
tff(fact_7936_uncurry__apply,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(C,fun(D,B)),A2: C,B2: D] : aa(product_prod(C,D),B,uncurry(C,D,B,F),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A2),B2)) = aa(D,B,aa(C,fun(D,B),F,A2),B2) ).

% uncurry_apply
tff(fact_7937_subset__mset_Osum__pos2,axiom,
    ! [C: $tType,B: $tType,I5: set(B),I: B,F: fun(B,multiset(C))] :
      ( aa(set(B),$o,finite_finite2(B),I5)
     => ( aa(set(B),$o,member(B,I),I5)
       => ( aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subset_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,I))
         => ( ! [I2: B] :
                ( aa(set(B),$o,member(B,I2),I5)
               => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,I2)) )
           => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subset_mset(C),zero_zero(multiset(C))),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,I5)) ) ) ) ) ).

% subset_mset.sum_pos2
tff(fact_7938_subset__mset_Oadd__less__cancel__right,axiom,
    ! [B: $tType,A2: multiset(B),C2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2) ) ).

% subset_mset.add_less_cancel_right
tff(fact_7939_subset__mset_Oadd__less__cancel__left,axiom,
    ! [B: $tType,C2: multiset(B),A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2) ) ).

% subset_mset.add_less_cancel_left
tff(fact_7940_subset__mset_Oless__add__same__cancel2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),A2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),B2) ) ).

% subset_mset.less_add_same_cancel2
tff(fact_7941_subset__mset_Oless__add__same__cancel1,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2))
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),B2) ) ).

% subset_mset.less_add_same_cancel1
tff(fact_7942_subset__mset_Oadd__less__same__cancel2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)),B2)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),zero_zero(multiset(B))) ) ).

% subset_mset.add_less_same_cancel2
tff(fact_7943_subset__mset_Oadd__less__same__cancel1,axiom,
    ! [B: $tType,B2: multiset(B),A2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),A2)),B2)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),zero_zero(multiset(B))) ) ).

% subset_mset.add_less_same_cancel1
tff(fact_7944_wf__subset__mset__rel,axiom,
    ! [B: $tType] : wf(multiset(B),aa(fun(product_prod(multiset(B),multiset(B)),$o),set(product_prod(multiset(B),multiset(B))),collect(product_prod(multiset(B),multiset(B))),aa(fun(multiset(B),fun(multiset(B),$o)),fun(product_prod(multiset(B),multiset(B)),$o),product_case_prod(multiset(B),multiset(B),$o),subset_mset(B)))) ).

% wf_subset_mset_rel
tff(fact_7945_subset__mset_Opos__add__strict,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),C2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)) ) ) ).

% subset_mset.pos_add_strict
tff(fact_7946_subset__mset_Oadd__pos__pos,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),B2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)) ) ) ).

% subset_mset.add_pos_pos
tff(fact_7947_subset__mset_Oadd__neg__neg,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),zero_zero(multiset(B)))
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)),zero_zero(multiset(B))) ) ) ).

% subset_mset.add_neg_neg
tff(fact_7948_subset__mset_OlessE,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2)
     => ~ ! [C4: multiset(B)] :
            ( ( B2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C4) )
           => ( C4 = zero_zero(multiset(B)) ) ) ) ).

% subset_mset.lessE
tff(fact_7949_subset__mset_Oadd__strict__increasing2,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),C2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)) ) ) ).

% subset_mset.add_strict_increasing2
tff(fact_7950_subset__mset_Oadd__strict__increasing,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),C2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)) ) ) ).

% subset_mset.add_strict_increasing
tff(fact_7951_subset__mset_Oadd__pos__nonneg,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),B2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)) ) ) ).

% subset_mset.add_pos_nonneg
tff(fact_7952_subset__mset_Oadd__nonpos__neg,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),B2),zero_zero(multiset(B)))
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)),zero_zero(multiset(B))) ) ) ).

% subset_mset.add_nonpos_neg
tff(fact_7953_subset__mset_Oadd__nonneg__pos,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),A2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),B2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)) ) ) ).

% subset_mset.add_nonneg_pos
tff(fact_7954_subset__mset_Oadd__neg__nonpos,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),zero_zero(multiset(B)))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),B2),zero_zero(multiset(B)))
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),B2)),zero_zero(multiset(B))) ) ) ).

% subset_mset.add_neg_nonpos
tff(fact_7955_mset__subset__size,axiom,
    ! [B: $tType,A4: multiset(B),B4: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A4),B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(multiset(B),nat,size_size(multiset(B)),A4)),aa(multiset(B),nat,size_size(multiset(B)),B4)) ) ).

% mset_subset_size
tff(fact_7956_size__psubset,axiom,
    ! [B: $tType,M: multiset(B),M7: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),M7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(multiset(B),nat,size_size(multiset(B)),M)),aa(multiset(B),nat,size_size(multiset(B)),M7))
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),M),M7) ) ) ).

% size_psubset
tff(fact_7957_subset__mset_Oadd__less__imp__less__right,axiom,
    ! [B: $tType,A2: multiset(B),C2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2) ) ).

% subset_mset.add_less_imp_less_right
tff(fact_7958_subset__mset_Oadd__less__imp__less__left,axiom,
    ! [B: $tType,C2: multiset(B),A2: multiset(B),B2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2))
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2) ) ).

% subset_mset.add_less_imp_less_left
tff(fact_7959_subset__mset_Oadd__strict__right__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),C2)) ) ).

% subset_mset.add_strict_right_mono
tff(fact_7960_subset__mset_Oadd__strict__left__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2)
     => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),A2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),C2),B2)) ) ).

% subset_mset.add_strict_left_mono
tff(fact_7961_subset__mset_Oadd__strict__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B),D2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),C2),D2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),D2)) ) ) ).

% subset_mset.add_strict_mono
tff(fact_7962_subset__mset_Oadd__less__le__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B),D2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),A2),B2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),C2),D2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),D2)) ) ) ).

% subset_mset.add_less_le_mono
tff(fact_7963_subset__mset_Oadd__le__less__mono,axiom,
    ! [B: $tType,A2: multiset(B),B2: multiset(B),C2: multiset(B),D2: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),A2),B2)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),C2),D2)
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),A2),C2)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),B2),D2)) ) ) ).

% subset_mset.add_le_less_mono
tff(fact_7964_subset__mset_Oasymp__greater,axiom,
    ! [B: $tType] : asymp(multiset(B),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o))) ).

% subset_mset.asymp_greater
tff(fact_7965_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
    ! [B: $tType] : lattic4895041142388067077er_set(multiset(B),union_mset(B),aTP_Lamp_aqi(multiset(B),fun(multiset(B),$o)),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o))) ).

% subset_mset.Sup_fin.semilattice_order_set_axioms
tff(fact_7966_subset__mset_Olexordp__def,axiom,
    ! [B: $tType] : lexordp2(multiset(B),subset_mset(B)) = complete_lattice_lfp(fun(list(multiset(B)),fun(list(multiset(B)),$o)),aTP_Lamp_aqj(fun(list(multiset(B)),fun(list(multiset(B)),$o)),fun(list(multiset(B)),fun(list(multiset(B)),$o)))) ).

% subset_mset.lexordp_def
tff(fact_7967_subset__mset_Olexordp_Omono,axiom,
    ! [B: $tType] : aa(fun(fun(list(multiset(B)),fun(list(multiset(B)),$o)),fun(list(multiset(B)),fun(list(multiset(B)),$o))),$o,order_mono(fun(list(multiset(B)),fun(list(multiset(B)),$o)),fun(list(multiset(B)),fun(list(multiset(B)),$o))),aTP_Lamp_aqj(fun(list(multiset(B)),fun(list(multiset(B)),$o)),fun(list(multiset(B)),fun(list(multiset(B)),$o)))) ).

% subset_mset.lexordp.mono
tff(fact_7968_subset__mset_Oordering__top__axioms,axiom,
    ! [B: $tType] : ordering_top(multiset(B),aTP_Lamp_aqi(multiset(B),fun(multiset(B),$o)),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o)),zero_zero(multiset(B))) ).

% subset_mset.ordering_top_axioms
tff(fact_7969_subset__mset_Osemilattice__neutr__order__axioms,axiom,
    ! [B: $tType] : semila1105856199041335345_order(multiset(B),union_mset(B),zero_zero(multiset(B)),aTP_Lamp_aqi(multiset(B),fun(multiset(B),$o)),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o))) ).

% subset_mset.semilattice_neutr_order_axioms
tff(fact_7970_uncurry__def,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,fun(C,D))] : uncurry(B,C,D,F) = aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),F) ).

% uncurry_def
tff(fact_7971_mset__subset__add__iff1,axiom,
    ! [B: $tType,J: nat,I: nat,U: multiset(B),Ma: multiset(B),N: multiset(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),I),U)),Ma)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),J),U)),N))
      <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma)),N) ) ) ).

% mset_subset_add_iff1
tff(fact_7972_mset__subset__add__iff2,axiom,
    ! [B: $tType,I: nat,J: nat,U: multiset(B),Ma: multiset(B),N: multiset(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),I),U)),Ma)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),J),U)),N))
      <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),Ma),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(nat,fun(multiset(B),multiset(B)),repeat_mset(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N)) ) ) ).

% mset_subset_add_iff2
tff(fact_7973_subset__mset_OcSUP__lessD,axiom,
    ! [C: $tType,B: $tType,F: fun(C,multiset(B)),A4: set(C),Y: multiset(B),I: C] :
      ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))),Y)
       => ( aa(set(C),$o,member(C,I),A4)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(C,multiset(B),F,I)),Y) ) ) ) ).

% subset_mset.cSUP_lessD
tff(fact_7974_subset__mset_Oless__cINF__D,axiom,
    ! [B: $tType,C: $tType,F: fun(C,multiset(B)),A4: set(C),Y: multiset(B),I: C] :
      ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4))
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),Y),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),A4)))
       => ( aa(set(C),$o,member(C,I),A4)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),Y),aa(C,multiset(B),F,I)) ) ) ) ).

% subset_mset.less_cINF_D
tff(fact_7975_subset__mset_Osum__strict__mono,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,multiset(C)),G: fun(B,multiset(C))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ( A4 != bot_bot(set(B)) )
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),A4)
             => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subset_mset(C),aa(B,multiset(C),F,X2)),aa(B,multiset(C),G,X2)) )
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subset_mset(C),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,A4)),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),G,A4)) ) ) ) ).

% subset_mset.sum_strict_mono
tff(fact_7976_subset__mset_Osum__pos,axiom,
    ! [C: $tType,B: $tType,I5: set(B),F: fun(B,multiset(C))] :
      ( aa(set(B),$o,finite_finite2(B),I5)
     => ( ( I5 != bot_bot(set(B)) )
       => ( ! [I2: B] :
              ( aa(set(B),$o,member(B,I2),I5)
             => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subset_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),F,I2)) )
         => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subset_mset(C),zero_zero(multiset(C))),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),F,I5)) ) ) ) ).

% subset_mset.sum_pos
tff(fact_7977_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_ar(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_7978_rel__filter_Ocases,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),F3: filter(B),G5: filter(C)] :
      ( rel_filter(B,C,R2,F3,G5)
     => ~ ! [Z6: filter(product_prod(B,C))] :
            ( eventually(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2),Z6)
           => ( ( map_filter_on(product_prod(B,C),B,aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2)),product_fst(B,C),Z6) = F3 )
             => ( map_filter_on(product_prod(B,C),C,aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2)),product_snd(B,C),Z6) != G5 ) ) ) ) ).

% rel_filter.cases
tff(fact_7979_rel__filter_Osimps,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),F3: filter(B),G5: filter(C)] :
      ( rel_filter(B,C,R2,F3,G5)
    <=> ? [Z8: filter(product_prod(B,C))] :
          ( eventually(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2),Z8)
          & ( map_filter_on(product_prod(B,C),B,aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2)),product_fst(B,C),Z8) = F3 )
          & ( map_filter_on(product_prod(B,C),C,aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2)),product_snd(B,C),Z8) = G5 ) ) ) ).

% rel_filter.simps
tff(fact_7980_rel__filter_Ointros,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),Z7: filter(product_prod(B,C)),F3: filter(B),G5: filter(C)] :
      ( eventually(product_prod(B,C),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2),Z7)
     => ( ( map_filter_on(product_prod(B,C),B,aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2)),product_fst(B,C),Z7) = F3 )
       => ( ( map_filter_on(product_prod(B,C),C,aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),R2)),product_snd(B,C),Z7) = G5 )
         => rel_filter(B,C,R2,F3,G5) ) ) ) ).

% rel_filter.intros
tff(fact_7981_map__comp__def,axiom,
    ! [C: $tType,D: $tType,B: $tType,F: fun(D,option(C)),G: fun(B,option(D)),X4: B] : map_comp(D,C,B,F,G,X4) = case_option(option(C),D,none(C),F,aa(B,option(D),G,X4)) ).

% map_comp_def
tff(fact_7982_rel__fun__iff__geq__image2p,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,R2: fun(B,fun(C,$o)),S: fun(D,fun(E,$o)),F: fun(B,D),G: fun(C,E)] :
      ( aa(fun(C,E),$o,aa(fun(B,D),fun(fun(C,E),$o),bNF_rel_fun(B,C,D,E,R2,S),F),G)
    <=> aa(fun(D,fun(E,$o)),$o,aa(fun(D,fun(E,$o)),fun(fun(D,fun(E,$o)),$o),ord_less_eq(fun(D,fun(E,$o))),bNF_Greatest_image2p(B,D,C,E,F,G,R2)),S) ) ).

% rel_fun_iff_geq_image2p
tff(fact_7983_map__comp__empty_I2_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,Ma: fun(B,option(D)),X4: B] : map_comp(D,C,B,aTP_Lamp_aqk(D,option(C)),Ma,X4) = none(C) ).

% map_comp_empty(2)
tff(fact_7984_map__comp__empty_I1_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,Ma: fun(D,option(C)),X4: B] : map_comp(D,C,B,Ma,aTP_Lamp_kb(B,option(D)),X4) = none(C) ).

% map_comp_empty(1)
tff(fact_7985_image2pI,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,R2: fun(B,fun(C,$o)),X: B,Y: C,F: fun(B,D),G: fun(C,E)] :
      ( aa(C,$o,aa(B,fun(C,$o),R2,X),Y)
     => aa(E,$o,aa(D,fun(E,$o),bNF_Greatest_image2p(B,D,C,E,F,G,R2),aa(B,D,F,X)),aa(C,E,G,Y)) ) ).

% image2pI
tff(fact_7986_image2pE,axiom,
    ! [E: $tType,C: $tType,B: $tType,D: $tType,F: fun(B,C),G: fun(D,E),R2: fun(B,fun(D,$o)),Fx: C,Gy: E] :
      ( aa(E,$o,aa(C,fun(E,$o),bNF_Greatest_image2p(B,C,D,E,F,G,R2),Fx),Gy)
     => ~ ! [X2: B] :
            ( ( Fx = aa(B,C,F,X2) )
           => ! [Y2: D] :
                ( ( Gy = aa(D,E,G,Y2) )
               => ~ aa(D,$o,aa(B,fun(D,$o),R2,X2),Y2) ) ) ) ).

% image2pE
tff(fact_7987_rel__fun__image2p,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,R2: fun(B,fun(C,$o)),F: fun(B,D),G: fun(C,E)] : aa(fun(C,E),$o,aa(fun(B,D),fun(fun(C,E),$o),bNF_rel_fun(B,C,D,E,R2,bNF_Greatest_image2p(B,D,C,E,F,G,R2)),F),G) ).

% rel_fun_image2p
tff(fact_7988_image2p__def,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,F: fun(D,B),G: fun(E,C),R2: fun(D,fun(E,$o)),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),bNF_Greatest_image2p(D,B,E,C,F,G,R2),X4),Xa)
    <=> ? [X11: D,Y8: E] :
          ( aa(E,$o,aa(D,fun(E,$o),R2,X11),Y8)
          & ( aa(D,B,F,X11) = X4 )
          & ( aa(E,C,G,Y8) = Xa ) ) ) ).

% image2p_def
tff(fact_7989_subset__mset_Oinf__Sup1__distrib,axiom,
    ! [B: $tType,A4: set(multiset(B)),X: multiset(B)] :
      ( aa(set(multiset(B)),$o,finite_finite2(multiset(B)),A4)
     => ( ( A4 != bot_bot(set(multiset(B))) )
       => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),X),aa(set(multiset(B)),multiset(B),lattic4630905495605216202up_fin(multiset(B),union_mset(B)),A4)) = aa(set(multiset(B)),multiset(B),lattic4630905495605216202up_fin(multiset(B),union_mset(B)),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aql(set(multiset(B)),fun(multiset(B),fun(multiset(B),$o)),A4),X))) ) ) ) ).

% subset_mset.inf_Sup1_distrib
tff(fact_7990_subset__mset_Oinf__Sup2__distrib,axiom,
    ! [B: $tType,A4: set(multiset(B)),B4: set(multiset(B))] :
      ( aa(set(multiset(B)),$o,finite_finite2(multiset(B)),A4)
     => ( ( A4 != bot_bot(set(multiset(B))) )
       => ( aa(set(multiset(B)),$o,finite_finite2(multiset(B)),B4)
         => ( ( B4 != bot_bot(set(multiset(B))) )
           => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),lattic4630905495605216202up_fin(multiset(B),union_mset(B)),A4)),aa(set(multiset(B)),multiset(B),lattic4630905495605216202up_fin(multiset(B),union_mset(B)),B4)) = aa(set(multiset(B)),multiset(B),lattic4630905495605216202up_fin(multiset(B),union_mset(B)),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aa(set(multiset(B)),fun(multiset(B),$o),aTP_Lamp_aqm(set(multiset(B)),fun(set(multiset(B)),fun(multiset(B),$o)),A4),B4))) ) ) ) ) ) ).

% subset_mset.inf_Sup2_distrib
tff(fact_7991_subset__mset_OSup__fin_Oeq__fold_H,axiom,
    ! [B: $tType,A4: set(multiset(B))] : aa(set(multiset(B)),multiset(B),lattic4630905495605216202up_fin(multiset(B),union_mset(B)),A4) = aa(option(multiset(B)),multiset(B),the2(multiset(B)),finite_fold(multiset(B),option(multiset(B)),aTP_Lamp_aqn(multiset(B),fun(option(multiset(B)),option(multiset(B)))),none(multiset(B)),A4)) ).

% subset_mset.Sup_fin.eq_fold'
tff(fact_7992_semilattice__sup_OSup__fin_Ocong,axiom,
    ! [B: $tType,Sup2: fun(B,fun(B,B))] : lattic4630905495605216202up_fin(B,Sup2) = lattic4630905495605216202up_fin(B,Sup2) ).

% semilattice_sup.Sup_fin.cong
tff(fact_7993_subset__mset_Osup__Inf2__distrib,axiom,
    ! [B: $tType,A4: set(multiset(B)),B4: set(multiset(B))] :
      ( aa(set(multiset(B)),$o,finite_finite2(multiset(B)),A4)
     => ( ( A4 != bot_bot(set(multiset(B))) )
       => ( aa(set(multiset(B)),$o,finite_finite2(multiset(B)),B4)
         => ( ( B4 != bot_bot(set(multiset(B))) )
           => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),lattic8678736583308907530nf_fin(multiset(B),inter_mset(B)),A4)),aa(set(multiset(B)),multiset(B),lattic8678736583308907530nf_fin(multiset(B),inter_mset(B)),B4)) = aa(set(multiset(B)),multiset(B),lattic8678736583308907530nf_fin(multiset(B),inter_mset(B)),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aa(set(multiset(B)),fun(multiset(B),$o),aTP_Lamp_aqo(set(multiset(B)),fun(set(multiset(B)),fun(multiset(B),$o)),A4),B4))) ) ) ) ) ) ).

% subset_mset.sup_Inf2_distrib
tff(fact_7994_subset__mset_Osup__Inf1__distrib,axiom,
    ! [B: $tType,A4: set(multiset(B)),X: multiset(B)] :
      ( aa(set(multiset(B)),$o,finite_finite2(multiset(B)),A4)
     => ( ( A4 != bot_bot(set(multiset(B))) )
       => ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),X),aa(set(multiset(B)),multiset(B),lattic8678736583308907530nf_fin(multiset(B),inter_mset(B)),A4)) = aa(set(multiset(B)),multiset(B),lattic8678736583308907530nf_fin(multiset(B),inter_mset(B)),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aqp(set(multiset(B)),fun(multiset(B),fun(multiset(B),$o)),A4),X))) ) ) ) ).

% subset_mset.sup_Inf1_distrib
tff(fact_7995_semilattice__inf_OInf__fin_Ocong,axiom,
    ! [B: $tType,Inf2: fun(B,fun(B,B))] : lattic8678736583308907530nf_fin(B,Inf2) = lattic8678736583308907530nf_fin(B,Inf2) ).

% semilattice_inf.Inf_fin.cong
tff(fact_7996_subset__mset_OInf__fin_Oeq__fold_H,axiom,
    ! [B: $tType,A4: set(multiset(B))] : aa(set(multiset(B)),multiset(B),lattic8678736583308907530nf_fin(multiset(B),inter_mset(B)),A4) = aa(option(multiset(B)),multiset(B),the2(multiset(B)),finite_fold(multiset(B),option(multiset(B)),aTP_Lamp_aqq(multiset(B),fun(option(multiset(B)),option(multiset(B)))),none(multiset(B)),A4)) ).

% subset_mset.Inf_fin.eq_fold'
tff(fact_7997_Code__Numeral_Osub__code_I4_J,axiom,
    ! [N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),aa(num,num,bit0,N)) = code_Neg(bitM(N)) ).

% Code_Numeral.sub_code(4)
tff(fact_7998_flip__pred,axiom,
    ! [B: $tType,C: $tType,A4: set(product_prod(B,C)),R2: fun(C,fun(B,$o))] :
      ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),A4),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),conversep(C,B,R2))))
     => aa(set(product_prod(C,B)),$o,aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),$o),ord_less_eq(set(product_prod(C,B))),aa(set(product_prod(B,C)),set(product_prod(C,B)),image2(product_prod(B,C),product_prod(C,B),aa(fun(B,fun(C,product_prod(C,B))),fun(product_prod(B,C),product_prod(C,B)),product_case_prod(B,C,product_prod(C,B)),aTP_Lamp_te(B,fun(C,product_prod(C,B))))),A4)),aa(fun(product_prod(C,B),$o),set(product_prod(C,B)),collect(product_prod(C,B)),aa(fun(C,fun(B,$o)),fun(product_prod(C,B),$o),product_case_prod(C,B,$o),R2))) ) ).

% flip_pred
tff(fact_7999_conversep__noteq,axiom,
    ! [B: $tType,X4: B,Xa: B] :
      ( aa(B,$o,aa(B,fun(B,$o),conversep(B,B,aTP_Lamp_ajz(B,fun(B,$o))),X4),Xa)
    <=> ( X4 != Xa ) ) ).

% conversep_noteq
tff(fact_8000_less__eq__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_Neg(K)),code_Neg(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),L),K) ) ).

% less_eq_integer_code(9)
tff(fact_8001_less__eq__integer__code_I3_J,axiom,
    ! [L: num] : ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),code_Neg(L)) ).

% less_eq_integer_code(3)
tff(fact_8002_less__eq__integer__code_I7_J,axiom,
    ! [K: num] : aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_Neg(K)),zero_zero(code_integer)) ).

% less_eq_integer_code(7)
tff(fact_8003_minus__integer__code_I6_J,axiom,
    ! [Ma: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_Neg(Ma)),code_Neg(N)) = aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,N),Ma) ).

% minus_integer_code(6)
tff(fact_8004_Code__Numeral_Odup__code_I3_J,axiom,
    ! [N: num] : aa(code_integer,code_integer,code_dup,code_Neg(N)) = code_Neg(aa(num,num,bit0,N)) ).

% Code_Numeral.dup_code(3)
tff(fact_8005_less__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Neg(K)),code_Neg(L))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),L),K) ) ).

% less_integer_code(9)
tff(fact_8006_plus__integer__code_I6_J,axiom,
    ! [Ma: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_Neg(Ma)),code_Neg(N)) = code_Neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),Ma),N)) ).

% plus_integer_code(6)
tff(fact_8007_less__integer__code_I7_J,axiom,
    ! [K: num] : aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_Neg(K)),zero_zero(code_integer)) ).

% less_integer_code(7)
tff(fact_8008_less__integer__code_I3_J,axiom,
    ! [L: num] : ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),code_Neg(L)) ).

% less_integer_code(3)
tff(fact_8009_conversep__converse__eq,axiom,
    ! [B: $tType,C: $tType,R: set(product_prod(C,B)),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),conversep(C,B,aTP_Lamp_mh(set(product_prod(C,B)),fun(C,fun(B,$o)),R)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),converse(C,B,R)) ) ).

% conversep_converse_eq
tff(fact_8010_converse__def,axiom,
    ! [C: $tType,B: $tType,X4: set(product_prod(B,C))] : converse(B,C,X4) = aa(fun(product_prod(C,B),$o),set(product_prod(C,B)),collect(product_prod(C,B)),aa(fun(C,fun(B,$o)),fun(product_prod(C,B),$o),product_case_prod(C,B,$o),conversep(B,C,aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),X4)))) ).

% converse_def
tff(fact_8011_divmod__abs__code_I2_J,axiom,
    ! [K: num,L: num] : code_divmod_abs(code_Neg(K),code_Neg(L)) = unique8689654367752047608divmod(code_integer,K,L) ).

% divmod_abs_code(2)
tff(fact_8012_Code__Numeral_Osub__code_I5_J,axiom,
    ! [N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),aa(num,num,bit1,N)) = code_Neg(aa(num,num,bit0,N)) ).

% Code_Numeral.sub_code(5)
tff(fact_8013_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),Xs: list(B)] :
      ( transp(B,Pa)
     => ( sorted_wrt(B,Pa,Xs)
      <=> ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(B),nat,size_size(list(B)),Xs))
           => aa(B,$o,aa(B,fun(B,$o),Pa,aa(nat,B,nth(B,Xs),I3)),aa(nat,B,nth(B,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_8014_prod__set__simps_I2_J,axiom,
    ! [C: $tType,B: $tType,X: C,Y: B] : basic_snds(C,B,aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(set(B),set(B),insert(B,Y),bot_bot(set(B))) ).

% prod_set_simps(2)
tff(fact_8015_transp__trans,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o))] :
      ( transp(B,R)
    <=> trans(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),R))) ) ).

% transp_trans
tff(fact_8016_transp__trans__eq,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( transp(B,aTP_Lamp_aab(set(product_prod(B,B)),fun(B,fun(B,$o)),R))
    <=> trans(B,R) ) ).

% transp_trans_eq
tff(fact_8017_transp__gr,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => transp(B,aTP_Lamp_js(B,fun(B,$o))) ) ).

% transp_gr
tff(fact_8018_transp__empty,axiom,
    ! [B: $tType] : transp(B,aTP_Lamp_aqr(B,fun(B,$o))) ).

% transp_empty
tff(fact_8019_transp__singleton,axiom,
    ! [B: $tType,A2: B] : transp(B,aTP_Lamp_aqs(B,fun(B,fun(B,$o)),A2)) ).

% transp_singleton
tff(fact_8020_subset__mset_Otransp__ge,axiom,
    ! [B: $tType] : transp(multiset(B),aTP_Lamp_aqi(multiset(B),fun(multiset(B),$o))) ).

% subset_mset.transp_ge
tff(fact_8021_transp__le,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => transp(B,ord_less_eq(B)) ) ).

% transp_le
tff(fact_8022_transp__ge,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => transp(B,aTP_Lamp_aqt(B,fun(B,$o))) ) ).

% transp_ge
tff(fact_8023_subset__mset_Otransp__gr,axiom,
    ! [B: $tType] : transp(multiset(B),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o))) ).

% subset_mset.transp_gr
tff(fact_8024_multp__implies__one__step,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),M: multiset(B),N5: multiset(B)] :
      ( transp(B,R)
     => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),M),N5)
       => ? [I6: multiset(B),J5: multiset(B)] :
            ( ( N5 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I6),J5) )
            & ? [K6: multiset(B)] :
                ( ( M = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),I6),K6) )
                & ( J5 != zero_zero(multiset(B)) )
                & ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),aa(multiset(B),set(B),set_mset(B),K6))
                   => ? [Xa3: B] :
                        ( aa(set(B),$o,member(B,Xa3),aa(multiset(B),set(B),set_mset(B),J5))
                        & aa(B,$o,aa(B,fun(B,$o),R,X4),Xa3) ) ) ) ) ) ) ).

% multp_implies_one_step
tff(fact_8025_multp__cancel__max,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),X5: multiset(B),Y4: multiset(B)] :
      ( transp(B,R)
     => ( irreflp(B,R)
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),X5),Y4)
        <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),X5),Y4)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),Y4),X5)) ) ) ) ).

% multp_cancel_max
tff(fact_8026_multp__cancel,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),X5: multiset(B),Z7: multiset(B),Y4: multiset(B)] :
      ( transp(B,R)
     => ( irreflp(B,R)
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),X5),Z7)),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),Y4),Z7))
        <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),X5),Y4) ) ) ) ).

% multp_cancel
tff(fact_8027_irreflp__multp,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o))] :
      ( transp(B,R)
     => ( irreflp(B,R)
       => irreflp(multiset(B),multp(B,R)) ) ) ).

% irreflp_multp
tff(fact_8028_subset__mset_Oirreflp__greater,axiom,
    ! [B: $tType] : irreflp(multiset(B),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o))) ).

% subset_mset.irreflp_greater
tff(fact_8029_irreflp__greater,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => irreflp(B,aTP_Lamp_js(B,fun(B,$o))) ) ).

% irreflp_greater
tff(fact_8030_irreflp__irrefl__eq,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( irreflp(B,aTP_Lamp_aab(set(product_prod(B,B)),fun(B,fun(B,$o)),R2))
    <=> irrefl(B,R2) ) ).

% irreflp_irrefl_eq
tff(fact_8031_multp__cancel__add__mset,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),Uu: B,X5: multiset(B),Y4: multiset(B)] :
      ( transp(B,R)
     => ( irreflp(B,R)
       => ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Uu),X5)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Uu),Y4))
        <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),multp(B,R),X5),Y4) ) ) ) ).

% multp_cancel_add_mset
tff(fact_8032_prod__set__simps_I1_J,axiom,
    ! [C: $tType,B: $tType,X: B,Y: C] : basic_fsts(B,C,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y)) = aa(set(B),set(B),insert(B,X),bot_bot(set(B))) ).

% prod_set_simps(1)
tff(fact_8033_single__valuedp__single__valued__eq,axiom,
    ! [C: $tType,B: $tType,R: set(product_prod(B,C))] :
      ( single_valuedp(B,C,aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),R))
    <=> single_valued(B,C,R) ) ).

% single_valuedp_single_valued_eq
tff(fact_8034_single__valuedp__iff__Uniq,axiom,
    ! [C: $tType,B: $tType,R: fun(B,fun(C,$o))] :
      ( single_valuedp(B,C,R)
    <=> ! [X3: B] : uniq(C,aa(B,fun(C,$o),R,X3)) ) ).

% single_valuedp_iff_Uniq
tff(fact_8035_prod_Oin__rel,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,R12: fun(B,fun(C,$o)),R23: fun(D,fun(E,$o)),A2: product_prod(B,D),B2: product_prod(C,E)] :
      ( aa(product_prod(C,E),$o,aa(product_prod(B,D),fun(product_prod(C,E),$o),basic_rel_prod(B,C,D,E,R12,R23),A2),B2)
    <=> ? [Z5: product_prod(product_prod(B,C),product_prod(D,E))] :
          ( aa(set(product_prod(product_prod(B,C),product_prod(D,E))),$o,member(product_prod(product_prod(B,C),product_prod(D,E)),Z5),aa(fun(product_prod(product_prod(B,C),product_prod(D,E)),$o),set(product_prod(product_prod(B,C),product_prod(D,E))),collect(product_prod(product_prod(B,C),product_prod(D,E))),aa(fun(D,fun(E,$o)),fun(product_prod(product_prod(B,C),product_prod(D,E)),$o),aTP_Lamp_aqu(fun(B,fun(C,$o)),fun(fun(D,fun(E,$o)),fun(product_prod(product_prod(B,C),product_prod(D,E)),$o)),R12),R23)))
          & ( aa(product_prod(product_prod(B,C),product_prod(D,E)),product_prod(B,D),product_map_prod(product_prod(B,C),B,product_prod(D,E),D,product_fst(B,C),product_fst(D,E)),Z5) = A2 )
          & ( aa(product_prod(product_prod(B,C),product_prod(D,E)),product_prod(C,E),product_map_prod(product_prod(B,C),C,product_prod(D,E),E,product_snd(B,C),product_snd(D,E)),Z5) = B2 ) ) ) ).

% prod.in_rel
tff(fact_8036_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Gcd(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_gcd(int),Xs),zero_zero(int)) ).

% Gcd_int_set_eq_fold
tff(fact_8037_fold__filter,axiom,
    ! [B: $tType,C: $tType,F: fun(C,fun(B,B)),Pa: fun(C,$o),Xs: list(C)] : fold(C,B,F,filter2(C,Pa,Xs)) = fold(C,B,aa(fun(C,$o),fun(C,fun(B,B)),aTP_Lamp_rw(fun(C,fun(B,B)),fun(fun(C,$o),fun(C,fun(B,B))),F),Pa),Xs) ).

% fold_filter
tff(fact_8038_snd__transfer,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,A4: fun(B,fun(D,$o)),B4: fun(C,fun(E,$o))] : aa(fun(product_prod(D,E),E),$o,aa(fun(product_prod(B,C),C),fun(fun(product_prod(D,E),E),$o),bNF_rel_fun(product_prod(B,C),product_prod(D,E),C,E,basic_rel_prod(B,D,C,E,A4,B4),B4),product_snd(B,C)),product_snd(D,E)) ).

% snd_transfer
tff(fact_8039_int_Oid__abs__transfer,axiom,
    aa(fun(product_prod(nat,nat),int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),int),$o),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_aqv(product_prod(nat,nat),product_prod(nat,nat))),abs_Integ) ).

% int.id_abs_transfer
tff(fact_8040_prod_Orel__map_I1_J,axiom,
    ! [F6: $tType,G4: $tType,B: $tType,D: $tType,E: $tType,C: $tType,S1b: fun(B,fun(C,$o)),S2b: fun(D,fun(E,$o)),I1: fun(F6,B),I22: fun(G4,D),X: product_prod(F6,G4),Y: product_prod(C,E)] :
      ( aa(product_prod(C,E),$o,aa(product_prod(B,D),fun(product_prod(C,E),$o),basic_rel_prod(B,C,D,E,S1b,S2b),aa(product_prod(F6,G4),product_prod(B,D),product_map_prod(F6,B,G4,D,I1,I22),X)),Y)
    <=> aa(product_prod(C,E),$o,aa(product_prod(F6,G4),fun(product_prod(C,E),$o),basic_rel_prod(F6,C,G4,E,aa(fun(F6,B),fun(F6,fun(C,$o)),aTP_Lamp_aqw(fun(B,fun(C,$o)),fun(fun(F6,B),fun(F6,fun(C,$o))),S1b),I1),aa(fun(G4,D),fun(G4,fun(E,$o)),aTP_Lamp_aqx(fun(D,fun(E,$o)),fun(fun(G4,D),fun(G4,fun(E,$o))),S2b),I22)),X),Y) ) ).

% prod.rel_map(1)
tff(fact_8041_prod_Orel__map_I2_J,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType,G4: $tType,F6: $tType,S1a: fun(B,fun(C,$o)),S2a: fun(D,fun(E,$o)),X: product_prod(B,D),G1: fun(F6,C),G22: fun(G4,E),Y: product_prod(F6,G4)] :
      ( aa(product_prod(C,E),$o,aa(product_prod(B,D),fun(product_prod(C,E),$o),basic_rel_prod(B,C,D,E,S1a,S2a),X),aa(product_prod(F6,G4),product_prod(C,E),product_map_prod(F6,C,G4,E,G1,G22),Y))
    <=> aa(product_prod(F6,G4),$o,aa(product_prod(B,D),fun(product_prod(F6,G4),$o),basic_rel_prod(B,F6,D,G4,aa(fun(F6,C),fun(B,fun(F6,$o)),aTP_Lamp_aqy(fun(B,fun(C,$o)),fun(fun(F6,C),fun(B,fun(F6,$o))),S1a),G1),aa(fun(G4,E),fun(D,fun(G4,$o)),aTP_Lamp_aqz(fun(D,fun(E,$o)),fun(fun(G4,E),fun(D,fun(G4,$o))),S2a),G22)),X),Y) ) ).

% prod.rel_map(2)
tff(fact_8042_foldl__conv__fold,axiom,
    ! [C: $tType,B: $tType,F: fun(B,fun(C,B)),S2: B,Xs: list(C)] : aa(list(C),B,aa(B,fun(list(C),B),foldl(B,C,F),S2),Xs) = aa(B,B,fold(C,B,aTP_Lamp_kq(fun(B,fun(C,B)),fun(C,fun(B,B)),F),Xs),S2) ).

% foldl_conv_fold
tff(fact_8043_fold__plus__sum__list__rev,axiom,
    ! [B: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(B)] : fold(B,B,plus_plus(B),Xs) = aa(B,fun(B,B),plus_plus(B),aa(list(B),B,groups8242544230860333062m_list(B),rev(B,Xs))) ) ).

% fold_plus_sum_list_rev
tff(fact_8044_fst__transfer,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,A4: fun(B,fun(D,$o)),B4: fun(C,fun(E,$o))] : aa(fun(product_prod(D,E),D),$o,aa(fun(product_prod(B,C),B),fun(fun(product_prod(D,E),D),$o),bNF_rel_fun(product_prod(B,C),product_prod(D,E),B,D,basic_rel_prod(B,D,C,E,A4,B4),A4),product_fst(B,C)),product_fst(D,E)) ).

% fst_transfer
tff(fact_8045_rel__prod__conv,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,R12: fun(B,fun(D,$o)),R23: fun(C,fun(E,$o))] : basic_rel_prod(B,D,C,E,R12,R23) = aa(fun(B,fun(C,fun(product_prod(D,E),$o))),fun(product_prod(B,C),fun(product_prod(D,E),$o)),product_case_prod(B,C,fun(product_prod(D,E),$o)),aa(fun(C,fun(E,$o)),fun(B,fun(C,fun(product_prod(D,E),$o))),aTP_Lamp_arb(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(B,fun(C,fun(product_prod(D,E),$o)))),R12),R23)) ).

% rel_prod_conv
tff(fact_8046_Pair__transfer,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,A4: fun(B,fun(C,$o)),B4: fun(D,fun(E,$o))] : aa(fun(C,fun(E,product_prod(C,E))),$o,aa(fun(B,fun(D,product_prod(B,D))),fun(fun(C,fun(E,product_prod(C,E))),$o),bNF_rel_fun(B,C,fun(D,product_prod(B,D)),fun(E,product_prod(C,E)),A4,bNF_rel_fun(D,E,product_prod(B,D),product_prod(C,E),B4,basic_rel_prod(B,C,D,E,A4,B4))),product_Pair(B,D)),product_Pair(C,E)) ).

% Pair_transfer
tff(fact_8047_bounded__quasi__semilattice__set_Oset__eq__fold,axiom,
    ! [B: $tType,F: fun(B,fun(B,B)),Top: B,Bot: B,Normalize: fun(B,B),Xs: list(B)] :
      ( bounde6485984586167503788ce_set(B,F,Top,Bot,Normalize)
     => ( aa(set(B),B,bounde2362111253966948842tice_F(B,F,Top,Bot),aa(list(B),set(B),set2(B),Xs)) = aa(B,B,fold(B,B,F,Xs),Top) ) ) ).

% bounded_quasi_semilattice_set.set_eq_fold
tff(fact_8048_case__prod__transfer,axiom,
    ! [B: $tType,C: $tType,D: $tType,G4: $tType,F6: $tType,E: $tType,A4: fun(B,fun(E,$o)),B4: fun(C,fun(F6,$o)),C6: fun(D,fun(G4,$o))] : aa(fun(fun(E,fun(F6,G4)),fun(product_prod(E,F6),G4)),$o,aa(fun(fun(B,fun(C,D)),fun(product_prod(B,C),D)),fun(fun(fun(E,fun(F6,G4)),fun(product_prod(E,F6),G4)),$o),bNF_rel_fun(fun(B,fun(C,D)),fun(E,fun(F6,G4)),fun(product_prod(B,C),D),fun(product_prod(E,F6),G4),bNF_rel_fun(B,E,fun(C,D),fun(F6,G4),A4,bNF_rel_fun(C,F6,D,G4,B4,C6)),bNF_rel_fun(product_prod(B,C),product_prod(E,F6),D,G4,basic_rel_prod(B,E,C,F6,A4,B4),C6)),product_case_prod(B,C,D)),product_case_prod(E,F6,G4)) ).

% case_prod_transfer
tff(fact_8049_rat_Odomain,axiom,
    ! [X4: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat),X4)
    <=> ? [Y5: product_prod(int,int)] :
          ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),X4),Y5)
          & aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Y5),Y5) ) ) ).

% rat.domain
tff(fact_8050_Gcd__set__eq__fold,axiom,
    ! [B: $tType] :
      ( semiring_Gcd(B)
     => ! [Xs: list(B)] : gcd_Gcd(B,aa(list(B),set(B),set2(B),Xs)) = aa(B,B,fold(B,B,gcd_gcd(B),Xs),zero_zero(B)) ) ).

% Gcd_set_eq_fold
tff(fact_8051_Gcd__fin_Oset__eq__fold,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [Xs: list(B)] : aa(set(B),B,semiring_gcd_Gcd_fin(B),aa(list(B),set(B),set2(B),Xs)) = aa(B,B,fold(B,B,gcd_gcd(B),Xs),zero_zero(B)) ) ).

% Gcd_fin.set_eq_fold
tff(fact_8052_rat_Odomain__par,axiom,
    ! [DR1: fun(int,$o),DR2: fun(int,$o),P23: fun(product_prod(int,int),$o)] :
      ( ( aa(fun(int,fun(int,$o)),fun(int,$o),domainp(int,int),fequal(int)) = DR1 )
     => ( ( aa(fun(int,fun(int,$o)),fun(int,$o),domainp(int,int),fequal(int)) = DR2 )
       => ( aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal($o)),P23),aTP_Lamp_arc(product_prod(int,int),$o))
         => ( aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat) = aa(fun(product_prod(int,int),$o),fun(product_prod(int,int),$o),aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),fun(product_prod(int,int),$o)),inf_inf(fun(product_prod(int,int),$o)),basic_pred_prod(int,int,DR1,DR2)),P23) ) ) ) ) ).

% rat.domain_par
tff(fact_8053_rat_Odomain__par__left__total,axiom,
    ! [P: fun(product_prod(int,int),$o)] :
      ( left_total(product_prod(int,int),product_prod(int,int),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)))
     => ( aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal($o)),P),aTP_Lamp_arc(product_prod(int,int),$o))
       => ( aa(fun(product_prod(int,int),fun(rat,$o)),fun(product_prod(int,int),$o),domainp(product_prod(int,int),rat),pcr_rat) = P ) ) ) ).

% rat.domain_par_left_total
tff(fact_8054_prod_Omap__cong__pred,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,X: product_prod(B,C),Ya: product_prod(B,C),F1: fun(B,D),G1: fun(B,D),F22: fun(C,E),G22: fun(C,E)] :
      ( ( X = Ya )
     => ( aa(product_prod(B,C),$o,basic_pred_prod(B,C,aa(fun(B,D),fun(B,$o),aTP_Lamp_ard(fun(B,D),fun(fun(B,D),fun(B,$o)),F1),G1),aa(fun(C,E),fun(C,$o),aTP_Lamp_are(fun(C,E),fun(fun(C,E),fun(C,$o)),F22),G22)),Ya)
       => ( aa(product_prod(B,C),product_prod(D,E),product_map_prod(B,D,C,E,F1,F22),X) = aa(product_prod(B,C),product_prod(D,E),product_map_prod(B,D,C,E,G1,G22),Ya) ) ) ) ).

% prod.map_cong_pred
tff(fact_8055_prod_Opred__True,axiom,
    ! [C: $tType,B: $tType,X4: product_prod(B,C)] : aa(product_prod(B,C),$o,basic_pred_prod(B,C,aTP_Lamp_jq(B,$o),aTP_Lamp_alj(C,$o)),X4) ).

% prod.pred_True
tff(fact_8056_sort__conv__fold,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,B,aTP_Lamp_hr(B,B)),Xs) = aa(list(B),list(B),fold(B,list(B),linorder_insort_key(B,B,aTP_Lamp_hr(B,B)),Xs),nil(B)) ) ).

% sort_conv_fold
tff(fact_8057_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_gcd(nat),Xs),zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_8058_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y2: set(product_prod(int,int))] :
          ( ( X = aa(set(product_prod(int,int)),rat,abs_rat,Y2) )
         => ~ aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y2),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o))) ) ).

% Abs_rat_cases
tff(fact_8059_Abs__rat__induct,axiom,
    ! [Pa: fun(rat,$o),X: rat] :
      ( ! [Y2: set(product_prod(int,int))] :
          ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y2),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o)))
         => aa(rat,$o,Pa,aa(set(product_prod(int,int)),rat,abs_rat,Y2)) )
     => aa(rat,$o,Pa,X) ) ).

% Abs_rat_induct
tff(fact_8060_int_Odomain,axiom,
    ! [DR1: fun(nat,$o),DR2: fun(nat,$o)] :
      ( ( aa(fun(nat,fun(nat,$o)),fun(nat,$o),domainp(nat,nat),fequal(nat)) = DR1 )
     => ( ( aa(fun(nat,fun(nat,$o)),fun(nat,$o),domainp(nat,nat),fequal(nat)) = DR2 )
       => ( aa(fun(product_prod(nat,nat),fun(int,$o)),fun(product_prod(nat,nat),$o),domainp(product_prod(nat,nat),int),pcr_int) = basic_pred_prod(nat,nat,DR1,DR2) ) ) ) ).

% int.domain
tff(fact_8061_Abs__rat__inject,axiom,
    ! [X: set(product_prod(int,int)),Y: set(product_prod(int,int))] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),X),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o)))
     => ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o)))
       => ( ( 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_8062_Abs__rat__inverse,axiom,
    ! [Y: set(product_prod(int,int))] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o)))
     => ( 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_8063_type__definition__rat,axiom,
    type_definition(rat,set(product_prod(int,int)),rep_rat,abs_rat,aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o))) ).

% type_definition_rat
tff(fact_8064_Rep__rat,axiom,
    ! [X: rat] : aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),aa(rat,set(product_prod(int,int)),rep_rat,X)),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o))) ).

% Rep_rat
tff(fact_8065_Rep__rat__cases,axiom,
    ! [Y: set(product_prod(int,int))] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o)))
     => ~ ! [X2: rat] : Y != aa(rat,set(product_prod(int,int)),rep_rat,X2) ) ).

% Rep_rat_cases
tff(fact_8066_Rep__rat__induct,axiom,
    ! [Y: set(product_prod(int,int)),Pa: fun(set(product_prod(int,int)),$o)] :
      ( aa(set(set(product_prod(int,int))),$o,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),$o),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_arf(set(product_prod(int,int)),$o)))
     => ( ! [X2: rat] : aa(set(product_prod(int,int)),$o,Pa,aa(rat,set(product_prod(int,int)),rep_rat,X2))
       => aa(set(product_prod(int,int)),$o,Pa,Y) ) ) ).

% Rep_rat_induct
tff(fact_8067_prod_Orel__compp__Grp,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,R12: fun(B,fun(D,$o)),R23: fun(C,fun(E,$o))] : basic_rel_prod(B,D,C,E,R12,R23) = aa(fun(product_prod(product_prod(B,D),product_prod(C,E)),fun(product_prod(D,E),$o)),fun(product_prod(B,C),fun(product_prod(D,E),$o)),aa(fun(product_prod(B,C),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o)),fun(fun(product_prod(product_prod(B,D),product_prod(C,E)),fun(product_prod(D,E),$o)),fun(product_prod(B,C),fun(product_prod(D,E),$o))),relcompp(product_prod(B,C),product_prod(product_prod(B,D),product_prod(C,E)),product_prod(D,E)),conversep(product_prod(product_prod(B,D),product_prod(C,E)),product_prod(B,C),bNF_Grp(product_prod(product_prod(B,D),product_prod(C,E)),product_prod(B,C),aa(fun(product_prod(product_prod(B,D),product_prod(C,E)),$o),set(product_prod(product_prod(B,D),product_prod(C,E))),collect(product_prod(product_prod(B,D),product_prod(C,E))),aa(fun(C,fun(E,$o)),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o),aTP_Lamp_arg(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o)),R12),R23)),product_map_prod(product_prod(B,D),B,product_prod(C,E),C,product_fst(B,D),product_fst(C,E))))),bNF_Grp(product_prod(product_prod(B,D),product_prod(C,E)),product_prod(D,E),aa(fun(product_prod(product_prod(B,D),product_prod(C,E)),$o),set(product_prod(product_prod(B,D),product_prod(C,E))),collect(product_prod(product_prod(B,D),product_prod(C,E))),aa(fun(C,fun(E,$o)),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o),aTP_Lamp_arg(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o)),R12),R23)),product_map_prod(product_prod(B,D),D,product_prod(C,E),E,product_snd(B,D),product_snd(C,E)))) ).

% prod.rel_compp_Grp
tff(fact_8068_convol__transfer,axiom,
    ! [B: $tType,C: $tType,F6: $tType,G4: $tType,E: $tType,D: $tType,R2: fun(B,fun(D,$o)),S: fun(C,fun(E,$o)),T4: fun(F6,fun(G4,$o))] : aa(fun(fun(D,E),fun(fun(D,G4),fun(D,product_prod(E,G4)))),$o,aa(fun(fun(B,C),fun(fun(B,F6),fun(B,product_prod(C,F6)))),fun(fun(fun(D,E),fun(fun(D,G4),fun(D,product_prod(E,G4)))),$o),bNF_rel_fun(fun(B,C),fun(D,E),fun(fun(B,F6),fun(B,product_prod(C,F6))),fun(fun(D,G4),fun(D,product_prod(E,G4))),bNF_rel_fun(B,D,C,E,R2,S),bNF_rel_fun(fun(B,F6),fun(D,G4),fun(B,product_prod(C,F6)),fun(D,product_prod(E,G4)),bNF_rel_fun(B,D,F6,G4,R2,T4),bNF_rel_fun(B,D,product_prod(C,F6),product_prod(E,G4),R2,basic_rel_prod(C,E,F6,G4,S,T4)))),bNF_convol(B,C,F6)),bNF_convol(D,E,G4)) ).

% convol_transfer
tff(fact_8069_pcr__Domainp,axiom,
    ! [D: $tType,C: $tType,B: $tType,B4: fun(B,fun(C,$o)),Pa: fun(B,$o),A4: fun(D,fun(B,$o))] :
      ( ( aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),B4) = Pa )
     => ! [X4: D] :
          ( aa(D,$o,aa(fun(D,fun(C,$o)),fun(D,$o),domainp(D,C),aa(fun(B,fun(C,$o)),fun(D,fun(C,$o)),aa(fun(D,fun(B,$o)),fun(fun(B,fun(C,$o)),fun(D,fun(C,$o))),relcompp(D,B,C),A4),B4)),X4)
        <=> ? [Y5: B] :
              ( aa(B,$o,aa(D,fun(B,$o),A4,X4),Y5)
              & aa(B,$o,Pa,Y5) ) ) ) ).

% pcr_Domainp
tff(fact_8070_prod_Orel__Grp,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,A14: set(B),F1: fun(B,D),A25: set(C),F22: fun(C,E)] : basic_rel_prod(B,D,C,E,bNF_Grp(B,D,A14,F1),bNF_Grp(C,E,A25,F22)) = bNF_Grp(product_prod(B,C),product_prod(D,E),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(set(C),fun(product_prod(B,C),$o),aTP_Lamp_arh(set(B),fun(set(C),fun(product_prod(B,C),$o)),A14),A25)),product_map_prod(B,D,C,E,F1,F22)) ).

% prod.rel_Grp
tff(fact_8071_OO__Grp__alt,axiom,
    ! [B: $tType,D: $tType,C: $tType,A4: set(D),F: fun(D,B),G: fun(D,C),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),conversep(D,B,bNF_Grp(D,B,A4,F))),bNF_Grp(D,C,A4,G)),X4),Xa)
    <=> ? [Z5: D] :
          ( aa(set(D),$o,member(D,Z5),A4)
          & ( aa(D,B,F,Z5) = X4 )
          & ( aa(D,C,G,Z5) = Xa ) ) ) ).

% OO_Grp_alt
tff(fact_8072_OO__def,axiom,
    ! [B: $tType,D: $tType,C: $tType,R2: fun(B,fun(D,$o)),S: fun(D,fun(C,$o)),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),R2),S),X4),Xa)
    <=> ? [Y5: D] :
          ( aa(D,$o,aa(B,fun(D,$o),R2,X4),Y5)
          & aa(C,$o,aa(D,fun(C,$o),S,Y5),Xa) ) ) ).

% OO_def
tff(fact_8073_fun_Orel__Grp,axiom,
    ! [B: $tType,D: $tType,C: $tType,A4: set(C),F: fun(C,D)] : bNF_rel_fun(B,B,C,D,fequal(B),bNF_Grp(C,D,A4,F)) = bNF_Grp(fun(B,C),fun(B,D),aa(fun(fun(B,C),$o),set(fun(B,C)),collect(fun(B,C)),aTP_Lamp_ari(set(C),fun(fun(B,C),$o),A4)),aa(fun(C,D),fun(fun(B,C),fun(B,D)),comp(C,D,B),F)) ).

% fun.rel_Grp
tff(fact_8074_eq__le__Grp__id__iff,axiom,
    ! [B: $tType,R2: fun(B,$o)] :
      ( aa(fun(B,fun(B,$o)),$o,aa(fun(B,fun(B,$o)),fun(fun(B,fun(B,$o)),$o),ord_less_eq(fun(B,fun(B,$o))),fequal(B)),bNF_Grp(B,B,aa(fun(B,$o),set(B),collect(B),R2),id(B)))
    <=> ! [X_12: B] : aa(B,$o,R2,X_12) ) ).

% eq_le_Grp_id_iff
tff(fact_8075_snd__convol_H,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(D,C),G: fun(D,B),X: D] : aa(product_prod(C,B),B,product_snd(C,B),aa(D,product_prod(C,B),aa(fun(D,B),fun(D,product_prod(C,B)),aa(fun(D,C),fun(fun(D,B),fun(D,product_prod(C,B))),bNF_convol(D,C,B),F),G),X)) = aa(D,B,G,X) ).

% snd_convol'
tff(fact_8076_multiset_Orel__Grp,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C)] : rel_mset(B,C,bNF_Grp(B,C,A4,F)) = bNF_Grp(multiset(B),multiset(C),aa(fun(multiset(B),$o),set(multiset(B)),collect(multiset(B)),aTP_Lamp_arj(set(B),fun(multiset(B),$o),A4)),image_mset(B,C,F)) ).

% multiset.rel_Grp
tff(fact_8077_Grp__def,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),bNF_Grp(B,C,A4,F),X4),Xa)
    <=> ( ( Xa = aa(B,C,F,X4) )
        & aa(set(B),$o,member(B,X4),A4) ) ) ).

% Grp_def
tff(fact_8078_relcompp__SUP__distrib,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,S2: fun(B,fun(D,$o)),R: fun(E,fun(D,fun(C,$o))),I5: set(E)] : aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),S2),aa(set(fun(D,fun(C,$o))),fun(D,fun(C,$o)),complete_Sup_Sup(fun(D,fun(C,$o))),aa(set(E),set(fun(D,fun(C,$o))),image2(E,fun(D,fun(C,$o)),R),I5))) = aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(E),set(fun(B,fun(C,$o))),image2(E,fun(B,fun(C,$o)),aa(fun(E,fun(D,fun(C,$o))),fun(E,fun(B,fun(C,$o))),aTP_Lamp_ark(fun(B,fun(D,$o)),fun(fun(E,fun(D,fun(C,$o))),fun(E,fun(B,fun(C,$o)))),S2),R)),I5)) ).

% relcompp_SUP_distrib
tff(fact_8079_relcompp__SUP__distrib2,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,R: fun(E,fun(B,fun(D,$o))),I5: set(E),S2: fun(D,fun(C,$o))] : aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),aa(set(fun(B,fun(D,$o))),fun(B,fun(D,$o)),complete_Sup_Sup(fun(B,fun(D,$o))),aa(set(E),set(fun(B,fun(D,$o))),image2(E,fun(B,fun(D,$o)),R),I5))),S2) = aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(E),set(fun(B,fun(C,$o))),image2(E,fun(B,fun(C,$o)),aa(fun(D,fun(C,$o)),fun(E,fun(B,fun(C,$o))),aTP_Lamp_arl(fun(E,fun(B,fun(D,$o))),fun(fun(D,fun(C,$o)),fun(E,fun(B,fun(C,$o)))),R),S2)),I5)) ).

% relcompp_SUP_distrib2
tff(fact_8080_relpowp__add,axiom,
    ! [B: $tType,Ma: nat,N: nat,Pa: fun(B,fun(B,$o))] : aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ma),N)),Pa) = aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(fun(B,fun(B,$o)),fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),relcompp(B,B,B),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),Ma),Pa)),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aa(nat,fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),compow(fun(B,fun(B,$o))),N),Pa)) ).

% relpowp_add
tff(fact_8081_relcompp__relcomp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,R: set(product_prod(B,D)),S2: set(product_prod(D,C)),X4: B,Xa: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),aTP_Lamp_arm(set(product_prod(B,D)),fun(B,fun(D,$o)),R)),aTP_Lamp_arn(set(product_prod(D,C)),fun(D,fun(C,$o)),S2)),X4),Xa)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Xa)),relcomp(B,D,C,R,S2)) ) ).

% relcompp_relcomp_eq
tff(fact_8082_fst__convol_H,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(D,B),G: fun(D,C),X: D] : aa(product_prod(B,C),B,product_fst(B,C),aa(D,product_prod(B,C),aa(fun(D,C),fun(D,product_prod(B,C)),aa(fun(D,B),fun(fun(D,C),fun(D,product_prod(B,C))),bNF_convol(D,B,C),F),G),X)) = aa(D,B,F,X) ).

% fst_convol'
tff(fact_8083_list_Orel__Grp,axiom,
    ! [C: $tType,B: $tType,A4: set(B),F: fun(B,C)] : list_all2(B,C,bNF_Grp(B,C,A4,F)) = bNF_Grp(list(B),list(C),aa(fun(list(B),$o),set(list(B)),collect(list(B)),aTP_Lamp_aro(set(B),fun(list(B),$o),A4)),map(B,C,F)) ).

% list.rel_Grp
tff(fact_8084_relcomp__def,axiom,
    ! [B: $tType,D: $tType,C: $tType,X4: set(product_prod(B,C)),Xa: set(product_prod(C,D))] : relcomp(B,C,D,X4,Xa) = aa(fun(product_prod(B,D),$o),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,$o)),fun(product_prod(B,D),$o),product_case_prod(B,D,$o),aa(fun(C,fun(D,$o)),fun(B,fun(D,$o)),aa(fun(B,fun(C,$o)),fun(fun(C,fun(D,$o)),fun(B,fun(D,$o))),relcompp(B,C,D),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),X4)),aTP_Lamp_arp(set(product_prod(C,D)),fun(C,fun(D,$o)),Xa)))) ).

% relcomp_def
tff(fact_8085_map__prod__o__convol,axiom,
    ! [E: $tType,C: $tType,D: $tType,F6: $tType,B: $tType,H1: fun(E,C),H22: fun(F6,D),F: fun(B,E),G: fun(B,F6)] : aa(fun(B,product_prod(E,F6)),fun(B,product_prod(C,D)),aa(fun(product_prod(E,F6),product_prod(C,D)),fun(fun(B,product_prod(E,F6)),fun(B,product_prod(C,D))),comp(product_prod(E,F6),product_prod(C,D),B),product_map_prod(E,C,F6,D,H1,H22)),aa(fun(B,F6),fun(B,product_prod(E,F6)),aa(fun(B,E),fun(fun(B,F6),fun(B,product_prod(E,F6))),bNF_convol(B,E,F6),F),G)) = aa(fun(B,D),fun(B,product_prod(C,D)),aa(fun(B,C),fun(fun(B,D),fun(B,product_prod(C,D))),bNF_convol(B,C,D),aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),H1),F)),aa(fun(B,F6),fun(B,D),aa(fun(F6,D),fun(fun(B,F6),fun(B,D)),comp(F6,D,B),H22),G)) ).

% map_prod_o_convol
tff(fact_8086_convol__o,axiom,
    ! [C: $tType,D: $tType,E: $tType,B: $tType,F: fun(E,C),G: fun(E,D),H: fun(B,E)] : aa(fun(B,E),fun(B,product_prod(C,D)),aa(fun(E,product_prod(C,D)),fun(fun(B,E),fun(B,product_prod(C,D))),comp(E,product_prod(C,D),B),aa(fun(E,D),fun(E,product_prod(C,D)),aa(fun(E,C),fun(fun(E,D),fun(E,product_prod(C,D))),bNF_convol(E,C,D),F),G)),H) = aa(fun(B,D),fun(B,product_prod(C,D)),aa(fun(B,C),fun(fun(B,D),fun(B,product_prod(C,D))),bNF_convol(B,C,D),aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),F),H)),aa(fun(B,E),fun(B,D),aa(fun(E,D),fun(fun(B,E),fun(B,D)),comp(E,D,B),G),H)) ).

% convol_o
tff(fact_8087_map__prod__o__convol__id,axiom,
    ! [C: $tType,B: $tType,D: $tType,F: fun(D,B),G: fun(D,C),X: D] : aa(D,product_prod(B,C),aa(fun(D,product_prod(D,C)),fun(D,product_prod(B,C)),aa(fun(product_prod(D,C),product_prod(B,C)),fun(fun(D,product_prod(D,C)),fun(D,product_prod(B,C))),comp(product_prod(D,C),product_prod(B,C),D),product_map_prod(D,B,C,C,F,id(C))),aa(fun(D,C),fun(D,product_prod(D,C)),aa(fun(D,D),fun(fun(D,C),fun(D,product_prod(D,C))),bNF_convol(D,D,C),id(D)),G)),X) = aa(D,product_prod(B,C),aa(fun(D,C),fun(D,product_prod(B,C)),aa(fun(D,B),fun(fun(D,C),fun(D,product_prod(B,C))),bNF_convol(D,B,C),aa(fun(D,B),fun(D,B),aa(fun(B,B),fun(fun(D,B),fun(D,B)),comp(B,B,D),id(B)),F)),G),X) ).

% map_prod_o_convol_id
tff(fact_8088_convol__expand__snd,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,product_prod(C,D)),G: fun(B,C)] :
      ( ( aa(fun(B,product_prod(C,D)),fun(B,C),aa(fun(product_prod(C,D),C),fun(fun(B,product_prod(C,D)),fun(B,C)),comp(product_prod(C,D),C,B),product_fst(C,D)),F) = G )
     => ( aa(fun(B,D),fun(B,product_prod(C,D)),aa(fun(B,C),fun(fun(B,D),fun(B,product_prod(C,D))),bNF_convol(B,C,D),G),aa(fun(B,product_prod(C,D)),fun(B,D),aa(fun(product_prod(C,D),D),fun(fun(B,product_prod(C,D)),fun(B,D)),comp(product_prod(C,D),D,B),product_snd(C,D)),F)) = F ) ) ).

% convol_expand_snd
tff(fact_8089_convol__expand__snd_H,axiom,
    ! [D: $tType,C: $tType,B: $tType,F: fun(B,product_prod(C,D)),G: fun(B,C),H: fun(B,D)] :
      ( ( aa(fun(B,product_prod(C,D)),fun(B,C),aa(fun(product_prod(C,D),C),fun(fun(B,product_prod(C,D)),fun(B,C)),comp(product_prod(C,D),C,B),product_fst(C,D)),F) = G )
     => ( ( H = aa(fun(B,product_prod(C,D)),fun(B,D),aa(fun(product_prod(C,D),D),fun(fun(B,product_prod(C,D)),fun(B,D)),comp(product_prod(C,D),D,B),product_snd(C,D)),F) )
      <=> ( aa(fun(B,D),fun(B,product_prod(C,D)),aa(fun(B,C),fun(fun(B,D),fun(B,product_prod(C,D))),bNF_convol(B,C,D),G),H) = F ) ) ) ).

% convol_expand_snd'
tff(fact_8090_type__copy__vimage2p__Grp__Abs,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,Rep: fun(B,C),Abs: fun(C,B),G: fun(D,E),Pa: fun(E,$o),H: fun(E,B)] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ( aa(fun(E,fun(B,$o)),fun(D,fun(C,$o)),bNF_vimage2p(D,E,C,B,$o,G,Abs),bNF_Grp(E,B,aa(fun(E,$o),set(E),collect(E),Pa),H)) = bNF_Grp(D,C,aa(fun(D,$o),set(D),collect(D),aa(fun(E,$o),fun(D,$o),aTP_Lamp_arq(fun(D,E),fun(fun(E,$o),fun(D,$o)),G),Pa)),aa(fun(D,E),fun(D,C),aa(fun(E,C),fun(fun(D,E),fun(D,C)),comp(E,C,D),aa(fun(E,B),fun(E,C),aa(fun(B,C),fun(fun(E,B),fun(E,C)),comp(B,C,E),Rep),H)),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_8091_type__copy__vimage2p__Grp__Rep,axiom,
    ! [C: $tType,B: $tType,E: $tType,D: $tType,Rep: fun(B,C),Abs: fun(C,B),F: fun(D,E),Pa: fun(E,$o),H: fun(E,C)] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ( aa(fun(E,fun(C,$o)),fun(D,fun(B,$o)),bNF_vimage2p(D,E,B,C,$o,F,Rep),bNF_Grp(E,C,aa(fun(E,$o),set(E),collect(E),Pa),H)) = bNF_Grp(D,B,aa(fun(D,$o),set(D),collect(D),aa(fun(E,$o),fun(D,$o),aTP_Lamp_arq(fun(D,E),fun(fun(E,$o),fun(D,$o)),F),Pa)),aa(fun(D,E),fun(D,B),aa(fun(E,B),fun(fun(D,E),fun(D,B)),comp(E,B,D),aa(fun(E,C),fun(E,B),aa(fun(C,B),fun(fun(E,C),fun(E,B)),comp(C,B,E),Abs),H)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_8092_vimage2p__rel__fun,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,F: fun(B,D),G: fun(C,E),R2: fun(D,fun(E,$o))] : aa(fun(C,E),$o,aa(fun(B,D),fun(fun(C,E),$o),bNF_rel_fun(B,C,D,E,aa(fun(D,fun(E,$o)),fun(B,fun(C,$o)),bNF_vimage2p(B,D,C,E,$o,F,G),R2),R2),F),G) ).

% vimage2p_rel_fun
tff(fact_8093_vimage2p__def,axiom,
    ! [B: $tType,E: $tType,D: $tType,F6: $tType,C: $tType,F: fun(B,E),G: fun(C,F6),R2: fun(E,fun(F6,D)),X4: B,Xa: C] : aa(C,D,aa(B,fun(C,D),aa(fun(E,fun(F6,D)),fun(B,fun(C,D)),bNF_vimage2p(B,E,C,F6,D,F,G),R2),X4),Xa) = aa(F6,D,aa(E,fun(F6,D),R2,aa(B,E,F,X4)),aa(C,F6,G,Xa)) ).

% vimage2p_def
tff(fact_8094_vimage2p__refl,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(B,$o)),F: fun(C,B),X: C] :
      ( ! [X2: B] : aa(B,$o,aa(B,fun(B,$o),R2,X2),X2)
     => aa(C,$o,aa(C,fun(C,$o),aa(fun(B,fun(B,$o)),fun(C,fun(C,$o)),bNF_vimage2p(C,B,C,B,$o,F,F),R2),X),X) ) ).

% vimage2p_refl
tff(fact_8095_vimage2p__id,axiom,
    ! [D: $tType,C: $tType,B: $tType,R2: fun(B,fun(C,D))] : aa(fun(B,fun(C,D)),fun(B,fun(C,D)),bNF_vimage2p(B,B,C,C,D,id(B),id(C)),R2) = R2 ).

% vimage2p_id
tff(fact_8096_vimage2p__comp,axiom,
    ! [F6: $tType,E: $tType,G4: $tType,B: $tType,D: $tType,C: $tType,H8: $tType,F1: fun(G4,B),F22: fun(E,G4),G1: fun(H8,C),G22: fun(F6,H8)] : bNF_vimage2p(E,B,F6,C,D,aa(fun(E,G4),fun(E,B),aa(fun(G4,B),fun(fun(E,G4),fun(E,B)),comp(G4,B,E),F1),F22),aa(fun(F6,H8),fun(F6,C),aa(fun(H8,C),fun(fun(F6,H8),fun(F6,C)),comp(H8,C,F6),G1),G22)) = aa(fun(fun(B,fun(C,D)),fun(G4,fun(H8,D))),fun(fun(B,fun(C,D)),fun(E,fun(F6,D))),aa(fun(fun(G4,fun(H8,D)),fun(E,fun(F6,D))),fun(fun(fun(B,fun(C,D)),fun(G4,fun(H8,D))),fun(fun(B,fun(C,D)),fun(E,fun(F6,D)))),comp(fun(G4,fun(H8,D)),fun(E,fun(F6,D)),fun(B,fun(C,D))),bNF_vimage2p(E,G4,F6,H8,D,F22,G22)),bNF_vimage2p(G4,B,H8,C,D,F1,G1)) ).

% vimage2p_comp
tff(fact_8097_vimage2p__mono,axiom,
    ! [B: $tType,E: $tType,C: $tType,D: $tType,F: fun(B,C),G: fun(D,E),R2: fun(C,fun(E,$o)),X: B,Y: D,S: fun(C,fun(E,$o))] :
      ( aa(D,$o,aa(B,fun(D,$o),aa(fun(C,fun(E,$o)),fun(B,fun(D,$o)),bNF_vimage2p(B,C,D,E,$o,F,G),R2),X),Y)
     => ( aa(fun(C,fun(E,$o)),$o,aa(fun(C,fun(E,$o)),fun(fun(C,fun(E,$o)),$o),ord_less_eq(fun(C,fun(E,$o))),R2),S)
       => aa(D,$o,aa(B,fun(D,$o),aa(fun(C,fun(E,$o)),fun(B,fun(D,$o)),bNF_vimage2p(B,C,D,E,$o,F,G),S),X),Y) ) ) ).

% vimage2p_mono
tff(fact_8098_list_Orel__compp__Grp,axiom,
    ! [C: $tType,B: $tType,R2: fun(B,fun(C,$o))] : list_all2(B,C,R2) = aa(fun(list(product_prod(B,C)),fun(list(C),$o)),fun(list(B),fun(list(C),$o)),aa(fun(list(B),fun(list(product_prod(B,C)),$o)),fun(fun(list(product_prod(B,C)),fun(list(C),$o)),fun(list(B),fun(list(C),$o))),relcompp(list(B),list(product_prod(B,C)),list(C)),conversep(list(product_prod(B,C)),list(B),bNF_Grp(list(product_prod(B,C)),list(B),aa(fun(list(product_prod(B,C)),$o),set(list(product_prod(B,C))),collect(list(product_prod(B,C))),aTP_Lamp_agr(fun(B,fun(C,$o)),fun(list(product_prod(B,C)),$o),R2)),map(product_prod(B,C),B,product_fst(B,C))))),bNF_Grp(list(product_prod(B,C)),list(C),aa(fun(list(product_prod(B,C)),$o),set(list(product_prod(B,C))),collect(list(product_prod(B,C))),aTP_Lamp_agr(fun(B,fun(C,$o)),fun(list(product_prod(B,C)),$o),R2)),map(product_prod(B,C),C,product_snd(B,C)))) ).

% list.rel_compp_Grp
tff(fact_8099_Abs__transfer,axiom,
    ! [C: $tType,B: $tType,D: $tType,E: $tType,Rep1: fun(B,C),Abs1: fun(C,B),Rep22: fun(D,E),Abs22: fun(E,D),R2: fun(C,fun(E,$o))] :
      ( type_definition(B,C,Rep1,Abs1,top_top(set(C)))
     => ( type_definition(D,E,Rep22,Abs22,top_top(set(E)))
       => aa(fun(E,D),$o,aa(fun(C,B),fun(fun(E,D),$o),bNF_rel_fun(C,E,B,D,R2,aa(fun(C,fun(E,$o)),fun(B,fun(D,$o)),bNF_vimage2p(B,C,D,E,$o,Rep1,Rep22),R2)),Abs1),Abs22) ) ) ).

% Abs_transfer
tff(fact_8100_fun_Orel__compp__Grp,axiom,
    ! [B: $tType,D: $tType,C: $tType,R2: fun(C,fun(D,$o))] : bNF_rel_fun(B,B,C,D,fequal(B),R2) = aa(fun(fun(B,product_prod(C,D)),fun(fun(B,D),$o)),fun(fun(B,C),fun(fun(B,D),$o)),aa(fun(fun(B,C),fun(fun(B,product_prod(C,D)),$o)),fun(fun(fun(B,product_prod(C,D)),fun(fun(B,D),$o)),fun(fun(B,C),fun(fun(B,D),$o))),relcompp(fun(B,C),fun(B,product_prod(C,D)),fun(B,D)),conversep(fun(B,product_prod(C,D)),fun(B,C),bNF_Grp(fun(B,product_prod(C,D)),fun(B,C),aa(fun(fun(B,product_prod(C,D)),$o),set(fun(B,product_prod(C,D))),collect(fun(B,product_prod(C,D))),aTP_Lamp_agd(fun(C,fun(D,$o)),fun(fun(B,product_prod(C,D)),$o),R2)),aa(fun(product_prod(C,D),C),fun(fun(B,product_prod(C,D)),fun(B,C)),comp(product_prod(C,D),C,B),product_fst(C,D))))),bNF_Grp(fun(B,product_prod(C,D)),fun(B,D),aa(fun(fun(B,product_prod(C,D)),$o),set(fun(B,product_prod(C,D))),collect(fun(B,product_prod(C,D))),aTP_Lamp_agd(fun(C,fun(D,$o)),fun(fun(B,product_prod(C,D)),$o),R2)),aa(fun(product_prod(C,D),D),fun(fun(B,product_prod(C,D)),fun(B,D)),comp(product_prod(C,D),D,B),product_snd(C,D)))) ).

% fun.rel_compp_Grp
tff(fact_8101_multiset_Orel__compp__Grp,axiom,
    ! [C: $tType,B: $tType,R2: fun(B,fun(C,$o))] : rel_mset(B,C,R2) = aa(fun(multiset(product_prod(B,C)),fun(multiset(C),$o)),fun(multiset(B),fun(multiset(C),$o)),aa(fun(multiset(B),fun(multiset(product_prod(B,C)),$o)),fun(fun(multiset(product_prod(B,C)),fun(multiset(C),$o)),fun(multiset(B),fun(multiset(C),$o))),relcompp(multiset(B),multiset(product_prod(B,C)),multiset(C)),conversep(multiset(product_prod(B,C)),multiset(B),bNF_Grp(multiset(product_prod(B,C)),multiset(B),aa(fun(multiset(product_prod(B,C)),$o),set(multiset(product_prod(B,C))),collect(multiset(product_prod(B,C))),aTP_Lamp_ahi(fun(B,fun(C,$o)),fun(multiset(product_prod(B,C)),$o),R2)),image_mset(product_prod(B,C),B,product_fst(B,C))))),bNF_Grp(multiset(product_prod(B,C)),multiset(C),aa(fun(multiset(product_prod(B,C)),$o),set(multiset(product_prod(B,C))),collect(multiset(product_prod(B,C))),aTP_Lamp_ahi(fun(B,fun(C,$o)),fun(multiset(product_prod(B,C)),$o),R2)),image_mset(product_prod(B,C),C,product_snd(B,C)))) ).

% multiset.rel_compp_Grp
tff(fact_8102_pcr__int__def,axiom,
    pcr_int = aa(fun(product_prod(nat,nat),fun(int,$o)),fun(product_prod(nat,nat),fun(int,$o)),aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(int,$o)),fun(product_prod(nat,nat),fun(int,$o))),relcompp(product_prod(nat,nat),product_prod(nat,nat),int),basic_rel_prod(nat,nat,nat,nat,fequal(nat),fequal(nat))),cr_int) ).

% pcr_int_def
tff(fact_8103_right__total__relcompp__transfer,axiom,
    ! [B: $tType,F6: $tType,D: $tType,G4: $tType,C: $tType,E: $tType,B4: fun(B,fun(C,$o)),A4: fun(D,fun(E,$o)),C6: fun(F6,fun(G4,$o))] :
      ( right_total(B,C,B4)
     => aa(fun(fun(E,fun(C,$o)),fun(fun(C,fun(G4,$o)),fun(E,fun(G4,$o)))),$o,aa(fun(fun(D,fun(B,$o)),fun(fun(B,fun(F6,$o)),fun(D,fun(F6,$o)))),fun(fun(fun(E,fun(C,$o)),fun(fun(C,fun(G4,$o)),fun(E,fun(G4,$o)))),$o),bNF_rel_fun(fun(D,fun(B,$o)),fun(E,fun(C,$o)),fun(fun(B,fun(F6,$o)),fun(D,fun(F6,$o))),fun(fun(C,fun(G4,$o)),fun(E,fun(G4,$o))),bNF_rel_fun(D,E,fun(B,$o),fun(C,$o),A4,bNF_rel_fun(B,C,$o,$o,B4,fequal($o))),bNF_rel_fun(fun(B,fun(F6,$o)),fun(C,fun(G4,$o)),fun(D,fun(F6,$o)),fun(E,fun(G4,$o)),bNF_rel_fun(B,C,fun(F6,$o),fun(G4,$o),B4,bNF_rel_fun(F6,G4,$o,$o,C6,fequal($o))),bNF_rel_fun(D,E,fun(F6,$o),fun(G4,$o),A4,bNF_rel_fun(F6,G4,$o,$o,C6,fequal($o))))),aTP_Lamp_arr(fun(B,fun(C,$o)),fun(fun(D,fun(B,$o)),fun(fun(B,fun(F6,$o)),fun(D,fun(F6,$o)))),B4)),relcompp(E,C,G4)) ) ).

% right_total_relcompp_transfer
tff(fact_8104_finite__Collect__bex,axiom,
    ! [C: $tType,B: $tType,A4: set(B),Q: fun(C,fun(B,$o))] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_ars(set(B),fun(fun(C,fun(B,$o)),fun(C,$o)),A4),Q)))
      <=> ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
           => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Q),X3))) ) ) ) ).

% finite_Collect_bex
tff(fact_8105_Image__Collect__case__prod,axiom,
    ! [B: $tType,C: $tType,Pa: fun(C,fun(B,$o)),A4: set(C)] : aa(set(C),set(B),image(C,B,aa(fun(product_prod(C,B),$o),set(product_prod(C,B)),collect(product_prod(C,B)),aa(fun(C,fun(B,$o)),fun(product_prod(C,B),$o),product_case_prod(C,B,$o),Pa))),A4) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_art(fun(C,fun(B,$o)),fun(set(C),fun(B,$o)),Pa),A4)) ).

% Image_Collect_case_prod
tff(fact_8106_SUP__bool__eq,axiom,
    ! [B: $tType] : aTP_Lamp_aru(set(B),fun(fun(B,$o),$o)) = bex(B) ).

% SUP_bool_eq
tff(fact_8107_nths__nths,axiom,
    ! [B: $tType,Xs: list(B),A4: set(nat),B4: set(nat)] : nths(B,nths(B,Xs,A4),B4) = nths(B,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_arw(set(nat),fun(set(nat),fun(nat,$o)),A4),B4))) ).

% nths_nths
tff(fact_8108_Bex__fold,axiom,
    ! [B: $tType,A4: set(B),Pa: fun(B,$o)] :
      ( aa(set(B),$o,finite_finite2(B),A4)
     => ( ? [X3: B] :
            ( aa(set(B),$o,member(B,X3),A4)
            & aa(B,$o,Pa,X3) )
      <=> finite_fold(B,$o,aTP_Lamp_arx(fun(B,$o),fun(B,fun($o,$o)),Pa),$false,A4) ) ) ).

% Bex_fold
tff(fact_8109_rel__set__def,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(C,$o)),X4: set(B),Xa: set(C)] :
      ( aa(set(C),$o,aa(set(B),fun(set(C),$o),bNF_rel_set(B,C,R2),X4),Xa)
    <=> ( ! [Xb4: B] :
            ( aa(set(B),$o,member(B,Xb4),X4)
           => ? [Xc2: C] :
                ( aa(set(C),$o,member(C,Xc2),Xa)
                & aa(C,$o,aa(B,fun(C,$o),R2,Xb4),Xc2) ) )
        & ! [Xb4: C] :
            ( aa(set(C),$o,member(C,Xb4),Xa)
           => ? [Xc2: B] :
                ( aa(set(B),$o,member(B,Xc2),X4)
                & aa(C,$o,aa(B,fun(C,$o),R2,Xc2),Xb4) ) ) ) ) ).

% rel_set_def
tff(fact_8110_Union__eq,axiom,
    ! [B: $tType,A4: set(set(B))] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A4) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ary(set(set(B)),fun(B,$o),A4)) ).

% Union_eq
tff(fact_8111_UNION__eq,axiom,
    ! [B: $tType,C: $tType,B4: fun(C,set(B)),A4: set(C)] : aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B4),A4)) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_arz(fun(C,set(B)),fun(set(C),fun(B,$o)),B4),A4)) ).

% UNION_eq
tff(fact_8112_Collect__bex__eq,axiom,
    ! [B: $tType,C: $tType,A4: set(C),Pa: fun(B,fun(C,$o))] : aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),aTP_Lamp_asa(set(C),fun(fun(B,fun(C,$o)),fun(B,$o)),A4),Pa)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aTP_Lamp_yq(fun(B,fun(C,$o)),fun(C,set(B)),Pa)),A4)) ).

% Collect_bex_eq
tff(fact_8113_vimage__image__eq,axiom,
    ! [C: $tType,B: $tType,F: fun(B,C),A4: set(B)] : aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),F),aa(set(B),set(C),image2(B,C,F),A4)) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_asb(fun(B,C),fun(set(B),fun(B,$o)),F),A4)) ).

% vimage_image_eq
tff(fact_8114_image__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,B),A4: set(C)] : aa(set(C),set(B),image2(C,B,F),A4) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_asc(fun(C,B),fun(set(C),fun(B,$o)),F),A4)) ).

% image_def
tff(fact_8115_Image__def,axiom,
    ! [B: $tType,C: $tType,R: set(product_prod(C,B)),S2: set(C)] : aa(set(C),set(B),image(C,B,R),S2) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_asd(set(product_prod(C,B)),fun(set(C),fun(B,$o)),R),S2)) ).

% Image_def
tff(fact_8116_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_8117_right__total__Domainp__transfer,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType,B4: fun(B,fun(C,$o)),A4: fun(D,fun(E,$o))] :
      ( right_total(B,C,B4)
     => aa(fun(fun(E,fun(C,$o)),fun(E,$o)),$o,aa(fun(fun(D,fun(B,$o)),fun(D,$o)),fun(fun(fun(E,fun(C,$o)),fun(E,$o)),$o),bNF_rel_fun(fun(D,fun(B,$o)),fun(E,fun(C,$o)),fun(D,$o),fun(E,$o),bNF_rel_fun(D,E,fun(B,$o),fun(C,$o),A4,bNF_rel_fun(B,C,$o,$o,B4,fequal($o))),bNF_rel_fun(D,E,$o,$o,A4,fequal($o))),aTP_Lamp_ase(fun(B,fun(C,$o)),fun(fun(D,fun(B,$o)),fun(D,$o)),B4)),domainp(E,C)) ) ).

% right_total_Domainp_transfer
tff(fact_8118_cr__int__def,axiom,
    ! [X4: product_prod(nat,nat)] : aa(product_prod(nat,nat),fun(int,$o),cr_int,X4) = aa(int,fun(int,$o),fequal(int),aa(product_prod(nat,nat),int,abs_Integ,X4)) ).

% cr_int_def
tff(fact_8119_multp__code__def,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),N5: multiset(B),M: multiset(B)] :
      ( multp_code(B,Pa,N5,M)
    <=> $let(
          z: multiset(B),
          z:= aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),M),N5),
          $let(
            x: multiset(B),
            x:= aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),z),
            ( ( x != zero_zero(multiset(B)) )
            & ! [X3: B] :
                ( aa(set(B),$o,member(B,X3),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),N5),z)))
               => ? [Y5: B] :
                    ( aa(set(B),$o,member(B,Y5),aa(multiset(B),set(B),set_mset(B),x))
                    & aa(B,$o,aa(B,fun(B,$o),Pa,X3),Y5) ) ) ) ) ) ) ).

% multp_code_def
tff(fact_8120_multeqp__code__def,axiom,
    ! [B: $tType,Pa: fun(B,fun(B,$o)),N5: multiset(B),M: multiset(B)] :
      ( multeqp_code(B,Pa,N5,M)
    <=> $let(
          z: multiset(B),
          z:= aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),M),N5),
          ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),N5),z)))
           => ? [Y5: B] :
                ( aa(set(B),$o,member(B,Y5),aa(multiset(B),set(B),set_mset(B),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),minus_minus(multiset(B)),M),z)))
                & aa(B,$o,aa(B,fun(B,$o),Pa,X3),Y5) ) ) ) ) ).

% multeqp_code_def
tff(fact_8121_map__project__def,axiom,
    ! [B: $tType,C: $tType,F: fun(C,option(B)),A4: set(C)] : map_project(C,B,F,A4) = aa(fun(B,$o),set(B),collect(B),aa(set(C),fun(B,$o),aTP_Lamp_asf(fun(C,option(B)),fun(set(C),fun(B,$o)),F),A4)) ).

% map_project_def
tff(fact_8122_min__ext__def,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] : min_ext(B,R) = aa(fun(product_prod(set(B),set(B)),$o),set(product_prod(set(B),set(B))),collect(product_prod(set(B),set(B))),aTP_Lamp_asg(set(product_prod(B,B)),fun(product_prod(set(B),set(B)),$o),R)) ).

% min_ext_def
tff(fact_8123_max__ext__eq,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] : max_ext(B,R2) = aa(fun(product_prod(set(B),set(B)),$o),set(product_prod(set(B),set(B))),collect(product_prod(set(B),set(B))),aa(fun(set(B),fun(set(B),$o)),fun(product_prod(set(B),set(B)),$o),product_case_prod(set(B),set(B),$o),aTP_Lamp_ash(set(product_prod(B,B)),fun(set(B),fun(set(B),$o)),R2))) ).

% max_ext_eq
tff(fact_8124_max__extp_Omax__extI,axiom,
    ! [B: $tType,X5: set(B),Y4: set(B),R2: fun(B,fun(B,$o))] :
      ( aa(set(B),$o,finite_finite2(B),X5)
     => ( aa(set(B),$o,finite_finite2(B),Y4)
       => ( ( Y4 != aa(fun(B,$o),set(B),collect(B),bot_bot(fun(B,$o))) )
         => ( ! [X2: B] :
                ( aa(set(B),$o,member(B,X2),X5)
               => ? [Xa: B] :
                    ( aa(set(B),$o,member(B,Xa),Y4)
                    & aa(B,$o,aa(B,fun(B,$o),R2,X2),Xa) ) )
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),max_extp(B,R2),X5),Y4) ) ) ) ) ).

% max_extp.max_extI
tff(fact_8125_max__extp__max__ext__eq,axiom,
    ! [B: $tType,R2: set(product_prod(B,B)),X4: set(B),Xa: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),max_extp(B,aTP_Lamp_aab(set(product_prod(B,B)),fun(B,fun(B,$o)),R2)),X4),Xa)
    <=> aa(set(product_prod(set(B),set(B))),$o,member(product_prod(set(B),set(B)),aa(set(B),product_prod(set(B),set(B)),aa(set(B),fun(set(B),product_prod(set(B),set(B))),product_Pair(set(B),set(B)),X4),Xa)),max_ext(B,R2)) ) ).

% max_extp_max_ext_eq
tff(fact_8126_max__ext__def,axiom,
    ! [B: $tType,X4: set(product_prod(B,B))] : max_ext(B,X4) = aa(fun(product_prod(set(B),set(B)),$o),set(product_prod(set(B),set(B))),collect(product_prod(set(B),set(B))),aa(fun(set(B),fun(set(B),$o)),fun(product_prod(set(B),set(B)),$o),product_case_prod(set(B),set(B),$o),max_extp(B,aTP_Lamp_aab(set(product_prod(B,B)),fun(B,fun(B,$o)),X4)))) ).

% max_ext_def
tff(fact_8127_max__extp__eq,axiom,
    ! [B: $tType,R: fun(B,fun(B,$o)),X: set(B),Y: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),max_extp(B,R),X),Y)
    <=> aa(set(product_prod(set(B),set(B))),$o,member(product_prod(set(B),set(B)),aa(set(B),product_prod(set(B),set(B)),aa(set(B),fun(set(B),product_prod(set(B),set(B))),product_Pair(set(B),set(B)),X),Y)),max_ext(B,aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),R)))) ) ).

% max_extp_eq
tff(fact_8128_max__extp_Ocases,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),A12: set(B),A23: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),max_extp(B,R2),A12),A23)
     => ~ ( aa(set(B),$o,finite_finite2(B),A12)
         => ( aa(set(B),$o,finite_finite2(B),A23)
           => ( ( A23 != aa(fun(B,$o),set(B),collect(B),bot_bot(fun(B,$o))) )
             => ~ ! [X4: B] :
                    ( aa(set(B),$o,member(B,X4),A12)
                   => ? [Xa3: B] :
                        ( aa(set(B),$o,member(B,Xa3),A23)
                        & aa(B,$o,aa(B,fun(B,$o),R2,X4),Xa3) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_8129_max__extp_Osimps,axiom,
    ! [B: $tType,R2: fun(B,fun(B,$o)),A12: set(B),A23: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),max_extp(B,R2),A12),A23)
    <=> ( aa(set(B),$o,finite_finite2(B),A12)
        & aa(set(B),$o,finite_finite2(B),A23)
        & ( A23 != aa(fun(B,$o),set(B),collect(B),bot_bot(fun(B,$o))) )
        & ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),A12)
           => ? [Xa4: B] :
                ( aa(set(B),$o,member(B,Xa4),A23)
                & aa(B,$o,aa(B,fun(B,$o),R2,X3),Xa4) ) ) ) ) ).

% max_extp.simps
tff(fact_8130_Quotient__int,axiom,
    quotient(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_8131_lcm__code__integer,axiom,
    ! [A2: code_integer,B2: code_integer] : gcd_lcm(code_integer,A2,B2) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),A2)),aa(code_integer,code_integer,abs_abs(code_integer),B2))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),A2),B2)) ).

% lcm_code_integer
tff(fact_8132_lcm__left__idem,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : gcd_lcm(B,A2,gcd_lcm(B,A2,B2)) = gcd_lcm(B,A2,B2) ) ).

% lcm_left_idem
tff(fact_8133_lcm__right__idem,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : gcd_lcm(B,gcd_lcm(B,A2,B2),B2) = gcd_lcm(B,A2,B2) ) ).

% lcm_right_idem
tff(fact_8134_lcm_Obottom__left__bottom,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B] : gcd_lcm(B,zero_zero(B),A2) = zero_zero(B) ) ).

% lcm.bottom_left_bottom
tff(fact_8135_lcm_Obottom__right__bottom,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B] : gcd_lcm(B,A2,zero_zero(B)) = zero_zero(B) ) ).

% lcm.bottom_right_bottom
tff(fact_8136_lcm__neg2,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,B2: B] : gcd_lcm(B,A2,aa(B,B,uminus_uminus(B),B2)) = gcd_lcm(B,A2,B2) ) ).

% lcm_neg2
tff(fact_8137_lcm__neg1,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,B2: B] : gcd_lcm(B,aa(B,B,uminus_uminus(B),A2),B2) = gcd_lcm(B,A2,B2) ) ).

% lcm_neg1
tff(fact_8138_dvd__lcm1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),gcd_lcm(B,A2,B2)) ) ).

% dvd_lcm1
tff(fact_8139_dvd__lcm2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,A2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),gcd_lcm(B,A2,B2)) ) ).

% dvd_lcm2
tff(fact_8140_lcm__least__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_lcm(B,A2,B2)),C2)
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
            & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2) ) ) ) ).

% lcm_least_iff
tff(fact_8141_lcm__neg__numeral__2,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [A2: B,N: num] : gcd_lcm(B,A2,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N))) = gcd_lcm(B,A2,aa(num,B,numeral_numeral(B),N)) ) ).

% lcm_neg_numeral_2
tff(fact_8142_lcm__neg__numeral__1,axiom,
    ! [B: $tType] :
      ( ring_gcd(B)
     => ! [N: num,A2: B] : gcd_lcm(B,aa(B,B,uminus_uminus(B),aa(num,B,numeral_numeral(B),N)),A2) = gcd_lcm(B,aa(num,B,numeral_numeral(B),N),A2) ) ).

% lcm_neg_numeral_1
tff(fact_8143_lcm__eq__1__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( gcd_lcm(B,A2,B2) = one_one(B) )
        <=> ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
            & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),one_one(B)) ) ) ) ).

% lcm_eq_1_iff
tff(fact_8144_open__typedef__to__Quotient,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),Pa: fun(C,$o),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,aa(fun(C,$o),set(C),collect(C),Pa))
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => quotient(C,B,bNF_eq_onp(C,Pa),Abs,Rep,T4) ) ) ).

% open_typedef_to_Quotient
tff(fact_8145_typedef__to__Quotient,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),S: set(C),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,S)
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => quotient(C,B,bNF_eq_onp(C,aTP_Lamp_al(set(C),fun(C,$o),S)),Abs,Rep,T4) ) ) ).

% typedef_to_Quotient
tff(fact_8146_UNIV__typedef__to__Quotient,axiom,
    ! [B: $tType,C: $tType,Rep: fun(B,C),Abs: fun(C,B),T4: fun(C,fun(B,$o))] :
      ( type_definition(B,C,Rep,Abs,top_top(set(C)))
     => ( ! [X2: C,Xa3: B] :
            ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
          <=> ( X2 = aa(B,C,Rep,Xa3) ) )
       => quotient(C,B,fequal(C),Abs,Rep,T4) ) ) ).

% UNIV_typedef_to_Quotient
tff(fact_8147_lcm__div__unit1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( gcd_lcm(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),B2),A2),C2) = gcd_lcm(B,B2,C2) ) ) ) ).

% lcm_div_unit1
tff(fact_8148_lcm__div__unit2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( gcd_lcm(B,B2,aa(B,B,aa(B,fun(B,B),divide_divide(B),C2),A2)) = gcd_lcm(B,B2,C2) ) ) ) ).

% lcm_div_unit2
tff(fact_8149_zero__eq__lcm__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( zero_zero(B) = gcd_lcm(B,A2,B2) )
        <=> ( ( A2 = zero_zero(B) )
            | ( B2 = zero_zero(B) ) ) ) ) ).

% zero_eq_lcm_iff
tff(fact_8150_lcm__eq__0__iff,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] :
          ( ( gcd_lcm(B,A2,B2) = zero_zero(B) )
        <=> ( ( A2 = zero_zero(B) )
            | ( B2 = zero_zero(B) ) ) ) ) ).

% lcm_eq_0_iff
tff(fact_8151_Quotient__cr__rel,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(B,$o)),Abs: fun(B,C),Rep: fun(C,B),T4: fun(B,fun(C,$o))] :
      ( quotient(B,C,R2,Abs,Rep,T4)
     => ! [X4: B,Xa: C] :
          ( aa(C,$o,aa(B,fun(C,$o),T4,X4),Xa)
        <=> ( aa(B,$o,aa(B,fun(B,$o),R2,X4),X4)
            & ( aa(B,C,Abs,X4) = Xa ) ) ) ) ).

% Quotient_cr_rel
tff(fact_8152_Quotient__def,axiom,
    ! [B: $tType,C: $tType,R2: fun(B,fun(B,$o)),Abs: fun(B,C),Rep: fun(C,B),T4: fun(B,fun(C,$o))] :
      ( quotient(B,C,R2,Abs,Rep,T4)
    <=> ( ! [A6: C] : aa(B,C,Abs,aa(C,B,Rep,A6)) = A6
        & ! [A6: C] : aa(B,$o,aa(B,fun(B,$o),R2,aa(C,B,Rep,A6)),aa(C,B,Rep,A6))
        & ! [R6: B,S7: B] :
            ( aa(B,$o,aa(B,fun(B,$o),R2,R6),S7)
          <=> ( aa(B,$o,aa(B,fun(B,$o),R2,R6),R6)
              & aa(B,$o,aa(B,fun(B,$o),R2,S7),S7)
              & ( aa(B,C,Abs,R6) = aa(B,C,Abs,S7) ) ) )
        & ! [X3: B,Xa4: C] :
            ( aa(C,$o,aa(B,fun(C,$o),T4,X3),Xa4)
          <=> ( aa(B,$o,aa(B,fun(B,$o),R2,X3),X3)
              & ( aa(B,C,Abs,X3) = Xa4 ) ) ) ) ) ).

% Quotient_def
tff(fact_8153_QuotientI,axiom,
    ! [B: $tType,C: $tType,Abs: fun(C,B),Rep: fun(B,C),R2: fun(C,fun(C,$o)),T4: fun(C,fun(B,$o))] :
      ( ! [A3: B] : aa(C,B,Abs,aa(B,C,Rep,A3)) = A3
     => ( ! [A3: B] : aa(C,$o,aa(C,fun(C,$o),R2,aa(B,C,Rep,A3)),aa(B,C,Rep,A3))
       => ( ! [R5: C,S3: C] :
              ( aa(C,$o,aa(C,fun(C,$o),R2,R5),S3)
            <=> ( aa(C,$o,aa(C,fun(C,$o),R2,R5),R5)
                & aa(C,$o,aa(C,fun(C,$o),R2,S3),S3)
                & ( aa(C,B,Abs,R5) = aa(C,B,Abs,S3) ) ) )
         => ( ! [X2: C,Xa3: B] :
                ( aa(B,$o,aa(C,fun(B,$o),T4,X2),Xa3)
              <=> ( aa(C,$o,aa(C,fun(C,$o),R2,X2),X2)
                  & ( aa(C,B,Abs,X2) = Xa3 ) ) )
           => quotient(C,B,R2,Abs,Rep,T4) ) ) ) ) ).

% QuotientI
tff(fact_8154_gcd__dvd__lcm,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(B,B,aa(B,fun(B,B),gcd_gcd(B),A2),B2)),gcd_lcm(B,A2,B2)) ) ).

% gcd_dvd_lcm
tff(fact_8155_lcm_Oassoc,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] : gcd_lcm(B,gcd_lcm(B,A2,B2),C2) = gcd_lcm(B,A2,gcd_lcm(B,B2,C2)) ) ).

% lcm.assoc
tff(fact_8156_lcm_Ocommute,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B] : gcd_lcm(B,A2,B2) = gcd_lcm(B,B2,A2) ) ).

% lcm.commute
tff(fact_8157_lcm_Oleft__commute,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [B2: B,A2: B,C2: B] : gcd_lcm(B,B2,gcd_lcm(B,A2,C2)) = gcd_lcm(B,A2,gcd_lcm(B,B2,C2)) ) ).

% lcm.left_commute
tff(fact_8158_lcm__mono,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B,D2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),D2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_lcm(B,A2,B2)),gcd_lcm(B,C2,D2)) ) ) ) ).

% lcm_mono
tff(fact_8159_dvd__lcmI1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),B2)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),gcd_lcm(B,B2,C2)) ) ) ).

% dvd_lcmI1
tff(fact_8160_dvd__lcmI2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),gcd_lcm(B,B2,C2)) ) ) ).

% dvd_lcmI2
tff(fact_8161_lcm__dvdD1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_lcm(B,A2,B2)),C2)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2) ) ) ).

% lcm_dvdD1
tff(fact_8162_lcm__dvdD2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_lcm(B,A2,B2)),C2)
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2) ) ) ).

% lcm_dvdD2
tff(fact_8163_lcm__least,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,C2: B,B2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),C2)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),C2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_lcm(B,A2,B2)),C2) ) ) ) ).

% lcm_least
tff(fact_8164_lcm__mult__unit1,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( gcd_lcm(B,aa(B,B,aa(B,fun(B,B),times_times(B),B2),A2),C2) = gcd_lcm(B,B2,C2) ) ) ) ).

% lcm_mult_unit1
tff(fact_8165_lcm__mult__unit2,axiom,
    ! [B: $tType] :
      ( semiring_gcd(B)
     => ! [A2: B,B2: B,C2: B] :
          ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),A2),one_one(B))
         => ( gcd_lcm(B,B2,aa(B,B,aa(B,fun(B,B),times_times(B),C2),A2)) = gcd_lcm(B,B2,C2) ) ) ) ).

% lcm_mult_unit2
tff(fact_8166_Quotient__eq__onp__typedef,axiom,
    ! [C: $tType,B: $tType,Pa: fun(B,$o),Abs: fun(B,C),Rep: fun(C,B),Cr: fun(B,fun(C,$o))] :
      ( quotient(B,C,bNF_eq_onp(B,Pa),Abs,Rep,Cr)
     => type_definition(C,B,Rep,Abs,aa(fun(B,$o),set(B),collect(B),Pa)) ) ).

% Quotient_eq_onp_typedef
tff(fact_8167_Quotient__crel__typedef,axiom,
    ! [B: $tType,C: $tType,Pa: fun(B,$o),Abs: fun(B,C),Rep: fun(C,B),T4: fun(B,fun(C,$o))] :
      ( quotient(B,C,bNF_eq_onp(B,Pa),Abs,Rep,T4)
     => ! [X4: B,Xa: C] :
          ( aa(C,$o,aa(B,fun(C,$o),T4,X4),Xa)
        <=> ( X4 = aa(C,B,Rep,Xa) ) ) ) ).

% Quotient_crel_typedef
tff(fact_8168_lcm__0__iff__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( ( gcd_lcm(nat,Ma,N) = zero_zero(nat) )
    <=> ( ( Ma = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% lcm_0_iff_nat
tff(fact_8169_lcm__0__iff__int,axiom,
    ! [Ma: int,N: int] :
      ( ( gcd_lcm(int,Ma,N) = zero_zero(int) )
    <=> ( ( Ma = zero_zero(int) )
        | ( N = zero_zero(int) ) ) ) ).

% lcm_0_iff_int
tff(fact_8170_lcm__proj2__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),Y)
     => ( gcd_lcm(nat,X,Y) = Y ) ) ).

% lcm_proj2_if_dvd_nat
tff(fact_8171_lcm__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),X),Y)
     => ( gcd_lcm(nat,Y,X) = Y ) ) ).

% lcm_proj1_if_dvd_nat
tff(fact_8172_lcm__proj2__iff__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( ( gcd_lcm(nat,Ma,N) = N )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Ma),N) ) ).

% lcm_proj2_iff_nat
tff(fact_8173_lcm__proj1__iff__nat,axiom,
    ! [Ma: nat,N: nat] :
      ( ( gcd_lcm(nat,Ma,N) = Ma )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),N),Ma) ) ).

% lcm_proj1_iff_nat
tff(fact_8174_lcm__int__int__eq,axiom,
    ! [Ma: nat,N: nat] : gcd_lcm(int,aa(nat,int,semiring_1_of_nat(int),Ma),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),gcd_lcm(nat,Ma,N)) ).

% lcm_int_int_eq
tff(fact_8175_abs__lcm__int,axiom,
    ! [I: int,J: int] : aa(int,int,abs_abs(int),gcd_lcm(int,I,J)) = gcd_lcm(int,I,J) ).

% abs_lcm_int
tff(fact_8176_lcm__abs1__int,axiom,
    ! [X: int,Y: int] : gcd_lcm(int,aa(int,int,abs_abs(int),X),Y) = gcd_lcm(int,X,Y) ).

% lcm_abs1_int
tff(fact_8177_lcm__abs2__int,axiom,
    ! [X: int,Y: int] : gcd_lcm(int,X,aa(int,int,abs_abs(int),Y)) = gcd_lcm(int,X,Y) ).

% lcm_abs2_int
tff(fact_8178_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] :
      aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_tt(product_prod(int,int),product_prod(int,int)),Uu) = $ite(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_8179_ATP_Olambda__2,axiom,
    ! [B: $tType,Uu: set(set(B))] : aa(set(set(B)),int,aTP_Lamp_ys(set(set(B)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(B)),nat,finite_card(set(B)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),Uu)))) ).

% ATP.lambda_2
tff(fact_8180_ATP_Olambda__3,axiom,
    ! [B: $tType,Uu: B] : aa(B,set(product_prod(B,B)),aTP_Lamp_aae(B,set(product_prod(B,B))),Uu) = aa(set(product_prod(B,B)),set(product_prod(B,B)),insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu)),bot_bot(set(product_prod(B,B)))) ).

% ATP.lambda_3
tff(fact_8181_ATP_Olambda__4,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Uu: B] :
          ( aa(B,$o,aTP_Lamp_acs(B,$o),Uu)
        <=> ( aa(set(B),$o,member(B,Uu),ring_1_Ints(B))
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),Uu) ) ) ) ).

% ATP.lambda_4
tff(fact_8182_ATP_Olambda__5,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),B,aTP_Lamp_tx(product_prod(int,int),B),Uu) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(int,B,ring_1_of_int(B),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(int,B,ring_1_of_int(B),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_5
tff(fact_8183_ATP_Olambda__6,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_tv(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_6
tff(fact_8184_ATP_Olambda__7,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_hy(nat,$o),Uu)
        <=> ( aa(nat,B,semiring_1_of_nat(B),Uu) = zero_zero(B) ) ) ) ).

% ATP.lambda_7
tff(fact_8185_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_kc(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_8
tff(fact_8186_ATP_Olambda__9,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_ty(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).

% ATP.lambda_9
tff(fact_8187_ATP_Olambda__10,axiom,
    ! [C: $tType,Uu: C] : aa(C,product_prod(C,C),aTP_Lamp_tf(C,product_prod(C,C)),Uu) = aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),Uu),Uu) ).

% ATP.lambda_10
tff(fact_8188_ATP_Olambda__11,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_sk(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_11
tff(fact_8189_ATP_Olambda__12,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_arc(product_prod(int,int),$o),Uu)
    <=> aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Uu),Uu) ) ).

% ATP.lambda_12
tff(fact_8190_ATP_Olambda__13,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_ag(B,B),Uu) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),one_one(B)) ) ).

% ATP.lambda_13
tff(fact_8191_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_agg(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_14
tff(fact_8192_ATP_Olambda__15,axiom,
    ! [B: $tType,Uu: B] : aa(B,multiset(B),aTP_Lamp_agk(B,multiset(B)),Uu) = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),Uu),zero_zero(multiset(B))) ).

% ATP.lambda_15
tff(fact_8193_ATP_Olambda__16,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B] : aa(B,list(B),aTP_Lamp_rk(B,list(B)),Uu) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),nil(B)) ) ).

% ATP.lambda_16
tff(fact_8194_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: B] : aa(B,list(B),aTP_Lamp_jp(B,list(B)),Uu) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),nil(B)) ).

% ATP.lambda_17
tff(fact_8195_ATP_Olambda__18,axiom,
    ! [B: $tType,Uu: B] : aa(B,set(B),aTP_Lamp_xj(B,set(B)),Uu) = aa(set(B),set(B),insert(B,Uu),bot_bot(set(B))) ).

% ATP.lambda_18
tff(fact_8196_ATP_Olambda__19,axiom,
    ! [D: $tType,B: $tType,Uu: fun(B,D)] : aa(fun(B,D),set(D),aTP_Lamp_aix(fun(B,D),set(D)),Uu) = aa(set(B),set(D),image2(B,D,Uu),top_top(set(B))) ).

% ATP.lambda_19
tff(fact_8197_ATP_Olambda__20,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C)] : aa(fun(B,C),set(C),aTP_Lamp_aiw(fun(B,C),set(C)),Uu) = aa(set(B),set(C),image2(B,C,Uu),top_top(set(B))) ).

% ATP.lambda_20
tff(fact_8198_ATP_Olambda__21,axiom,
    ! [C: $tType,Uu: list(C)] :
      ( aa(list(C),$o,aTP_Lamp_agt(list(C),$o),Uu)
    <=> ( Uu = nil(C) ) ) ).

% ATP.lambda_21
tff(fact_8199_ATP_Olambda__22,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ags(list(B),$o),Uu)
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_22
tff(fact_8200_ATP_Olambda__23,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_tr(product_prod(int,int),$o),Uu)
    <=> aa(int,$o,aa(int,fun(int,$o),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_23
tff(fact_8201_ATP_Olambda__24,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_apx(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_apw(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_24
tff(fact_8202_ATP_Olambda__25,axiom,
    ! [C: $tType,Uu: list(C)] : aa(list(C),fun(nat,nat),aTP_Lamp_po(list(C),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(C),nat,size_size(list(C)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_25
tff(fact_8203_ATP_Olambda__26,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(set(product_prod(B,B)),set(product_prod(B,B))),aTP_Lamp_acd(B,fun(set(product_prod(B,B)),set(product_prod(B,B)))),Uu) = insert(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu)) ).

% ATP.lambda_26
tff(fact_8204_ATP_Olambda__27,axiom,
    ! [C: $tType,Uu: list(C)] :
      ( aa(list(C),$o,aTP_Lamp_pp(list(C),$o),Uu)
    <=> ( Uu != nil(C) ) ) ).

% ATP.lambda_27
tff(fact_8205_ATP_Olambda__28,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_pt(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_28
tff(fact_8206_ATP_Olambda__29,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(C,D),product_prod(product_prod(B,C),D)),aTP_Lamp_aou(B,fun(product_prod(C,D),product_prod(product_prod(B,C),D))),Uu) = aa(fun(C,fun(D,product_prod(product_prod(B,C),D))),fun(product_prod(C,D),product_prod(product_prod(B,C),D)),product_case_prod(C,D,product_prod(product_prod(B,C),D)),aTP_Lamp_aot(B,fun(C,fun(D,product_prod(product_prod(B,C),D))),Uu)) ).

% ATP.lambda_29
tff(fact_8207_ATP_Olambda__30,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,option(C))] : aa(fun(B,option(C)),fun(product_prod(B,C),fun(B,option(C))),aTP_Lamp_ss(fun(B,option(C)),fun(product_prod(B,C),fun(B,option(C)))),Uu) = aa(fun(B,fun(C,fun(B,option(C)))),fun(product_prod(B,C),fun(B,option(C))),product_case_prod(B,C,fun(B,option(C))),aTP_Lamp_sr(fun(B,option(C)),fun(B,fun(C,fun(B,option(C)))),Uu)) ).

% ATP.lambda_30
tff(fact_8208_ATP_Olambda__31,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: C] : aa(C,fun(product_prod(B,D),product_prod(B,product_prod(C,D))),aTP_Lamp_sh(C,fun(product_prod(B,D),product_prod(B,product_prod(C,D)))),Uu) = aa(fun(B,fun(D,product_prod(B,product_prod(C,D)))),fun(product_prod(B,D),product_prod(B,product_prod(C,D))),product_case_prod(B,D,product_prod(B,product_prod(C,D))),aTP_Lamp_sg(C,fun(B,fun(D,product_prod(B,product_prod(C,D)))),Uu)) ).

% ATP.lambda_31
tff(fact_8209_ATP_Olambda__32,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),B),aTP_Lamp_ni(B,fun(product_prod(heap_ext(product_unit),nat),B)),Uu) = aa(fun(heap_ext(product_unit),fun(nat,B)),fun(product_prod(heap_ext(product_unit),nat),B),product_case_prod(heap_ext(product_unit),nat,B),aTP_Lamp_nh(B,fun(heap_ext(product_unit),fun(nat,B)),Uu)) ).

% ATP.lambda_32
tff(fact_8210_ATP_Olambda__33,axiom,
    ! [B: $tType,Uu: multiset(B)] : aa(multiset(B),set(B),aTP_Lamp_aht(multiset(B),set(B)),Uu) = aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ahk(multiset(B),fun(B,$o),Uu)) ).

% ATP.lambda_33
tff(fact_8211_ATP_Olambda__34,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_aaj(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ga(nat,fun(nat,$o),Uu)) ).

% ATP.lambda_34
tff(fact_8212_ATP_Olambda__35,axiom,
    ! [C: $tType,Uu: fun(C,nat)] :
      ( aa(fun(C,nat),$o,aTP_Lamp_ali(fun(C,nat),$o),Uu)
    <=> aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aTP_Lamp_alh(fun(C,nat),fun(C,$o),Uu))) ) ).

% ATP.lambda_35
tff(fact_8213_ATP_Olambda__36,axiom,
    ! [B: $tType,Uu: fun(B,nat)] :
      ( aa(fun(B,nat),$o,aTP_Lamp_ahn(fun(B,nat),$o),Uu)
    <=> aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ft(fun(B,nat),fun(B,$o),Uu))) ) ).

% ATP.lambda_36
tff(fact_8214_ATP_Olambda__37,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C)] : aa(fun(B,C),set(product_prod(B,C)),aTP_Lamp_afj(fun(B,C),set(product_prod(B,C))),Uu) = aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aTP_Lamp_afi(fun(B,C),fun(B,fun(C,$o)),Uu))) ).

% ATP.lambda_37
tff(fact_8215_ATP_Olambda__38,axiom,
    ! [C: $tType,Uu: list(C)] : aa(list(C),fun(nat,nat),aTP_Lamp_pn(list(C),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(C),nat,size_size(list(C)),Uu)) ).

% ATP.lambda_38
tff(fact_8216_ATP_Olambda__39,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ps(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_39
tff(fact_8217_ATP_Olambda__40,axiom,
    ! [B: $tType] :
      ( semiring_1(B)
     => ! [Uu: nat] : aa(nat,heap_Time_Heap(B),aTP_Lamp_alg(nat,heap_Time_Heap(B)),Uu) = aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),aa(nat,B,semiring_1_of_nat(B),Uu)) ) ).

% ATP.lambda_40
tff(fact_8218_ATP_Olambda__41,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_jb(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_41
tff(fact_8219_ATP_Olambda__42,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_jo(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_42
tff(fact_8220_ATP_Olambda__43,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_om(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_43
tff(fact_8221_ATP_Olambda__44,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_on(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_44
tff(fact_8222_ATP_Olambda__45,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_mg(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_45
tff(fact_8223_ATP_Olambda__46,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B] : aa(B,filter(B),aTP_Lamp_aeu(B,filter(B)),Uu) = principal(B,aa(B,set(B),set_ord_atLeast(B),Uu)) ) ).

% ATP.lambda_46
tff(fact_8224_ATP_Olambda__47,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Uu: B] : aa(B,filter(B),aTP_Lamp_aes(B,filter(B)),Uu) = principal(B,aa(B,set(B),set_ord_atLeast(B),Uu)) ) ).

% ATP.lambda_47
tff(fact_8225_ATP_Olambda__48,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B] : aa(B,filter(B),aTP_Lamp_aev(B,filter(B)),Uu) = principal(B,aa(B,set(B),set_ord_atMost(B),Uu)) ) ).

% ATP.lambda_48
tff(fact_8226_ATP_Olambda__49,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Uu: B] : aa(B,filter(B),aTP_Lamp_aer(B,filter(B)),Uu) = principal(B,aa(B,set(B),set_ord_atMost(B),Uu)) ) ).

% ATP.lambda_49
tff(fact_8227_ATP_Olambda__50,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_jk(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_50
tff(fact_8228_ATP_Olambda__51,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Uu: B] :
          ( aa(B,$o,aTP_Lamp_act(B,$o),Uu)
        <=> ? [N4: int] :
              ( ( Uu = aa(int,B,ring_1_of_int(B),N4) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N4) ) ) ) ).

% ATP.lambda_51
tff(fact_8229_ATP_Olambda__52,axiom,
    ! [Uu: set(product_prod(int,int))] :
      ( aa(set(product_prod(int,int)),$o,aTP_Lamp_arf(set(product_prod(int,int)),$o),Uu)
    <=> ? [X3: product_prod(int,int)] :
          ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X3),X3)
          & ( Uu = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X3)) ) ) ) ).

% ATP.lambda_52
tff(fact_8230_ATP_Olambda__53,axiom,
    ! [B: $tType,C: $tType,Uu: product_prod(product_prod($o,B),product_prod($o,C))] :
      ( aa(product_prod(product_prod($o,B),product_prod($o,C)),$o,aTP_Lamp_abu(product_prod(product_prod($o,B),product_prod($o,C)),$o),Uu)
    <=> ? [X3: B,Y5: C] : Uu = aa(product_prod($o,C),product_prod(product_prod($o,B),product_prod($o,C)),aa(product_prod($o,B),fun(product_prod($o,C),product_prod(product_prod($o,B),product_prod($o,C))),product_Pair(product_prod($o,B),product_prod($o,C)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$true),X3)),aa(C,product_prod($o,C),aa($o,fun(C,product_prod($o,C)),product_Pair($o,C),$false),Y5)) ) ).

% ATP.lambda_53
tff(fact_8231_ATP_Olambda__54,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),aTP_Lamp_nl(B,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit))),Uu) = aa(fun(heap_ext(product_unit),fun(nat,heap_ext(product_unit))),fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),product_case_prod(heap_ext(product_unit),nat,heap_ext(product_unit)),aTP_Lamp_nk(heap_ext(product_unit),fun(nat,heap_ext(product_unit)))) ).

% ATP.lambda_54
tff(fact_8232_ATP_Olambda__55,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),nat),aTP_Lamp_mz(B,fun(product_prod(heap_ext(product_unit),nat),nat)),Uu) = aa(fun(heap_ext(product_unit),fun(nat,nat)),fun(product_prod(heap_ext(product_unit),nat),nat),product_case_prod(heap_ext(product_unit),nat,nat),aTP_Lamp_my(heap_ext(product_unit),fun(nat,nat))) ).

% ATP.lambda_55
tff(fact_8233_ATP_Olambda__56,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: array(B)] : aa(array(B),heap_Time_Heap(product_unit),aTP_Lamp_aps(array(B),heap_Time_Heap(product_unit)),Uu) = aa(product_unit,heap_Time_Heap(product_unit),heap_Time_ureturn(product_unit),product_Unity) ) ).

% ATP.lambda_56
tff(fact_8234_ATP_Olambda__57,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_mb(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),aa(fun(num,option(num)),fun(num,option(num)),aa(fun(num,option(num)),fun(fun(num,option(num)),fun(num,option(num))),aa(option(num),fun(fun(num,option(num)),fun(fun(num,option(num)),fun(num,option(num)))),case_num(option(num)),aa(num,option(num),some(num),one2)),aTP_Lamp_lz(nat,fun(num,option(num)),Uua)),aTP_Lamp_ma(nat,fun(num,option(num)),Uua)),Uu) ).

% ATP.lambda_57
tff(fact_8235_ATP_Olambda__58,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,B,aTP_Lamp_by(nat,fun(nat,B),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(Uu),Uua)),zero_zero(B)) ) ).

% ATP.lambda_58
tff(fact_8236_ATP_Olambda__59,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: product_prod(B,D),Uua: product_prod(D,C)] :
      aa(product_prod(D,C),list(product_prod(B,C)),aTP_Lamp_ua(product_prod(B,D),fun(product_prod(D,C),list(product_prod(B,C))),Uu),Uua) = $ite(aa(product_prod(B,D),D,product_snd(B,D),Uu) = aa(product_prod(D,C),D,product_fst(D,C),Uua),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),aa(product_prod(B,D),B,product_fst(B,D),Uu)),aa(product_prod(D,C),C,product_snd(D,C),Uua))),nil(product_prod(B,C))),nil(product_prod(B,C))) ).

% ATP.lambda_59
tff(fact_8237_ATP_Olambda__60,axiom,
    ! [B: $tType,Uu: list(B),Uua: nat] :
      aa(nat,option(B),aTP_Lamp_jw(list(B),fun(nat,option(B)),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),aa(list(B),nat,size_size(list(B)),Uu)),aa(B,option(B),some(B),aa(nat,B,nth(B,Uu),Uua)),none(B)) ).

% ATP.lambda_60
tff(fact_8238_ATP_Olambda__61,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ow(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_61
tff(fact_8239_ATP_Olambda__62,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_tu(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(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_62
tff(fact_8240_ATP_Olambda__63,axiom,
    ! [B: $tType,Uu: set(fun(B,nat)),Uua: B] :
      aa(B,nat,aa(set(fun(B,nat)),fun(B,nat),aTP_Lamp_ahw(set(fun(B,nat)),fun(B,nat)),Uu),Uua) = $ite(Uu = bot_bot(set(fun(B,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(B,nat)),set(nat),image2(fun(B,nat),nat,aTP_Lamp_aho(B,fun(fun(B,nat),nat),Uua)),Uu))) ).

% ATP.lambda_63
tff(fact_8241_ATP_Olambda__64,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,B,aTP_Lamp_bx(nat,fun(nat,B),Uu),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(Uu),Uua)),zero_zero(B)) ) ).

% ATP.lambda_64
tff(fact_8242_ATP_Olambda__65,axiom,
    ! [B: $tType] :
      ( ( lattice(B)
        & order_top(B) )
     => ! [Uu: fun(B,B),Uua: nat] : aa(nat,B,aTP_Lamp_agh(fun(B,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Uua),Uu),top_top(B)) ) ).

% ATP.lambda_65
tff(fact_8243_ATP_Olambda__66,axiom,
    ! [B: $tType] :
      ( ( lattice(B)
        & order_bot(B) )
     => ! [Uu: fun(B,B),Uua: nat] : aa(nat,B,aTP_Lamp_ady(fun(B,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Uua),Uu),bot_bot(B)) ) ).

% ATP.lambda_66
tff(fact_8244_ATP_Olambda__67,axiom,
    ! [B: $tType,Uu: list(list(B)),Uua: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_agx(list(list(B)),fun(list(B),$o),Uu),Uua)
    <=> aa(list(list(B)),$o,aa(list(B),fun(list(list(B)),$o),list_all2(B,list(B),aTP_Lamp_agw(B,fun(list(B),$o))),Uua),Uu) ) ).

% ATP.lambda_67
tff(fact_8245_ATP_Olambda__68,axiom,
    ! [B: $tType] :
      ( inf(B)
     => ! [Uu: option(B),Uua: B] : aa(B,option(B),aTP_Lamp_ko(option(B),fun(B,option(B)),Uu),Uua) = case_option(option(B),B,none(B),aTP_Lamp_kn(B,fun(B,option(B)),Uua),Uu) ) ).

% ATP.lambda_68
tff(fact_8246_ATP_Olambda__69,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [Uu: option(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_jt(option(B),fun(B,$o),Uu),Uua)
        <=> case_option($o,B,$true,aa(B,fun(B,$o),aTP_Lamp_js(B,fun(B,$o)),Uua),Uu) ) ) ).

% ATP.lambda_69
tff(fact_8247_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_mc(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_mb(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_70
tff(fact_8248_ATP_Olambda__71,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [Uu: option(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ju(option(B),fun(B,$o),Uu),Uua)
        <=> case_option($o,B,$false,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_71
tff(fact_8249_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_lz(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_jo(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_72
tff(fact_8250_ATP_Olambda__73,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aTP_Lamp_tm(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),Uu),Uua)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,aa(product_prod(B,C),B,product_fst(B,C),Uua)),aa(product_prod(B,C),C,product_snd(B,C),Uua)) ) ).

% ATP.lambda_73
tff(fact_8251_ATP_Olambda__74,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(B,fun(B,B)),Uua: B] : aa(B,B,aTP_Lamp_aed(fun(B,fun(B,B)),fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),Uu,Uua),Uua) ) ).

% ATP.lambda_74
tff(fact_8252_ATP_Olambda__75,axiom,
    ! [B: $tType,Uu: list(list(B)),Uua: nat] : aa(nat,list(B),aTP_Lamp_qx(list(list(B)),fun(nat,list(B)),Uu),Uua) = aa(list(nat),list(B),map(nat,B,aa(nat,fun(nat,B),aTP_Lamp_qw(list(list(B)),fun(nat,fun(nat,B)),Uu),Uua)),upt(zero_zero(nat),aa(list(list(B)),nat,size_size(list(list(B))),Uu))) ).

% ATP.lambda_75
tff(fact_8253_ATP_Olambda__76,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat] : aa(nat,B,aTP_Lamp_li(fun(nat,fun(nat,B)),fun(nat,B),Uu),Uua) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_lh(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_76
tff(fact_8254_ATP_Olambda__77,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat] : aa(nat,B,aTP_Lamp_lg(fun(nat,fun(nat,B)),fun(nat,B),Uu),Uua) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_lf(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_77
tff(fact_8255_ATP_Olambda__78,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_ts(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_78
tff(fact_8256_ATP_Olambda__79,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [Uu: nat,Uua: nat] : aa(nat,B,aTP_Lamp_db(nat,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),Uua)),aa(nat,B,semiring_1_of_nat(B),Uua))),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_79
tff(fact_8257_ATP_Olambda__80,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: nat,Uua: nat] : aa(nat,B,aTP_Lamp_dd(nat,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_80
tff(fact_8258_ATP_Olambda__81,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ld(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_81
tff(fact_8259_ATP_Olambda__82,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_tw(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_82
tff(fact_8260_ATP_Olambda__83,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_rq(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_83
tff(fact_8261_ATP_Olambda__84,axiom,
    ! [B: $tType,Uu: set(B),Uua: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_hg(set(B),fun(list(B),$o),Uu),Uua)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uua)),Uu)
        & distinct(B,Uua) ) ) ).

% ATP.lambda_84
tff(fact_8262_ATP_Olambda__85,axiom,
    ! [B: $tType,Uu: set(B),Uua: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_hf(set(B),fun(list(B),$o),Uu),Uua)
    <=> ( ( aa(list(B),set(B),set2(B),Uua) = Uu )
        & distinct(B,Uua) ) ) ).

% ATP.lambda_85
tff(fact_8263_ATP_Olambda__86,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [Uu: nat,Uua: nat] : aa(nat,B,aTP_Lamp_dh(nat,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),Uua)),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_86
tff(fact_8264_ATP_Olambda__87,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_aez(rat,fun(product_prod(int,int),$o),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)) )
        & aa(int,$o,aa(int,fun(int,$o),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_87
tff(fact_8265_ATP_Olambda__88,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B] : aa(B,set(set(B)),aTP_Lamp_afz(set(product_prod(B,B)),fun(B,set(set(B))),Uu),Uua) = aa(set(set(B)),set(set(B)),insert(set(B),aa(set(B),set(B),image(B,B,Uu),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))))),bot_bot(set(set(B)))) ).

% ATP.lambda_88
tff(fact_8266_ATP_Olambda__89,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_ck(B,fun(nat,B),Uu),Uua) = aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uua))),Uua) ) ).

% ATP.lambda_89
tff(fact_8267_ATP_Olambda__90,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_bz(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,Uu),Uua)),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),Uu),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))),aa(nat,B,semiring_1_of_nat(B),Uua))) ) ).

% ATP.lambda_90
tff(fact_8268_ATP_Olambda__91,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_cr(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,Uu),Uua)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),one_one(B))),Uua)) ) ).

% ATP.lambda_91
tff(fact_8269_ATP_Olambda__92,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: array(B),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(list(B),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ald(array(B),fun(heap_ext(product_unit),product_prod(list(B),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(list(B),product_prod(heap_ext(product_unit),nat)),aa(list(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(B),product_prod(heap_ext(product_unit),nat))),product_Pair(list(B),product_prod(heap_ext(product_unit),nat)),array_get(B,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(B,Uua,Uu)))) ) ).

% ATP.lambda_92
tff(fact_8270_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ba(nat,fun(nat,$o)),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_93
tff(fact_8271_ATP_Olambda__94,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(set(B))] :
      ( aa(set(set(B)),$o,aTP_Lamp_yt(set(set(B)),fun(set(set(B)),$o),Uu),Uua)
    <=> ( aa(set(set(B)),$o,aa(set(set(B)),fun(set(set(B)),$o),ord_less_eq(set(set(B))),Uua),Uu)
        & ( Uua != bot_bot(set(set(B))) ) ) ) ).

% ATP.lambda_94
tff(fact_8272_ATP_Olambda__95,axiom,
    ! [B: $tType,Uu: set(option(B)),Uua: option(B)] :
      ( aa(option(B),$o,aTP_Lamp_aha(set(option(B)),fun(option(B),$o),Uu),Uua)
    <=> ( aa(set(option(B)),$o,member(option(B),Uua),Uu)
        & ( Uua != none(B) ) ) ) ).

% ATP.lambda_95
tff(fact_8273_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_cu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% ATP.lambda_96
tff(fact_8274_ATP_Olambda__97,axiom,
    ! [Uu: set(int),Uua: int] :
      ( aa(int,$o,aTP_Lamp_akl(set(int),fun(int,$o),Uu),Uua)
    <=> ( aa(set(int),$o,member(int,Uua),Uu)
        & ! [X3: int] :
            ( aa(set(int),$o,member(int,X3),Uu)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),Uua) ) ) ) ).

% ATP.lambda_97
tff(fact_8275_ATP_Olambda__98,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_akm(set(set(B)),fun(set(B),$o),Uu),Uua)
    <=> ( aa(set(set(B)),$o,member(set(B),Uua),Uu)
        & ! [X3: set(B)] :
            ( aa(set(set(B)),$o,member(set(B),X3),Uu)
           => ~ aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),Uua),X3) ) ) ) ).

% ATP.lambda_98
tff(fact_8276_ATP_Olambda__99,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),aTP_Lamp_adj(set(B),fun(set(B),$o)),Uu),Uua)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),Uu),Uua)
        & aa(set(B),$o,finite_finite2(B),Uua) ) ) ).

% ATP.lambda_99
tff(fact_8277_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aeg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).

% ATP.lambda_100
tff(fact_8278_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ci(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_101
tff(fact_8279_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ch(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_102
tff(fact_8280_ATP_Olambda__103,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: B] : aa(B,set(product_prod(C,B)),aTP_Lamp_abw(C,fun(B,set(product_prod(C,B))),Uu),Uua) = aa(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),Uua)),bot_bot(set(product_prod(C,B)))) ).

% ATP.lambda_103
tff(fact_8281_ATP_Olambda__104,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,set(product_prod(B,C)),aTP_Lamp_wb(B,fun(C,set(product_prod(B,C))),Uu),Uua) = aa(set(product_prod(B,C)),set(product_prod(B,C)),insert(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uua)),bot_bot(set(product_prod(B,C)))) ).

% ATP.lambda_104
tff(fact_8282_ATP_Olambda__105,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_gb(B,fun(B,$o),Uu),Uua)
        <=> ( aa(set(B),$o,member(B,Uua),ring_1_Ints(B))
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),Uua)),Uu) ) ) ) ).

% ATP.lambda_105
tff(fact_8283_ATP_Olambda__106,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(C,fun(D,$o)),Uua: fun(B,product_prod(C,D))] :
      ( aa(fun(B,product_prod(C,D)),$o,aTP_Lamp_agd(fun(C,fun(D,$o)),fun(fun(B,product_prod(C,D)),$o),Uu),Uua)
    <=> aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),aa(set(B),set(product_prod(C,D)),image2(B,product_prod(C,D),Uua),top_top(set(B)))),aa(fun(product_prod(C,D),$o),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),Uu))) ) ).

% ATP.lambda_106
tff(fact_8284_ATP_Olambda__107,axiom,
    ! [B: $tType,Uu: fun(set(B),$o),Uua: set(B)] :
      ( aa(set(B),$o,aa(fun(set(B),$o),fun(set(B),$o),aTP_Lamp_adx(fun(set(B),$o),fun(set(B),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(B)) )
        | ? [A11: set(B),A6: B] :
            ( ( Uua = aa(set(B),set(B),insert(B,A6),A11) )
            & aa(set(B),$o,Uu,A11) ) ) ) ).

% ATP.lambda_107
tff(fact_8285_ATP_Olambda__108,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,C)] :
      ( aa(fun(B,C),$o,aTP_Lamp_ari(set(C),fun(fun(B,C),$o),Uu),Uua)
    <=> aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,Uua),top_top(set(B)))),Uu) ) ).

% ATP.lambda_108
tff(fact_8286_ATP_Olambda__109,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B] : aa(B,set(set(B)),aTP_Lamp_afx(set(product_prod(B,B)),fun(B,set(set(B))),Uu),Uua) = equiv_quotient(B,aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))),Uu) ).

% ATP.lambda_109
tff(fact_8287_ATP_Olambda__110,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: set(B),Uua: list(B)] :
          ( aa(list(B),$o,aTP_Lamp_aqf(set(B),fun(list(B),$o),Uu),Uua)
        <=> ( sorted_wrt(B,ord_less(B),Uua)
            & ( aa(list(B),set(B),set2(B),Uua) = Uu ) ) ) ) ).

% ATP.lambda_110
tff(fact_8288_ATP_Olambda__111,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_zz(set(product_prod(B,B)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(B,B)),nat,finite_card(product_prod(B,B)),Uu)) ) ) ).

% ATP.lambda_111
tff(fact_8289_ATP_Olambda__112,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aad(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_112
tff(fact_8290_ATP_Olambda__113,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(D,product_prod(B,C)),Uua: B] : aa(B,set(C),aTP_Lamp_vo(fun(D,product_prod(B,C)),fun(B,set(C)),Uu),Uua) = aa(set(D),set(C),image2(D,C,aa(fun(D,product_prod(B,C)),fun(D,C),aa(fun(product_prod(B,C),C),fun(fun(D,product_prod(B,C)),fun(D,C)),comp(product_prod(B,C),C,D),product_snd(B,C)),Uu)),top_top(set(D))) ).

% ATP.lambda_113
tff(fact_8291_ATP_Olambda__114,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_ep(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,B,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,one2))),Uua)))) ) ).

% ATP.lambda_114
tff(fact_8292_ATP_Olambda__115,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_cq(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,B,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,one2))),Uua)))) ) ).

% ATP.lambda_115
tff(fact_8293_ATP_Olambda__116,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_cg(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,Uu,aa(nat,nat,suc,Uua))),aa(nat,B,Uu,Uua)) ) ).

% ATP.lambda_116
tff(fact_8294_ATP_Olambda__117,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat] : aa(nat,B,aTP_Lamp_dz(fun(nat,fun(nat,B)),fun(nat,B),Uu),Uua) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_117
tff(fact_8295_ATP_Olambda__118,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat] : aa(nat,B,aTP_Lamp_ej(fun(nat,fun(nat,B)),fun(nat,B),Uu),Uua) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_118
tff(fact_8296_ATP_Olambda__119,axiom,
    ! [B: $tType] :
      ( division_ring(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_ey(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uua)),aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),Uua)) ) ).

% ATP.lambda_119
tff(fact_8297_ATP_Olambda__120,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_co(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,Uu,Uua)),aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_120
tff(fact_8298_ATP_Olambda__121,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_cn(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,Uu,Uua)),aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_121
tff(fact_8299_ATP_Olambda__122,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_cf(fun(nat,B),fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,Uu,Uua)),aa(nat,B,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_122
tff(fact_8300_ATP_Olambda__123,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: fun(B,$o),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ain(fun(B,$o),fun(B,$o),Uu),Uua)
        <=> ( aa(B,$o,Uu,Uua)
            & ! [Y5: B] :
                ( aa(B,$o,Uu,Y5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Y5) ) ) ) ) ).

% ATP.lambda_123
tff(fact_8301_ATP_Olambda__124,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Uu: fun(B,$o),Uua: B] :
          ( aa(B,$o,aTP_Lamp_aio(fun(B,$o),fun(B,$o),Uu),Uua)
        <=> ( aa(B,$o,Uu,Uua)
            & ! [Y5: B] :
                ( aa(B,$o,Uu,Y5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y5),Uua) ) ) ) ) ).

% ATP.lambda_124
tff(fact_8302_ATP_Olambda__125,axiom,
    ! [B: $tType,Uu: fun(multiset(B),$o),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aTP_Lamp_aiq(fun(multiset(B),$o),fun(multiset(B),$o),Uu),Uua)
    <=> ( aa(multiset(B),$o,Uu,Uua)
        & ! [Y5: multiset(B)] :
            ( aa(multiset(B),$o,Uu,Y5)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Uua),Y5) ) ) ) ).

% ATP.lambda_125
tff(fact_8303_ATP_Olambda__126,axiom,
    ! [B: $tType,Uu: fun(multiset(B),$o),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aTP_Lamp_aiv(fun(multiset(B),$o),fun(multiset(B),$o),Uu),Uua)
    <=> ( aa(multiset(B),$o,Uu,Uua)
        & ! [Y5: multiset(B)] :
            ( aa(multiset(B),$o,Uu,Y5)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Y5),Uua) ) ) ) ).

% ATP.lambda_126
tff(fact_8304_ATP_Olambda__127,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat] : aa(nat,B,aTP_Lamp_ii(fun(nat,fun(nat,B)),fun(nat,B),Uu),Uua) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_127
tff(fact_8305_ATP_Olambda__128,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat] : aa(nat,B,aTP_Lamp_ih(fun(nat,fun(nat,B)),fun(nat,B),Uu),Uua) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_128
tff(fact_8306_ATP_Olambda__129,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(B,B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_aee(fun(B,B),fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,Uu,Uua)),Uua) ) ) ).

% ATP.lambda_129
tff(fact_8307_ATP_Olambda__130,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: B] : aa(B,product_prod(C,B),aTP_Lamp_afd(fun(B,C),fun(B,product_prod(C,B)),Uu),Uua) = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(B,C,Uu,Uua)),Uua) ).

% ATP.lambda_130
tff(fact_8308_ATP_Olambda__131,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,B),Uua: C] : aa(C,list(B),aTP_Lamp_px(fun(C,B),fun(C,list(B)),Uu),Uua) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(C,B,Uu,Uua)),nil(B)) ).

% ATP.lambda_131
tff(fact_8309_ATP_Olambda__132,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,B),Uua: C] : aa(C,set(B),aTP_Lamp_yr(fun(C,B),fun(C,set(B)),Uu),Uua) = aa(set(B),set(B),insert(B,aa(C,B,Uu,Uua)),bot_bot(set(B))) ).

% ATP.lambda_132
tff(fact_8310_ATP_Olambda__133,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(B,C),Uua: B] :
          ( aa(B,$o,aTP_Lamp_fr(fun(B,C),fun(B,$o),Uu),Uua)
        <=> ( aa(B,C,Uu,Uua) = zero_zero(C) ) ) ) ).

% ATP.lambda_133
tff(fact_8311_ATP_Olambda__134,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(B,C),Uua: B] :
          ( aa(B,$o,aTP_Lamp_fw(fun(B,C),fun(B,$o),Uu),Uua)
        <=> ( aa(B,C,Uu,Uua) = one_one(C) ) ) ) ).

% ATP.lambda_134
tff(fact_8312_ATP_Olambda__135,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: list(B),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_mt(list(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(B),heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ms(list(B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(B,Uu,Uua)) ) ).

% ATP.lambda_135
tff(fact_8313_ATP_Olambda__136,axiom,
    ! [Uu: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_app(code_integer,fun($o,char)),Uu),(Uua)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_apo($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu)) ).

% ATP.lambda_136
tff(fact_8314_ATP_Olambda__137,axiom,
    ! [B: $tType,Uu: list(B),Uua: list(B)] : aa(list(B),list(list(B)),aTP_Lamp_qk(list(B),fun(list(B),list(list(B))),Uu),Uua) = aa(list(B),list(list(B)),map(B,list(B),aa(list(B),fun(B,list(B)),aTP_Lamp_qj(list(B),fun(B,list(B))),Uua)),Uu) ).

% ATP.lambda_137
tff(fact_8315_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ru(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_138
tff(fact_8316_ATP_Olambda__139,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_hs(num,fun(num,int),Uu),Uua) = aa(int,int,aa(nat,fun(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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_139
tff(fact_8317_ATP_Olambda__140,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_ajq(set(B),fun(set(B),$o),Uu),Uua)
    <=> ( aa(set(B),$o,finite_finite2(B),Uua)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uua),Uu) ) ) ).

% ATP.lambda_140
tff(fact_8318_ATP_Olambda__141,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: multiset(product_prod(B,C))] :
      ( aa(multiset(product_prod(B,C)),$o,aTP_Lamp_ahi(fun(B,fun(C,$o)),fun(multiset(product_prod(B,C)),$o),Uu),Uua)
    <=> aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),aa(multiset(product_prod(B,C)),set(product_prod(B,C)),set_mset(product_prod(B,C)),Uua)),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Uu))) ) ).

% ATP.lambda_141
tff(fact_8319_ATP_Olambda__142,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: list(product_prod(B,C))] :
      ( aa(list(product_prod(B,C)),$o,aTP_Lamp_agr(fun(B,fun(C,$o)),fun(list(product_prod(B,C)),$o),Uu),Uua)
    <=> aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),aa(list(product_prod(B,C)),set(product_prod(B,C)),set2(product_prod(B,C)),Uua)),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Uu))) ) ).

% ATP.lambda_142
tff(fact_8320_ATP_Olambda__143,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: set(set(B))] : aa(set(set(B)),set(B),aTP_Lamp_aml(fun(B,fun(C,$o)),fun(set(set(B)),set(B)),Uu),Uua) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),Uua)),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),Uu))) ).

% ATP.lambda_143
tff(fact_8321_ATP_Olambda__144,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: set(B)] : aa(set(B),set(B),aTP_Lamp_amp(fun(B,fun(C,$o)),fun(set(B),set(B)),Uu),Uua) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(B),set(B),uminus_uminus(set(B)),Uua)),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),Uu))) ).

% ATP.lambda_144
tff(fact_8322_ATP_Olambda__145,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: product_prod(B,C),Uua: product_prod(B,C)] :
          ( aa(product_prod(B,C),$o,aa(product_prod(B,C),fun(product_prod(B,C),$o),aTP_Lamp_sx(product_prod(B,C),fun(product_prod(B,C),$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(product_prod(B,C),B,product_fst(B,C),Uu)),aa(product_prod(B,C),B,product_fst(B,C),Uua)) ) ) ).

% ATP.lambda_145
tff(fact_8323_ATP_Olambda__146,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: product_prod(B,C),Uua: product_prod(B,C)] :
          ( aa(product_prod(B,C),$o,aa(product_prod(B,C),fun(product_prod(B,C),$o),aTP_Lamp_sy(product_prod(B,C),fun(product_prod(B,C),$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(product_prod(B,C),B,product_fst(B,C),Uua)),aa(product_prod(B,C),B,product_fst(B,C),Uu)) ) ) ).

% ATP.lambda_146
tff(fact_8324_ATP_Olambda__147,axiom,
    ! [B: $tType] :
      ( ring_1(B)
     => ! [Uu: nat,Uua: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_nr(nat,fun(nat,B)),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(nat,B,semiring_1_of_nat(B),Uu)),aa(nat,B,semiring_1_of_nat(B),Uua)) ) ).

% ATP.lambda_147
tff(fact_8325_ATP_Olambda__148,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_uc(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_148
tff(fact_8326_ATP_Olambda__149,axiom,
    ! [B: $tType,Uu: list(list(B)),Uua: B] : aa(B,list(list(B)),aTP_Lamp_ql(list(list(B)),fun(B,list(list(B))),Uu),Uua) = aa(list(list(B)),list(list(B)),map(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uua)),product_lists(B,Uu)) ).

% ATP.lambda_149
tff(fact_8327_ATP_Olambda__150,axiom,
    ! [B: $tType,Uu: list(B),Uua: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_aki(list(B),fun(list(B),$o)),Uu),Uua)
    <=> ( aa(list(B),nat,size_size(list(B)),Uu) = aa(list(B),nat,size_size(list(B)),Uua) ) ) ).

% ATP.lambda_150
tff(fact_8328_ATP_Olambda__151,axiom,
    ! [B: $tType,Uu: list(B),Uua: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_km(list(B),fun(list(B),$o),Uu),Uua)
    <=> ( mset(B,Uua) = mset(B,Uu) ) ) ).

% ATP.lambda_151
tff(fact_8329_ATP_Olambda__152,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_lc(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= aa(code_integer,num,code_num_of_integer,Uu),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_152
tff(fact_8330_ATP_Olambda__153,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_kz(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= aa(code_integer,nat,code_nat_of_integer,Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_153
tff(fact_8331_ATP_Olambda__154,axiom,
    ! [B: $tType,Uu: set(B),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aTP_Lamp_arj(set(B),fun(multiset(B),$o),Uu),Uua)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(multiset(B),set(B),set_mset(B),Uua)),Uu) ) ).

% ATP.lambda_154
tff(fact_8332_ATP_Olambda__155,axiom,
    ! [B: $tType,Uu: set(B),Uua: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_aro(set(B),fun(list(B),$o),Uu),Uua)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uua)),Uu) ) ).

% ATP.lambda_155
tff(fact_8333_ATP_Olambda__156,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: nat,Uua: nat] : aa(nat,B,aTP_Lamp_cz(nat,fun(nat,B),Uu),Uua) = aa(nat,B,gbinomial(B,aa(nat,B,semiring_1_of_nat(B),Uua)),Uu) ) ).

% ATP.lambda_156
tff(fact_8334_ATP_Olambda__157,axiom,
    ! [B: $tType,C: $tType,Uu: list(C),Uua: B] : aa(B,list(product_prod(B,C)),aTP_Lamp_qi(list(C),fun(B,list(product_prod(B,C))),Uu),Uua) = aa(list(C),list(product_prod(B,C)),map(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua)),Uu) ).

% ATP.lambda_157
tff(fact_8335_ATP_Olambda__158,axiom,
    ! [B: $tType,Uu: set(nat),Uua: product_prod(B,nat)] :
      ( aa(product_prod(B,nat),$o,aTP_Lamp_tp(set(nat),fun(product_prod(B,nat),$o),Uu),Uua)
    <=> aa(set(nat),$o,member(nat,aa(product_prod(B,nat),nat,product_snd(B,nat),Uua)),Uu) ) ).

% ATP.lambda_158
tff(fact_8336_ATP_Olambda__159,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ry(set(nat),fun(nat,$o),Uu),Uua)
    <=> aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uua)),Uu) ) ).

% ATP.lambda_159
tff(fact_8337_ATP_Olambda__160,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Uu: B,Uua: B] : aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),aTP_Lamp_md(B,fun(B,product_prod(B,B))),Uu),Uua) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Uua)),one_one(B))) ) ).

% ATP.lambda_160
tff(fact_8338_ATP_Olambda__161,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Uu: B,Uua: B] : aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),aTP_Lamp_me(B,fun(B,product_prod(B,B))),Uu),Uua) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_161
tff(fact_8339_ATP_Olambda__162,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_dj(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,semiring_1_of_nat(B),Uua)),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_162
tff(fact_8340_ATP_Olambda__163,axiom,
    ! [B: $tType,Uu: B,Uua: set(set(B))] : aa(set(set(B)),set(set(B)),aa(B,fun(set(set(B)),set(set(B))),aTP_Lamp_acr(B,fun(set(set(B)),set(set(B)))),Uu),Uua) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),Uua),aa(set(set(B)),set(set(B)),image2(set(B),set(B),insert(B,Uu)),Uua)) ).

% ATP.lambda_163
tff(fact_8341_ATP_Olambda__164,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: list(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_pq(list(B),fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(nat,B,nth(B,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% ATP.lambda_164
tff(fact_8342_ATP_Olambda__165,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: list(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_pr(list(B),fun(B,$o),Uu),Uua)
        <=> ( Uua = aa(nat,B,nth(B,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% ATP.lambda_165
tff(fact_8343_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_df(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_166
tff(fact_8344_ATP_Olambda__167,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: multiset(B),Uua: B] : aa(B,B,aTP_Lamp_aij(multiset(B),fun(B,B),Uu),Uua) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uu),Uua)) ) ).

% ATP.lambda_167
tff(fact_8345_ATP_Olambda__168,axiom,
    ! [B: $tType,Uu: B,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_mi(B,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = 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)),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_168
tff(fact_8346_ATP_Olambda__169,axiom,
    ! [B: $tType,Uu: B,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_mj(B,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = 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)),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_169
tff(fact_8347_ATP_Olambda__170,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: C] : aa(C,set(B),aTP_Lamp_agb(set(product_prod(C,B)),fun(C,set(B)),Uu),Uua) = aa(set(C),set(B),image(C,B,Uu),aa(set(C),set(C),insert(C,Uua),bot_bot(set(C)))) ).

% ATP.lambda_170
tff(fact_8348_ATP_Olambda__171,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,C),Uua: C] : aa(C,set(B),aTP_Lamp_adu(fun(B,C),fun(C,set(B)),Uu),Uua) = aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),Uu),aa(set(C),set(C),insert(C,Uua),bot_bot(set(C)))) ).

% ATP.lambda_171
tff(fact_8349_ATP_Olambda__172,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_rt(set(B),fun(B,$o),Uu),Uua)
    <=> ( Uu = aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))) ) ) ).

% ATP.lambda_172
tff(fact_8350_ATP_Olambda__173,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(B,B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ann(fun(B,B),fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(B,B,Uu,Uua)) ) ) ).

% ATP.lambda_173
tff(fact_8351_ATP_Olambda__174,axiom,
    ! [B: $tType,Uu: fun(nat,B),Uua: nat] : aa(nat,product_prod(nat,B),aTP_Lamp_qt(fun(nat,B),fun(nat,product_prod(nat,B)),Uu),Uua) = aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),Uua),aa(nat,B,Uu,Uua)) ).

% ATP.lambda_174
tff(fact_8352_ATP_Olambda__175,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: B] : aa(B,product_prod(B,C),aTP_Lamp_rx(fun(B,C),fun(B,product_prod(B,C)),Uu),Uua) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),aa(B,C,Uu,Uua)) ).

% ATP.lambda_175
tff(fact_8353_ATP_Olambda__176,axiom,
    ! [B: $tType] :
      ( bit_un5681908812861735899ations(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_eo(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),Uua),aa($o,B,zero_neq_one_of_bool(B),aa(nat,$o,aa(B,fun(nat,$o),bit_se5641148757651400278ts_bit(B),Uu),Uua))) ) ).

% ATP.lambda_176
tff(fact_8354_ATP_Olambda__177,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,option(C)),Uua: B] : aa(B,product_prod(B,C),aTP_Lamp_ans(fun(B,option(C)),fun(B,product_prod(B,C)),Uu),Uua) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),aa(option(C),C,the2(C),aa(B,option(C),Uu,Uua))) ).

% ATP.lambda_177
tff(fact_8355_ATP_Olambda__178,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_lm(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_178
tff(fact_8356_ATP_Olambda__179,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aac(set(product_prod(B,B)),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(B,B)),nat,finite_card(product_prod(B,B)),Uu)) ) ).

% ATP.lambda_179
tff(fact_8357_ATP_Olambda__180,axiom,
    ! [B: $tType,Uu: nat,Uua: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_pw(nat,fun(list(B),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(B),nat,size_size(list(B)),Uua)) ) ).

% ATP.lambda_180
tff(fact_8358_ATP_Olambda__181,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_dl(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uua)) ) ).

% ATP.lambda_181
tff(fact_8359_ATP_Olambda__182,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_eh(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uua)) ) ).

% ATP.lambda_182
tff(fact_8360_ATP_Olambda__183,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_ed(B,fun(nat,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uua)) ) ).

% ATP.lambda_183
tff(fact_8361_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_aqg(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_184
tff(fact_8362_ATP_Olambda__185,axiom,
    ! [C: $tType,B: $tType,Uu: list(product_prod(B,C)),Uua: fun(B,option(C))] : aa(fun(B,option(C)),fun(B,option(C)),aa(list(product_prod(B,C)),fun(fun(B,option(C)),fun(B,option(C))),aTP_Lamp_uf(list(product_prod(B,C)),fun(fun(B,option(C)),fun(B,option(C)))),Uu),Uua) = map_add(B,C,Uua,map_of(B,C,Uu)) ).

% ATP.lambda_185
tff(fact_8363_ATP_Olambda__186,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] :
      ( aa(list(B),$o,aa(B,fun(list(B),$o),aTP_Lamp_agw(B,fun(list(B),$o)),Uu),Uua)
    <=> aa(set(B),$o,member(B,Uu),aa(list(B),set(B),set2(B),Uua)) ) ).

% ATP.lambda_186
tff(fact_8364_ATP_Olambda__187,axiom,
    ! [B: $tType,Uu: list(set(B)),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_abo(list(set(B)),fun(set(B),$o),Uu),Uua)
    <=> aa(set(set(B)),$o,member(set(B),Uua),aa(list(set(B)),set(set(B)),set2(set(B)),Uu)) ) ).

% ATP.lambda_187
tff(fact_8365_ATP_Olambda__188,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_pi(list(B),fun(B,$o),Uu),Uua)
    <=> aa(set(B),$o,member(B,Uua),aa(list(B),set(B),set2(B),Uu)) ) ).

% ATP.lambda_188
tff(fact_8366_ATP_Olambda__189,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aep(list(B),fun(B,$o),Uu),Uua)
    <=> ( Uua = aa(list(B),B,hd(B),Uu) ) ) ).

% ATP.lambda_189
tff(fact_8367_ATP_Olambda__190,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aeo(list(B),fun(B,$o),Uu),Uua)
    <=> ( Uua = last(B,Uu) ) ) ).

% ATP.lambda_190
tff(fact_8368_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_adm(nat,fun(nat,$o)),Uu),Uua)
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_191
tff(fact_8369_ATP_Olambda__192,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_gc(set(B),fun(set(B),$o),Uu),Uua)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uua),Uu) ) ).

% ATP.lambda_192
tff(fact_8370_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ap(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_193
tff(fact_8371_ATP_Olambda__194,axiom,
    ! [B: $tType] :
      ( bounde4967611905675639751up_bot(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_abp(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_194
tff(fact_8372_ATP_Olambda__195,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_apg(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_195
tff(fact_8373_ATP_Olambda__196,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_gy(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_196
tff(fact_8374_ATP_Olambda__197,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aqt(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_197
tff(fact_8375_ATP_Olambda__198,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_kh(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_198
tff(fact_8376_ATP_Olambda__199,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_ca(B,fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Uu) ) ) ).

% ATP.lambda_199
tff(fact_8377_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_vn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).

% ATP.lambda_200
tff(fact_8378_ATP_Olambda__201,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_io(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),divide_divide(B),Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_8379_ATP_Olambda__202,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_afa(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),divide_divide(B),Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_8380_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aq(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_203
tff(fact_8381_ATP_Olambda__204,axiom,
    ! [B: $tType] :
      ( bounde4967611905675639751up_bot(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_abq(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_204
tff(fact_8382_ATP_Olambda__205,axiom,
    ! [B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_alt(B,fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_205
tff(fact_8383_ATP_Olambda__206,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aph(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_206
tff(fact_8384_ATP_Olambda__207,axiom,
    ! [B: $tType] :
      ( order_bot(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_gz(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_207
tff(fact_8385_ATP_Olambda__208,axiom,
    ! [B: $tType] :
      ( preorder(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_js(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_208
tff(fact_8386_ATP_Olambda__209,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_apf(B,fun(B,$o)),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_209
tff(fact_8387_ATP_Olambda__210,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_ia(B,fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),Uu) ) ) ).

% ATP.lambda_210
tff(fact_8388_ATP_Olambda__211,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_jc(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),Uu) ) ).

% ATP.lambda_211
tff(fact_8389_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_212
tff(fact_8390_ATP_Olambda__213,axiom,
    ! [B: $tType] :
      ( linordered_idom(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_in(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),Uua),Uu) ) ).

% ATP.lambda_213
tff(fact_8391_ATP_Olambda__214,axiom,
    ! [B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_jj(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),minus_minus(B),Uua),Uu) ) ).

% ATP.lambda_214
tff(fact_8392_ATP_Olambda__215,axiom,
    ! [B: $tType,Uu: B,Uua: multiset(B)] : aa(multiset(B),nat,aTP_Lamp_ahp(B,fun(multiset(B),nat),Uu),Uua) = aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uua),Uu) ).

% ATP.lambda_215
tff(fact_8393_ATP_Olambda__216,axiom,
    ! [B: $tType,Uu: multiset(B),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aqi(multiset(B),fun(multiset(B),$o)),Uu),Uua)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Uua),Uu) ) ).

% ATP.lambda_216
tff(fact_8394_ATP_Olambda__217,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_abi(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),sup_sup(B),Uua),Uu) ) ).

% ATP.lambda_217
tff(fact_8395_ATP_Olambda__218,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B)] : aa(set(B),set(B),aTP_Lamp_ya(set(B),fun(set(B),set(B)),Uu),Uua) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uua),Uu) ).

% ATP.lambda_218
tff(fact_8396_ATP_Olambda__219,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_wq(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uua),Uu) ) ).

% ATP.lambda_219
tff(fact_8397_ATP_Olambda__220,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_jm(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_220
tff(fact_8398_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_221
tff(fact_8399_ATP_Olambda__222,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_jl(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_222
tff(fact_8400_ATP_Olambda__223,axiom,
    ! [B: $tType] :
      ( cancel_semigroup_add(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_afg(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uua),Uu) ) ).

% ATP.lambda_223
tff(fact_8401_ATP_Olambda__224,axiom,
    ! [B: $tType] :
      ( linordered_semidom(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_im(B,fun(B,B),Uu),Uua) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uua),Uu) ) ).

% ATP.lambda_224
tff(fact_8402_ATP_Olambda__225,axiom,
    ! [B: $tType,Uu: multiset(B),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aqh(multiset(B),fun(multiset(B),$o)),Uu),Uua)
    <=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),Uua),Uu) ) ).

% ATP.lambda_225
tff(fact_8403_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ga(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu) ) ).

% ATP.lambda_226
tff(fact_8404_ATP_Olambda__227,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_gj(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uua),Uu) ) ).

% ATP.lambda_227
tff(fact_8405_ATP_Olambda__228,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_az(B,fun(B,$o),Uu),Uua)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),Uu) ) ) ).

% ATP.lambda_228
tff(fact_8406_ATP_Olambda__229,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: set(B)] : aa(set(B),set(product_prod(B,C)),aTP_Lamp_we(fun(B,set(C)),fun(set(B),set(product_prod(B,C))),Uu),Uua) = product_Sigma(B,C,Uua,Uu) ).

% ATP.lambda_229
tff(fact_8407_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_nq(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_230
tff(fact_8408_ATP_Olambda__231,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: B] : aa(B,product_prod(B,C),aa(C,fun(B,product_prod(B,C)),aTP_Lamp_sj(C,fun(B,product_prod(B,C))),Uu),Uua) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uu) ).

% ATP.lambda_231
tff(fact_8409_ATP_Olambda__232,axiom,
    ! [B: $tType,Uu: B,Uua: nat] : aa(nat,product_prod(nat,B),aTP_Lamp_qv(B,fun(nat,product_prod(nat,B)),Uu),Uua) = aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),Uua),Uu) ).

% ATP.lambda_232
tff(fact_8410_ATP_Olambda__233,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,product_prod(C,B),aa(B,fun(C,product_prod(C,B)),aTP_Lamp_te(B,fun(C,product_prod(C,B))),Uu),Uua) = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uua),Uu) ).

% ATP.lambda_233
tff(fact_8411_ATP_Olambda__234,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ce(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_234
tff(fact_8412_ATP_Olambda__235,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: array(B),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),nat,aTP_Lamp_alf(array(B),fun(heap_ext(product_unit),nat),Uu),Uua) = array_length(B,Uua,Uu) ) ).

% ATP.lambda_235
tff(fact_8413_ATP_Olambda__236,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] : aa(B,list(B),aa(list(B),fun(B,list(B)),aTP_Lamp_qj(list(B),fun(B,list(B))),Uu),Uua) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uua),Uu) ).

% ATP.lambda_236
tff(fact_8414_ATP_Olambda__237,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: ref(B),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),B,aTP_Lamp_nf(ref(B),fun(heap_ext(product_unit),B),Uu),Uua) = ref_get(B,Uua,Uu) ) ).

% ATP.lambda_237
tff(fact_8415_ATP_Olambda__238,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ahh(set(nat),fun(nat,$o),Uu),Uua)
    <=> aa(set(nat),$o,member(nat,Uua),Uu) ) ).

% ATP.lambda_238
tff(fact_8416_ATP_Olambda__239,axiom,
    ! [C: $tType,Uu: set(C),Uua: C] :
      ( aa(C,$o,aTP_Lamp_al(set(C),fun(C,$o),Uu),Uua)
    <=> aa(set(C),$o,member(C,Uua),Uu) ) ).

% ATP.lambda_239
tff(fact_8417_ATP_Olambda__240,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Uu: set(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ahe(set(B),fun(B,$o),Uu),Uua)
        <=> aa(set(B),$o,member(B,Uua),Uu) ) ) ).

% ATP.lambda_240
tff(fact_8418_ATP_Olambda__241,axiom,
    ! [B: $tType] :
      ( order(B)
     => ! [Uu: set(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ahg(set(B),fun(B,$o),Uu),Uua)
        <=> aa(set(B),$o,member(B,Uua),Uu) ) ) ).

% ATP.lambda_241
tff(fact_8419_ATP_Olambda__242,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ab(set(B),fun(B,$o)),Uu),Uua)
    <=> aa(set(B),$o,member(B,Uua),Uu) ) ).

% ATP.lambda_242
tff(fact_8420_ATP_Olambda__243,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: nat] : aa(nat,set(product_prod(B,B)),aTP_Lamp_zy(set(product_prod(B,B)),fun(nat,set(product_prod(B,B))),Uu),Uua) = aa(set(product_prod(B,B)),set(product_prod(B,B)),aa(nat,fun(set(product_prod(B,B)),set(product_prod(B,B))),compow(set(product_prod(B,B))),Uua),Uu) ).

% ATP.lambda_243
tff(fact_8421_ATP_Olambda__244,axiom,
    ! [B: $tType,Uu: list(B),Uua: nat] : aa(nat,list(B),aTP_Lamp_lb(list(B),fun(nat,list(B)),Uu),Uua) = drop(B,Uua,Uu) ).

% ATP.lambda_244
tff(fact_8422_ATP_Olambda__245,axiom,
    ! [B: $tType,Uu: nat,Uua: list(B)] : aa(list(B),B,aTP_Lamp_qd(nat,fun(list(B),B),Uu),Uua) = aa(nat,B,nth(B,Uua),Uu) ).

% ATP.lambda_245
tff(fact_8423_ATP_Olambda__246,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,D),Uua: fun(D,set(C))] : aa(fun(D,set(C)),fun(B,set(C)),aTP_Lamp_aex(fun(B,D),fun(fun(D,set(C)),fun(B,set(C))),Uu),Uua) = aa(fun(B,D),fun(B,set(C)),aa(fun(D,set(C)),fun(fun(B,D),fun(B,set(C))),comp(D,set(C),B),Uua),Uu) ).

% ATP.lambda_246
tff(fact_8424_ATP_Olambda__247,axiom,
    ! [B: $tType,Uu: B,Uua: B] :
      ( aa(B,$o,aTP_Lamp_af(B,fun(B,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_247
tff(fact_8425_ATP_Olambda__248,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),aTP_Lamp_qa(B,fun(list(B),list(B))),Uu),Uua) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),nil(B)) ).

% ATP.lambda_248
tff(fact_8426_ATP_Olambda__249,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),list(list(B)),aa(B,fun(list(B),list(list(B))),aTP_Lamp_qn(B,fun(list(B),list(list(B)))),Uu),Uua) = aa(list(list(B)),list(list(B)),aa(list(B),fun(list(list(B)),list(list(B))),cons(list(B)),Uua),nil(list(B))) ).

% ATP.lambda_249
tff(fact_8427_ATP_Olambda__250,axiom,
    ! [B: $tType,Uu: B,Uua: B] : aa(B,set(B),aTP_Lamp_adh(B,fun(B,set(B)),Uu),Uua) = aa(set(B),set(B),insert(B,Uu),bot_bot(set(B))) ).

% ATP.lambda_250
tff(fact_8428_ATP_Olambda__251,axiom,
    ! [B: $tType,Uu: multiset(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ahk(multiset(B),fun(B,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uu),Uua)) ) ).

% ATP.lambda_251
tff(fact_8429_ATP_Olambda__252,axiom,
    ! [C: $tType,Uu: fun(C,nat),Uua: C] :
      ( aa(C,$o,aTP_Lamp_alh(fun(C,nat),fun(C,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(C,nat,Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_8430_ATP_Olambda__253,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ft(fun(B,nat),fun(B,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8431_ATP_Olambda__254,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ahu(set(multiset(B)),fun(B,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(B)),set(nat),image2(multiset(B),nat,aTP_Lamp_ahp(B,fun(multiset(B),nat),Uua)),Uu))) ) ).

% ATP.lambda_254
tff(fact_8432_ATP_Olambda__255,axiom,
    ! [B: $tType,Uu: code_natural,Uua: B] :
      ( aa(B,$o,aa(code_natural,fun(B,$o),aTP_Lamp_aqb(code_natural,fun(B,$o)),Uu),Uua)
    <=> aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),zero_zero(code_natural)),Uu) ) ).

% ATP.lambda_255
tff(fact_8433_ATP_Olambda__256,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B] : aa(B,set(C),aTP_Lamp_xb(set(product_prod(B,C)),fun(B,set(C)),Uu),Uua) = aa(set(product_prod(B,C)),set(C),image2(product_prod(B,C),C,product_snd(B,C)),Uu) ).

% ATP.lambda_256
tff(fact_8434_ATP_Olambda__257,axiom,
    ! [B: $tType,Uu: fun(nat,$o),Uua: product_prod(B,nat)] :
      ( aa(product_prod(B,nat),$o,aTP_Lamp_tn(fun(nat,$o),fun(product_prod(B,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,aa(product_prod(B,nat),nat,product_snd(B,nat),Uua))) ) ).

% ATP.lambda_257
tff(fact_8435_ATP_Olambda__258,axiom,
    ! [B: $tType,Uu: fun(nat,$o),Uua: product_prod(B,nat)] :
      ( aa(product_prod(B,nat),$o,aTP_Lamp_to(fun(nat,$o),fun(product_prod(B,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(product_prod(B,nat),nat,product_snd(B,nat),Uua)) ) ).

% ATP.lambda_258
tff(fact_8436_ATP_Olambda__259,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: product_prod(B,C)] : aa(product_prod(B,C),D,aTP_Lamp_th(fun(C,D),fun(product_prod(B,C),D),Uu),Uua) = aa(C,D,Uu,aa(product_prod(B,C),C,product_snd(B,C),Uua)) ).

% ATP.lambda_259
tff(fact_8437_ATP_Olambda__260,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,nat),Uua: product_prod(C,B)] : aa(product_prod(C,B),nat,aTP_Lamp_tj(fun(B,nat),fun(product_prod(C,B),nat),Uu),Uua) = aa(B,nat,Uu,aa(product_prod(C,B),B,product_snd(C,B),Uua)) ).

% ATP.lambda_260
tff(fact_8438_ATP_Olambda__261,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,nat),Uua: product_prod(B,C)] : aa(product_prod(B,C),nat,aTP_Lamp_sz(fun(B,nat),fun(product_prod(B,C),nat),Uu),Uua) = aa(B,nat,Uu,aa(product_prod(B,C),B,product_fst(B,C),Uua)) ).

% ATP.lambda_261
tff(fact_8439_ATP_Olambda__262,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: fun(B,D),Uua: product_prod(B,C)] : aa(product_prod(B,C),D,aTP_Lamp_ta(fun(B,D),fun(product_prod(B,C),D),Uu),Uua) = aa(B,D,Uu,aa(product_prod(B,C),B,product_fst(B,C),Uua)) ).

% ATP.lambda_262
tff(fact_8440_ATP_Olambda__263,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ahc(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_263
tff(fact_8441_ATP_Olambda__264,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_dq(fun(nat,B),fun(nat,B),Uu),Uua) = aa(nat,B,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_264
tff(fact_8442_ATP_Olambda__265,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_cb(fun(nat,B),fun(nat,B),Uu),Uua) = aa(nat,B,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_265
tff(fact_8443_ATP_Olambda__266,axiom,
    ! [B: $tType,Uu: fun(nat,B),Uua: nat] : aa(nat,B,aTP_Lamp_qu(fun(nat,B),fun(nat,B),Uu),Uua) = aa(nat,B,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_266
tff(fact_8444_ATP_Olambda__267,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: fun(C,set(B))] : aa(fun(C,set(B)),set(B),aTP_Lamp_aew(C,fun(fun(C,set(B)),set(B)),Uu),Uua) = aa(C,set(B),Uua,Uu) ).

% ATP.lambda_267
tff(fact_8445_ATP_Olambda__268,axiom,
    ! [B: $tType,C: $tType] :
      ( complete_Sup(B)
     => ! [Uu: C,Uua: fun(C,B)] : aa(fun(C,B),B,aTP_Lamp_xm(C,fun(fun(C,B),B),Uu),Uua) = aa(C,B,Uua,Uu) ) ).

% ATP.lambda_268
tff(fact_8446_ATP_Olambda__269,axiom,
    ! [B: $tType,C: $tType] :
      ( complete_Inf(B)
     => ! [Uu: C,Uua: fun(C,B)] : aa(fun(C,B),B,aTP_Lamp_yz(C,fun(fun(C,B),B),Uu),Uua) = aa(C,B,Uua,Uu) ) ).

% ATP.lambda_269
tff(fact_8447_ATP_Olambda__270,axiom,
    ! [B: $tType,Uu: B,Uua: fun(B,nat)] : aa(fun(B,nat),nat,aTP_Lamp_aho(B,fun(fun(B,nat),nat),Uu),Uua) = aa(B,nat,Uua,Uu) ).

% ATP.lambda_270
tff(fact_8448_ATP_Olambda__271,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [Uu: B,Uua: fun(B,C)] : aa(fun(B,C),C,aTP_Lamp_xu(B,fun(fun(B,C),C),Uu),Uua) = aa(B,C,Uua,Uu) ) ).

% ATP.lambda_271
tff(fact_8449_ATP_Olambda__272,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [Uu: B,Uua: fun(B,C)] : aa(fun(B,C),C,aTP_Lamp_zg(B,fun(fun(B,C),C),Uu),Uua) = aa(B,C,Uua,Uu) ) ).

% ATP.lambda_272
tff(fact_8450_ATP_Olambda__273,axiom,
    ! [B: $tType,Uu: fun(product_unit,B),Uua: product_unit] : aa(product_unit,B,aTP_Lamp_apq(fun(product_unit,B),fun(product_unit,B),Uu),Uua) = aa(product_unit,B,Uu,product_Unity) ).

% ATP.lambda_273
tff(fact_8451_ATP_Olambda__274,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: option(B)] : aa(option(B),option(B),aa(B,fun(option(B),option(B)),aTP_Lamp_ada(B,fun(option(B),option(B))),Uu),Uua) = aa(B,option(B),some(B),case_option(B,B,Uu,aa(B,fun(B,B),ord_min(B),Uu),Uua)) ) ).

% ATP.lambda_274
tff(fact_8452_ATP_Olambda__275,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: option(B)] : aa(option(B),option(B),aa(B,fun(option(B),option(B)),aTP_Lamp_acn(B,fun(option(B),option(B))),Uu),Uua) = aa(B,option(B),some(B),case_option(B,B,Uu,aa(B,fun(B,B),ord_max(B),Uu),Uua)) ) ).

% ATP.lambda_275
tff(fact_8453_ATP_Olambda__276,axiom,
    ! [B: $tType] :
      ( semilattice_sup(B)
     => ! [Uu: B,Uua: option(B)] : aa(option(B),option(B),aa(B,fun(option(B),option(B)),aTP_Lamp_acu(B,fun(option(B),option(B))),Uu),Uua) = aa(B,option(B),some(B),case_option(B,B,Uu,aa(B,fun(B,B),sup_sup(B),Uu),Uua)) ) ).

% ATP.lambda_276
tff(fact_8454_ATP_Olambda__277,axiom,
    ! [B: $tType] :
      ( semilattice_inf(B)
     => ! [Uu: B,Uua: option(B)] : aa(option(B),option(B),aa(B,fun(option(B),option(B)),aTP_Lamp_acv(B,fun(option(B),option(B))),Uu),Uua) = aa(B,option(B),some(B),case_option(B,B,Uu,aa(B,fun(B,B),inf_inf(B),Uu),Uua)) ) ).

% ATP.lambda_277
tff(fact_8455_ATP_Olambda__278,axiom,
    ! [B: $tType,Uu: multiset(B),Uua: option(multiset(B))] : aa(option(multiset(B)),option(multiset(B)),aa(multiset(B),fun(option(multiset(B)),option(multiset(B))),aTP_Lamp_aqn(multiset(B),fun(option(multiset(B)),option(multiset(B)))),Uu),Uua) = aa(multiset(B),option(multiset(B)),some(multiset(B)),case_option(multiset(B),multiset(B),Uu,aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),Uu),Uua)) ).

% ATP.lambda_278
tff(fact_8456_ATP_Olambda__279,axiom,
    ! [B: $tType,Uu: multiset(B),Uua: option(multiset(B))] : aa(option(multiset(B)),option(multiset(B)),aa(multiset(B),fun(option(multiset(B)),option(multiset(B))),aTP_Lamp_aqq(multiset(B),fun(option(multiset(B)),option(multiset(B)))),Uu),Uua) = aa(multiset(B),option(multiset(B)),some(multiset(B)),case_option(multiset(B),multiset(B),Uu,aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),Uu),Uua)) ).

% ATP.lambda_279
tff(fact_8457_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ma(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_280
tff(fact_8458_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_agj(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_agi(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_281
tff(fact_8459_ATP_Olambda__282,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(D,fun(B,D)),Uua: D] : aa(D,fun(product_prod(C,B),D),aTP_Lamp_su(fun(D,fun(B,D)),fun(D,fun(product_prod(C,B),D)),Uu),Uua) = aa(fun(C,fun(B,D)),fun(product_prod(C,B),D),product_case_prod(C,B,D),aa(D,fun(C,fun(B,D)),aTP_Lamp_st(fun(D,fun(B,D)),fun(D,fun(C,fun(B,D))),Uu),Uua)) ).

% ATP.lambda_282
tff(fact_8460_ATP_Olambda__283,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),list(B)),aTP_Lamp_of(fun(B,C),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_oe(fun(B,C),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua)) ) ).

% ATP.lambda_283
tff(fact_8461_ATP_Olambda__284,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_nz(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_ny(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_284
tff(fact_8462_ATP_Olambda__285,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_nx(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_nw(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_285
tff(fact_8463_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nv(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_nu(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_286
tff(fact_8464_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_nt(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_ns(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_287
tff(fact_8465_ATP_Olambda__288,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_np(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_no(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_288
tff(fact_8466_ATP_Olambda__289,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,heap_Time_Heap(B)),Uua: C] : aa(C,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_nb(fun(C,heap_Time_Heap(B)),fun(C,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(C,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_na(fun(C,heap_Time_Heap(B)),fun(C,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).

% ATP.lambda_289
tff(fact_8467_ATP_Olambda__290,axiom,
    ! [B: $tType,Uu: list(B),Uua: list(B)] : aa(list(B),fun(product_prod(B,list(B)),option($o)),aTP_Lamp_mx(list(B),fun(list(B),fun(product_prod(B,list(B)),option($o))),Uu),Uua) = aa(fun(B,fun(list(B),option($o))),fun(product_prod(B,list(B)),option($o)),product_case_prod(B,list(B),option($o)),aa(list(B),fun(B,fun(list(B),option($o))),aTP_Lamp_mw(list(B),fun(list(B),fun(B,fun(list(B),option($o)))),Uu),Uua)) ).

% ATP.lambda_290
tff(fact_8468_ATP_Olambda__291,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),fun(product_prod(B,list(B)),option(product_prod(list(B),product_prod(B,list(B))))),aTP_Lamp_mr(B,fun(list(B),fun(product_prod(B,list(B)),option(product_prod(list(B),product_prod(B,list(B)))))),Uu),Uua) = aa(fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B)))))),fun(product_prod(B,list(B)),option(product_prod(list(B),product_prod(B,list(B))))),product_case_prod(B,list(B),option(product_prod(list(B),product_prod(B,list(B))))),aa(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B)))))),aTP_Lamp_mq(B,fun(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B))))))),Uu),Uua)) ).

% ATP.lambda_291
tff(fact_8469_ATP_Olambda__292,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(B,$o)] : aa(fun(B,$o),set(B),aTP_Lamp_amo(fun(B,fun(C,$o)),fun(fun(B,$o),set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(fun(B,$o),fun(B,$o),aTP_Lamp_amn(fun(B,fun(C,$o)),fun(fun(B,$o),fun(B,$o)),Uu),Uua)) ).

% ATP.lambda_292
tff(fact_8470_ATP_Olambda__293,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,$o)),Uua: C] : aa(C,set(B),aTP_Lamp_yq(fun(B,fun(C,$o)),fun(C,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aTP_Lamp_yp(fun(B,fun(C,$o)),fun(C,fun(B,$o)),Uu),Uua)) ).

% ATP.lambda_293
tff(fact_8471_ATP_Olambda__294,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: fun(C,D)] : aa(fun(C,D),B,aTP_Lamp_zs(fun(D,fun(C,B)),fun(fun(C,D),B),Uu),Uua) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,D),fun(C,B),aTP_Lamp_zr(fun(D,fun(C,B)),fun(fun(C,D),fun(C,B)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_294
tff(fact_8472_ATP_Olambda__295,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: C] : aa(C,B,aTP_Lamp_zt(fun(D,fun(C,B)),fun(C,B),Uu),Uua) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),aTP_Lamp_zp(fun(D,fun(C,B)),fun(C,fun(D,B)),Uu),Uua)),top_top(set(D)))) ) ).

% ATP.lambda_295
tff(fact_8473_ATP_Olambda__296,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: fun(C,D)] : aa(fun(C,D),B,aTP_Lamp_zu(fun(D,fun(C,B)),fun(fun(C,D),B),Uu),Uua) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,D),fun(C,B),aTP_Lamp_zr(fun(D,fun(C,B)),fun(fun(C,D),fun(C,B)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_296
tff(fact_8474_ATP_Olambda__297,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: C] : aa(C,B,aTP_Lamp_zq(fun(D,fun(C,B)),fun(C,B),Uu),Uua) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),aTP_Lamp_zp(fun(D,fun(C,B)),fun(C,fun(D,B)),Uu),Uua)),top_top(set(D)))) ) ).

% ATP.lambda_297
tff(fact_8475_ATP_Olambda__298,axiom,
    ! [B: $tType,Uu: fun(heap_ext(product_unit),B),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nm(fun(heap_ext(product_unit),B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = 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)),aa(heap_ext(product_unit),B,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_298
tff(fact_8476_ATP_Olambda__299,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,option(B)),Uua: C] :
      ( aa(C,$o,aTP_Lamp_rr(fun(C,option(B)),fun(C,$o),Uu),Uua)
    <=> ( aa(C,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_299
tff(fact_8477_ATP_Olambda__300,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,option(C)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_anp(fun(B,option(C)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(C),Uu,Uua) != none(C) ) ) ).

% ATP.lambda_300
tff(fact_8478_ATP_Olambda__301,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(B,C)),aTP_Lamp_wc(fun(B,set(C)),fun(B,set(product_prod(B,C))),Uu),Uua) = aa(set(set(product_prod(B,C))),set(product_prod(B,C)),complete_Sup_Sup(set(product_prod(B,C))),aa(set(C),set(set(product_prod(B,C))),image2(C,set(product_prod(B,C)),aTP_Lamp_wb(B,fun(C,set(product_prod(B,C))),Uua)),aa(B,set(C),Uu,Uua))) ).

% ATP.lambda_301
tff(fact_8479_ATP_Olambda__302,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: B] : aa(B,nat,aTP_Lamp_ahx(set(multiset(B)),fun(B,nat),Uu),Uua) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(B)),set(nat),image2(multiset(B),nat,aTP_Lamp_ahp(B,fun(multiset(B),nat),Uua)),Uu)) ).

% ATP.lambda_302
tff(fact_8480_ATP_Olambda__303,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: list(product_prod(D,C)),Uua: product_prod(B,D)] : aa(product_prod(B,D),list(product_prod(B,C)),aTP_Lamp_ub(list(product_prod(D,C)),fun(product_prod(B,D),list(product_prod(B,C))),Uu),Uua) = concat(product_prod(B,C),aa(list(product_prod(D,C)),list(list(product_prod(B,C))),map(product_prod(D,C),list(product_prod(B,C)),aTP_Lamp_ua(product_prod(B,D),fun(product_prod(D,C),list(product_prod(B,C))),Uua)),Uu)) ).

% ATP.lambda_303
tff(fact_8481_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bb(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ) ).

% ATP.lambda_304
tff(fact_8482_ATP_Olambda__305,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_xr(set(set(B)),fun(B,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(set(B)),set($o),image2(set(B),$o,member(B,Uua)),Uu)) ) ).

% ATP.lambda_305
tff(fact_8483_ATP_Olambda__306,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_zx(set(set(B)),fun(B,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Inf_Inf($o),aa(set(set(B)),set($o),image2(set(B),$o,member(B,Uua)),Uu)) ) ).

% ATP.lambda_306
tff(fact_8484_ATP_Olambda__307,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_apw(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,one2)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_307
tff(fact_8485_ATP_Olambda__308,axiom,
    ! [B: $tType,Uu: fun(nat,set(B)),Uua: nat] : aa(nat,set(B),aTP_Lamp_wy(fun(nat,set(B)),fun(nat,set(B)),Uu),Uua) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(nat),set(set(B)),image2(nat,set(B),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_308
tff(fact_8486_ATP_Olambda__309,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat] : aa(nat,fun(B,B),aTP_Lamp_aw(B,fun(nat,fun(B,B)),Uu),Uua) = aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uua))) ) ).

% ATP.lambda_309
tff(fact_8487_ATP_Olambda__310,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: B,Uua: nat] : aa(nat,fun(B,B),aTP_Lamp_ax(B,fun(nat,fun(B,B)),Uu),Uua) = aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uua))) ) ).

% ATP.lambda_310
tff(fact_8488_ATP_Olambda__311,axiom,
    ! [B: $tType,Uu: list(B),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural))),aTP_Lamp_aqd(list(B),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(B,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural))),product_Pair(B,product_prod(code_natural,code_natural)),aa(nat,B,nth(B,Uu),aa(code_natural,nat,code_nat_of_natural,Uua))) ).

% ATP.lambda_311
tff(fact_8489_ATP_Olambda__312,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_pj(list(B),fun(B,$o),Uu),Uua)
    <=> ~ aa(set(B),$o,member(B,Uua),aa(list(B),set(B),set2(B),Uu)) ) ).

% ATP.lambda_312
tff(fact_8490_ATP_Olambda__313,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ri(list(B),fun(B,$o),Uu),Uua)
    <=> ( Uua != last(B,Uu) ) ) ).

% ATP.lambda_313
tff(fact_8491_ATP_Olambda__314,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: set(C)] : aa(set(C),set(B),aTP_Lamp_yc(fun(C,set(B)),fun(set(C),set(B)),Uu),Uua) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),Uu),Uua)) ).

% ATP.lambda_314
tff(fact_8492_ATP_Olambda__315,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: set(C)] : aa(set(C),B,aTP_Lamp_akr(fun(C,B),fun(set(C),B),Uu),Uua) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,Uu),Uua)) ) ).

% ATP.lambda_315
tff(fact_8493_ATP_Olambda__316,axiom,
    ! [E: $tType,C: $tType,Uu: set(C),Uua: fun(C,set(E))] : aa(fun(C,set(E)),set(E),aa(set(C),fun(fun(C,set(E)),set(E)),aTP_Lamp_ajd(set(C),fun(fun(C,set(E)),set(E))),Uu),Uua) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(C),set(set(E)),image2(C,set(E),Uua),Uu)) ).

% ATP.lambda_316
tff(fact_8494_ATP_Olambda__317,axiom,
    ! [D: $tType,C: $tType] :
      ( complete_Sup(D)
     => ! [Uu: set(C),Uua: fun(C,D)] : aa(fun(C,D),D,aa(set(C),fun(fun(C,D),D),aTP_Lamp_ajb(set(C),fun(fun(C,D),D)),Uu),Uua) = aa(set(D),D,complete_Sup_Sup(D),aa(set(C),set(D),image2(C,D,Uua),Uu)) ) ).

% ATP.lambda_317
tff(fact_8495_ATP_Olambda__318,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(set(B),fun(fun(B,$o),$o),aTP_Lamp_aru(set(B),fun(fun(B,$o),$o)),Uu),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(B),set($o),image2(B,$o,Uua),Uu)) ) ).

% ATP.lambda_318
tff(fact_8496_ATP_Olambda__319,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_ajc(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_319
tff(fact_8497_ATP_Olambda__320,axiom,
    ! [D: $tType,B: $tType] :
      ( complete_Sup(D)
     => ! [Uu: set(B),Uua: fun(B,D)] : aa(fun(B,D),D,aa(set(B),fun(fun(B,D),D),aTP_Lamp_aja(set(B),fun(fun(B,D),D)),Uu),Uua) = aa(set(D),D,complete_Sup_Sup(D),aa(set(B),set(D),image2(B,D,Uua),Uu)) ) ).

% ATP.lambda_320
tff(fact_8498_ATP_Olambda__321,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: set(C)] : aa(set(C),set(B),aTP_Lamp_zo(fun(C,set(B)),fun(set(C),set(B)),Uu),Uua) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),Uu),Uua)) ).

% ATP.lambda_321
tff(fact_8499_ATP_Olambda__322,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: set(C)] : aa(set(C),B,aTP_Lamp_aks(fun(C,B),fun(set(C),B),Uu),Uua) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,Uu),Uua)) ) ).

% ATP.lambda_322
tff(fact_8500_ATP_Olambda__323,axiom,
    ! [D: $tType,C: $tType] :
      ( complete_Inf(D)
     => ! [Uu: set(C),Uua: fun(C,D)] : aa(fun(C,D),D,aa(set(C),fun(fun(C,D),D),aTP_Lamp_aiz(set(C),fun(fun(C,D),D)),Uu),Uua) = aa(set(D),D,complete_Inf_Inf(D),aa(set(C),set(D),image2(C,D,Uua),Uu)) ) ).

% ATP.lambda_323
tff(fact_8501_ATP_Olambda__324,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(set(B),fun(fun(B,$o),$o),aTP_Lamp_aju(set(B),fun(fun(B,$o),$o)),Uu),Uua)
    <=> aa(set($o),$o,complete_Inf_Inf($o),aa(set(B),set($o),image2(B,$o,Uua),Uu)) ) ).

% ATP.lambda_324
tff(fact_8502_ATP_Olambda__325,axiom,
    ! [D: $tType,B: $tType] :
      ( complete_Inf(D)
     => ! [Uu: set(B),Uua: fun(B,D)] : aa(fun(B,D),D,aa(set(B),fun(fun(B,D),D),aTP_Lamp_aiy(set(B),fun(fun(B,D),D)),Uu),Uua) = aa(set(D),D,complete_Inf_Inf(D),aa(set(B),set(D),image2(B,D,Uua),Uu)) ) ).

% ATP.lambda_325
tff(fact_8503_ATP_Olambda__326,axiom,
    ! [B: $tType] :
      ( sup(B)
     => ! [Uu: B,Uua: B] : aa(B,option(B),aTP_Lamp_abc(B,fun(B,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),Uu),Uua)) ) ).

% ATP.lambda_326
tff(fact_8504_ATP_Olambda__327,axiom,
    ! [B: $tType] :
      ( inf(B)
     => ! [Uu: B,Uua: B] : aa(B,option(B),aTP_Lamp_kn(B,fun(B,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),Uua)) ) ).

% ATP.lambda_327
tff(fact_8505_ATP_Olambda__328,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_apy(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),modulo_modulo(code_natural),Uua),Uu)) ).

% ATP.lambda_328
tff(fact_8506_ATP_Olambda__329,axiom,
    ! [D: $tType,B: $tType,C: $tType,Uu: B,Uua: C] : aa(C,fun(D,product_prod(product_prod(B,C),D)),aTP_Lamp_aot(B,fun(C,fun(D,product_prod(product_prod(B,C),D))),Uu),Uua) = aa(product_prod(B,C),fun(D,product_prod(product_prod(B,C),D)),product_Pair(product_prod(B,C),D),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uua)) ).

% ATP.lambda_329
tff(fact_8507_ATP_Olambda__330,axiom,
    ! [B: $tType,Uu: list(product_prod(code_natural,B)),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural))),aTP_Lamp_aqc(list(product_prod(code_natural,B)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(B,fun(product_prod(code_natural,code_natural),product_prod(B,product_prod(code_natural,code_natural))),product_Pair(B,product_prod(code_natural,code_natural)),aa(code_natural,B,pick(B,Uu),Uua)) ).

% ATP.lambda_330
tff(fact_8508_ATP_Olambda__331,axiom,
    ! [C: $tType,B: $tType,Uu: list(product_prod(B,C)),Uua: B] : aa(B,C,aTP_Lamp_vh(list(product_prod(B,C)),fun(B,C),Uu),Uua) = aa(option(C),C,the2(C),aa(B,option(C),map_of(B,C,Uu),Uua)) ).

% ATP.lambda_331
tff(fact_8509_ATP_Olambda__332,axiom,
    ! [C: $tType,B: $tType,Uu: C,Uua: B] : aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aTP_Lamp_abx(C,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),Uu),Uua) = insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uua)) ).

% ATP.lambda_332
tff(fact_8510_ATP_Olambda__333,axiom,
    ! [B: $tType,C: $tType,Uu: B,Uua: C] : aa(C,fun(set(product_prod(B,C)),set(product_prod(B,C))),aTP_Lamp_abz(B,fun(C,fun(set(product_prod(B,C)),set(product_prod(B,C)))),Uu),Uua) = insert(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uua)) ).

% ATP.lambda_333
tff(fact_8511_ATP_Olambda__334,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sp(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_334
tff(fact_8512_ATP_Olambda__335,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sq(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_335
tff(fact_8513_ATP_Olambda__336,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gk(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_336
tff(fact_8514_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gl(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_337
tff(fact_8515_ATP_Olambda__338,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_pv(B,fun(B,$o),Uu),Uua)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),Uua) ) ) ).

% ATP.lambda_338
tff(fact_8516_ATP_Olambda__339,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_jr(set(B),fun(B,$o),Uu),Uua)
    <=> ~ aa(set(B),$o,member(B,Uua),Uu) ) ).

% ATP.lambda_339
tff(fact_8517_ATP_Olambda__340,axiom,
    ! [B: $tType,Uu: B,Uua: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_ajz(B,fun(B,$o)),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_340
tff(fact_8518_ATP_Olambda__341,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_amb(B,fun(B,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_341
tff(fact_8519_ATP_Olambda__342,axiom,
    ! [B: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [Uu: B,Uua: B] :
          ( aa(B,$o,aTP_Lamp_alu(B,fun(B,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_342
tff(fact_8520_ATP_Olambda__343,axiom,
    ! [B: $tType,Uu: B,Uua: B] :
      ( aa(B,$o,aTP_Lamp_ph(B,fun(B,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_343
tff(fact_8521_ATP_Olambda__344,axiom,
    ! [B: $tType,Uu: B,Uua: B] : aa(B,set(B),aTP_Lamp_adi(B,fun(B,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),insert(B,Uu),bot_bot(set(B)))) ).

% ATP.lambda_344
tff(fact_8522_ATP_Olambda__345,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_apr(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_345
tff(fact_8523_ATP_Olambda__346,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(B,fun(B,B)),Uua: B] : aa(B,B,aTP_Lamp_aec(fun(B,fun(B,B)),fun(B,B),Uu),Uua) = complete_lattice_lfp(B,aa(B,fun(B,B),Uu,Uua)) ) ).

% ATP.lambda_346
tff(fact_8524_ATP_Olambda__347,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(B,fun(B,B)),Uua: B] : aa(B,B,aTP_Lamp_ano(fun(B,fun(B,B)),fun(B,B),Uu),Uua) = complete_lattice_gfp(B,aa(B,fun(B,B),Uu,Uua)) ) ).

% ATP.lambda_347
tff(fact_8525_ATP_Olambda__348,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(B,$o),Uua: B] : aa(B,C,aTP_Lamp_kg(fun(B,$o),fun(B,C),Uu),Uua) = aa($o,C,zero_neq_one_of_bool(C),aa(B,$o,Uu,Uua)) ) ).

% ATP.lambda_348
tff(fact_8526_ATP_Olambda__349,axiom,
    ! [B: $tType,C: $tType] :
      ( field_char_0(B)
     => ! [Uu: fun(C,rat),Uua: C] : aa(C,B,aTP_Lamp_tz(fun(C,rat),fun(C,B),Uu),Uua) = aa(rat,B,field_char_0_of_rat(B),aa(C,rat,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_8527_ATP_Olambda__350,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(C,nat),Uua: C] : aa(C,B,aTP_Lamp_dn(fun(C,nat),fun(C,B),Uu),Uua) = aa(nat,B,semiring_1_of_nat(B),aa(C,nat,Uu,Uua)) ) ).

% ATP.lambda_350
tff(fact_8528_ATP_Olambda__351,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(C,nat),Uua: C] : aa(C,B,aTP_Lamp_be(fun(C,nat),fun(C,B),Uu),Uua) = aa(nat,B,semiring_1_of_nat(B),aa(C,nat,Uu,Uua)) ) ).

% ATP.lambda_351
tff(fact_8529_ATP_Olambda__352,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_bk(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_352
tff(fact_8530_ATP_Olambda__353,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: C] : aa(C,set(B),aTP_Lamp_zc(fun(C,set(B)),fun(C,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),aa(C,set(B),Uu,Uua)) ).

% ATP.lambda_353
tff(fact_8531_ATP_Olambda__354,axiom,
    ! [B: $tType,C: $tType] :
      ( comple489889107523837845lgebra(B)
     => ! [Uu: fun(C,B),Uua: C] : aa(C,B,aTP_Lamp_zn(fun(C,B),fun(C,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8532_ATP_Olambda__355,axiom,
    ! [B: $tType,C: $tType] :
      ( ab_group_add(B)
     => ! [Uu: fun(C,B),Uua: C] : aa(C,B,aTP_Lamp_bu(fun(C,B),fun(C,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8533_ATP_Olambda__356,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(product_prod(C,C))),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_acq(fun(B,set(product_prod(C,C))),fun(B,set(product_prod(C,C))),Uu),Uua) = transitive_trancl(C,aa(B,set(product_prod(C,C)),Uu,Uua)) ).

% ATP.lambda_356
tff(fact_8534_ATP_Olambda__357,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,fun(B,B),aTP_Lamp_em(fun(nat,B),fun(nat,fun(B,B)),Uu),Uua) = aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uua)) ) ).

% ATP.lambda_357
tff(fact_8535_ATP_Olambda__358,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_ring_1(B)
     => ! [Uu: fun(C,int),Uua: C] : aa(C,B,aTP_Lamp_do(fun(C,int),fun(C,B),Uu),Uua) = aa(int,B,ring_1_of_int(B),aa(C,int,Uu,Uua)) ) ).

% ATP.lambda_358
tff(fact_8536_ATP_Olambda__359,axiom,
    ! [B: $tType,C: $tType] :
      ( ring_1(B)
     => ! [Uu: fun(C,int),Uua: C] : aa(C,B,aTP_Lamp_bd(fun(C,int),fun(C,B),Uu),Uua) = aa(int,B,ring_1_of_int(B),aa(C,int,Uu,Uua)) ) ).

% ATP.lambda_359
tff(fact_8537_ATP_Olambda__360,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: C] : aa(C,heap_Time_Heap(B),aTP_Lamp_aj(fun(C,B),fun(C,heap_Time_Heap(B)),Uu),Uua) = aa(B,heap_Time_Heap(B),heap_Time_return(B),aa(C,B,Uu,Uua)) ).

% ATP.lambda_360
tff(fact_8538_ATP_Olambda__361,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat] : aa(nat,fun(B,B),aTP_Lamp_cj(fun(nat,B),fun(nat,fun(B,B)),Uu),Uua) = aa(B,fun(B,B),plus_plus(B),aa(nat,B,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8539_ATP_Olambda__362,axiom,
    ! [B: $tType,C: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(C,B),Uua: C] : aa(C,B,aTP_Lamp_bh(fun(C,B),fun(C,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_362
tff(fact_8540_ATP_Olambda__363,axiom,
    ! [B: $tType,C: $tType] :
      ( linordered_field(B)
     => ! [Uu: fun(C,B),Uua: C] : aa(C,B,aTP_Lamp_dx(fun(C,B),fun(C,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8541_ATP_Olambda__364,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,B),Uua: C] : aa(C,option(B),aTP_Lamp_zh(fun(C,B),fun(C,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_364
tff(fact_8542_ATP_Olambda__365,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: C] : aa(C,option(B),aTP_Lamp_jy(fun(C,B),fun(C,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(C,B,Uu,Uua)) ).

% ATP.lambda_365
tff(fact_8543_ATP_Olambda__366,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [Uu: fun(B,C),Uua: B] : aa(B,option(C),aTP_Lamp_wd(fun(B,C),fun(B,option(C)),Uu),Uua) = aa(C,option(C),some(C),aa(B,C,Uu,Uua)) ) ).

% ATP.lambda_366
tff(fact_8544_ATP_Olambda__367,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: B] : aa(B,option(C),aTP_Lamp_ank(fun(B,C),fun(B,option(C)),Uu),Uua) = aa(C,option(C),some(C),aa(B,C,Uu,Uua)) ).

% ATP.lambda_367
tff(fact_8545_ATP_Olambda__368,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(D,set(product_prod(C,B))),Uua: D] : aa(D,set(product_prod(B,C)),aTP_Lamp_ajg(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(B,C))),Uu),Uua) = converse(C,B,aa(D,set(product_prod(C,B)),Uu,Uua)) ).

% ATP.lambda_368
tff(fact_8546_ATP_Olambda__369,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: D] : aa(D,fun(C,product_prod(B,C)),aTP_Lamp_so(fun(D,B),fun(D,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uu,Uua)) ).

% ATP.lambda_369
tff(fact_8547_ATP_Olambda__370,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: B] : aa(B,fun(D,product_prod(C,D)),aTP_Lamp_aey(fun(B,C),fun(B,fun(D,product_prod(C,D))),Uu),Uua) = aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(B,C,Uu,Uua)) ).

% ATP.lambda_370
tff(fact_8548_ATP_Olambda__371,axiom,
    ! [D: $tType,E: $tType,Uu: fun(E,set(D)),Uua: E] : aa(E,filter(D),aTP_Lamp_ajh(fun(E,set(D)),fun(E,filter(D)),Uu),Uua) = principal(D,aa(E,set(D),Uu,Uua)) ).

% ATP.lambda_371
tff(fact_8549_ATP_Olambda__372,axiom,
    ! [E: $tType,D: $tType,Uu: fun(D,set(E)),Uua: D] : aa(D,filter(E),aTP_Lamp_ajl(fun(D,set(E)),fun(D,filter(E)),Uu),Uua) = principal(E,aa(D,set(E),Uu,Uua)) ).

% ATP.lambda_372
tff(fact_8550_ATP_Olambda__373,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: C] : aa(C,filter(B),aTP_Lamp_aei(fun(C,set(B)),fun(C,filter(B)),Uu),Uua) = principal(B,aa(C,set(B),Uu,Uua)) ).

% ATP.lambda_373
tff(fact_8551_ATP_Olambda__374,axiom,
    ! [F6: $tType,B: $tType,Uu: fun(B,set(F6)),Uua: B] : aa(B,filter(F6),aTP_Lamp_ajk(fun(B,set(F6)),fun(B,filter(F6)),Uu),Uua) = principal(F6,aa(B,set(F6),Uu,Uua)) ).

% ATP.lambda_374
tff(fact_8552_ATP_Olambda__375,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,filter(C),aTP_Lamp_aeh(fun(B,set(C)),fun(B,filter(C)),Uu),Uua) = principal(C,aa(B,set(C),Uu,Uua)) ).

% ATP.lambda_375
tff(fact_8553_ATP_Olambda__376,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,nat,aTP_Lamp_vp(fun(B,set(C)),fun(B,nat),Uu),Uua) = aa(set(C),nat,finite_card(C),aa(B,set(C),Uu,Uua)) ).

% ATP.lambda_376
tff(fact_8554_ATP_Olambda__377,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,list(B)),Uua: C] : aa(C,set(B),aTP_Lamp_adl(fun(C,list(B)),fun(C,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),aa(C,list(B),Uu,Uua)) ).

% ATP.lambda_377
tff(fact_8555_ATP_Olambda__378,axiom,
    ! [B: $tType,Uu: fun(set(B),fun(B,$o)),Uua: set(B)] : aa(set(B),set(B),aTP_Lamp_anl(fun(set(B),fun(B,$o)),fun(set(B),set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),Uu,Uua)) ).

% ATP.lambda_378
tff(fact_8556_ATP_Olambda__379,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: B] : aa(B,set(C),aTP_Lamp_wm(fun(B,fun(C,$o)),fun(B,set(C)),Uu),Uua) = aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),Uu,Uua)) ).

% ATP.lambda_379
tff(fact_8557_ATP_Olambda__380,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: B] : aa(B,fun(set(C),set(C)),aTP_Lamp_aby(fun(B,C),fun(B,fun(set(C),set(C))),Uu),Uua) = insert(C,aa(B,C,Uu,Uua)) ).

% ATP.lambda_380
tff(fact_8558_ATP_Olambda__381,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: C] : aa(C,set(set(B)),aTP_Lamp_adc(fun(C,set(B)),fun(C,set(set(B))),Uu),Uua) = pow2(B,aa(C,set(B),Uu,Uua)) ).

% ATP.lambda_381
tff(fact_8559_ATP_Olambda__382,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,nat,aTP_Lamp_gu(fun(B,nat),fun(B,nat),Uu),Uua) = aa(nat,nat,suc,aa(B,nat,Uu,Uua)) ).

% ATP.lambda_382
tff(fact_8560_ATP_Olambda__383,axiom,
    ! [C: $tType,Uu: fun(C,$o),Uua: C] :
      ( aa(C,$o,aTP_Lamp_aav(fun(C,$o),fun(C,$o),Uu),Uua)
    <=> ~ aa(C,$o,Uu,Uua) ) ).

% ATP.lambda_383
tff(fact_8561_ATP_Olambda__384,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ae(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_384
tff(fact_8562_ATP_Olambda__385,axiom,
    ! [C: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(C,fun(B,$o)),Uua: C] :
          ( aa(C,$o,aTP_Lamp_ama(fun(C,fun(B,$o)),fun(C,$o),Uu),Uua)
        <=> ! [X_12: B] : aa(B,$o,aa(C,fun(B,$o),Uu,Uua),X_12) ) ) ).

% ATP.lambda_385
tff(fact_8563_ATP_Olambda__386,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,fun(B,$o)),Uua: C] :
      ( aa(C,$o,aTP_Lamp_ana(fun(C,fun(B,$o)),fun(C,$o),Uu),Uua)
    <=> ! [X_12: B] : aa(B,$o,aa(C,fun(B,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_386
tff(fact_8564_ATP_Olambda__387,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aim(fun(B,fun(C,$o)),fun(B,$o),Uu),Uua)
    <=> ! [X_12: C] : aa(C,$o,aa(B,fun(C,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_387
tff(fact_8565_ATP_Olambda__388,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ut(fun(B,fun(C,$o)),fun(B,$o),Uu),Uua)
    <=> ? [X_12: C] : aa(C,$o,aa(B,fun(C,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_388
tff(fact_8566_ATP_Olambda__389,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B)] : aa(set(B),filter(set(B)),aTP_Lamp_ajp(set(B),fun(set(B),filter(set(B))),Uu),Uua) = principal(set(B),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aa(set(B),fun(set(B),$o),aTP_Lamp_ajo(set(B),fun(set(B),fun(set(B),$o)),Uu),Uua))) ).

% ATP.lambda_389
tff(fact_8567_ATP_Olambda__390,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,$o),Uua: fun(C,$o)] : aa(fun(C,$o),filter(product_prod(B,C)),aa(fun(B,$o),fun(fun(C,$o),filter(product_prod(B,C))),aTP_Lamp_aow(fun(B,$o),fun(fun(C,$o),filter(product_prod(B,C)))),Uu),Uua) = principal(product_prod(B,C),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(fun(C,$o),fun(B,fun(C,$o)),aTP_Lamp_vq(fun(B,$o),fun(fun(C,$o),fun(B,fun(C,$o))),Uu),Uua)))) ).

% ATP.lambda_390
tff(fact_8568_ATP_Olambda__391,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,$o)),Uua: C] :
      ( aa(C,$o,aTP_Lamp_uu(fun(B,fun(C,$o)),fun(C,$o),Uu),Uua)
    <=> ? [A6: B] : aa(C,$o,aa(B,fun(C,$o),Uu,A6),Uua) ) ).

% ATP.lambda_391
tff(fact_8569_ATP_Olambda__392,axiom,
    ! [B: $tType,Uu: list(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_va(list(B),fun(B,$o),Uu),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,B,nth(B,Uu),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Uu)) ) ) ).

% ATP.lambda_392
tff(fact_8570_ATP_Olambda__393,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: set(set(option(B))),Uua: set(option(B))] :
          ( aa(set(option(B)),$o,aTP_Lamp_aku(set(set(option(B))),fun(set(option(B)),$o),Uu),Uua)
        <=> ? [F5: fun(set(option(B)),option(B))] :
              ( ( Uua = aa(set(set(option(B))),set(option(B)),image2(set(option(B)),option(B),F5),Uu) )
              & ! [X3: set(option(B))] :
                  ( aa(set(set(option(B))),$o,member(set(option(B)),X3),Uu)
                 => aa(set(option(B)),$o,member(option(B),aa(set(option(B)),option(B),F5,X3)),X3) ) ) ) ) ).

% ATP.lambda_393
tff(fact_8571_ATP_Olambda__394,axiom,
    ! [C: $tType,Uu: set(set(C)),Uua: set(C)] :
      ( aa(set(C),$o,aTP_Lamp_akt(set(set(C)),fun(set(C),$o),Uu),Uua)
    <=> ? [F5: fun(set(C),C)] :
          ( ( Uua = aa(set(set(C)),set(C),image2(set(C),C,F5),Uu) )
          & ! [X3: set(C)] :
              ( aa(set(set(C)),$o,member(set(C),X3),Uu)
             => aa(set(C),$o,member(C,aa(set(C),C,F5,X3)),X3) ) ) ) ).

% ATP.lambda_394
tff(fact_8572_ATP_Olambda__395,axiom,
    ! [B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: set(set(B)),Uua: set(B)] :
          ( aa(set(B),$o,aTP_Lamp_akn(set(set(B)),fun(set(B),$o),Uu),Uua)
        <=> ? [F5: fun(set(B),B)] :
              ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F5),Uu) )
              & ! [X3: set(B)] :
                  ( aa(set(set(B)),$o,member(set(B),X3),Uu)
                 => aa(set(B),$o,member(B,aa(set(B),B,F5,X3)),X3) ) ) ) ) ).

% ATP.lambda_395
tff(fact_8573_ATP_Olambda__396,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: set(set(B)),Uua: set(B)] :
          ( aa(set(B),$o,aTP_Lamp_akp(set(set(B)),fun(set(B),$o),Uu),Uua)
        <=> ? [F5: fun(set(B),B)] :
              ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F5),Uu) )
              & ! [X3: set(B)] :
                  ( aa(set(set(B)),$o,member(set(B),X3),Uu)
                 => aa(set(B),$o,member(B,aa(set(B),B,F5,X3)),X3) ) ) ) ) ).

% ATP.lambda_396
tff(fact_8574_ATP_Olambda__397,axiom,
    ! [B: $tType] :
      ( finite8700451911770168679attice(B)
     => ! [Uu: set(set(B)),Uua: set(B)] :
          ( aa(set(B),$o,aTP_Lamp_akq(set(set(B)),fun(set(B),$o),Uu),Uua)
        <=> ? [F5: fun(set(B),B)] :
              ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F5),Uu) )
              & ! [X3: set(B)] :
                  ( aa(set(set(B)),$o,member(set(B),X3),Uu)
                 => aa(set(B),$o,member(B,aa(set(B),B,F5,X3)),X3) ) ) ) ) ).

% ATP.lambda_397
tff(fact_8575_ATP_Olambda__398,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_adb(set(B),fun(set(B),$o),Uu),Uua)
    <=> ? [B8: set(B)] :
          ( ( Uua = aa(set(B),set(B),uminus_uminus(set(B)),B8) )
          & aa(set(set(B)),$o,member(set(B),Uu),pow2(B,B8)) ) ) ).

% ATP.lambda_398
tff(fact_8576_ATP_Olambda__399,axiom,
    ! [B: $tType,Uu: set(filter(B)),Uua: filter(B)] :
      ( aa(filter(B),$o,aTP_Lamp_ako(set(filter(B)),fun(filter(B),$o),Uu),Uua)
    <=> ! [X3: filter(B)] :
          ( aa(set(filter(B)),$o,member(filter(B),X3),Uu)
         => aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),Uua),X3) ) ) ).

% ATP.lambda_399
tff(fact_8577_ATP_Olambda__400,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: set(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_aon(set(B),fun(B,$o),Uu),Uua)
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),Uu)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),X3) ) ) ) ).

% ATP.lambda_400
tff(fact_8578_ATP_Olambda__401,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: set(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_akd(set(B),fun(B,$o),Uu),Uua)
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),Uu)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),X3) ) ) ) ).

% ATP.lambda_401
tff(fact_8579_ATP_Olambda__402,axiom,
    ! [B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: set(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_aod(set(B),fun(B,$o),Uu),Uua)
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),Uu)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Uua) ) ) ) ).

% ATP.lambda_402
tff(fact_8580_ATP_Olambda__403,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: set(B),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ake(set(B),fun(B,$o),Uu),Uua)
        <=> ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),Uu)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Uua) ) ) ) ).

% ATP.lambda_403
tff(fact_8581_ATP_Olambda__404,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aTP_Lamp_akw(set(multiset(B)),fun(multiset(B),$o),Uu),Uua)
    <=> ! [X3: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X3),Uu)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),Uua),X3) ) ) ).

% ATP.lambda_404
tff(fact_8582_ATP_Olambda__405,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: multiset(B)] :
      ( aa(multiset(B),$o,aTP_Lamp_akv(set(multiset(B)),fun(multiset(B),$o),Uu),Uua)
    <=> ! [X3: multiset(B)] :
          ( aa(set(multiset(B)),$o,member(multiset(B),X3),Uu)
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),X3),Uua) ) ) ).

% ATP.lambda_405
tff(fact_8583_ATP_Olambda__406,axiom,
    ! [B: $tType,Uu: set(filter(B)),Uua: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aTP_Lamp_ams(set(filter(B)),fun(fun(B,$o),$o),Uu),Uua)
    <=> ! [X3: filter(B)] :
          ( aa(set(filter(B)),$o,member(filter(B),X3),Uu)
         => eventually(B,Uua,X3) ) ) ).

% ATP.lambda_406
tff(fact_8584_ATP_Olambda__407,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_akk(set(set(B)),fun(B,$o),Uu),Uua)
    <=> ! [X3: set(B)] :
          ( aa(set(set(B)),$o,member(set(B),X3),Uu)
         => aa(set(B),$o,member(B,Uua),X3) ) ) ).

% ATP.lambda_407
tff(fact_8585_ATP_Olambda__408,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ary(set(set(B)),fun(B,$o),Uu),Uua)
    <=> ? [X3: set(B)] :
          ( aa(set(set(B)),$o,member(set(B),X3),Uu)
          & aa(set(B),$o,member(B,Uua),X3) ) ) ).

% ATP.lambda_408
tff(fact_8586_ATP_Olambda__409,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(B,$o),Uua: B] :
          ( aa(B,$o,aTP_Lamp_amc(fun(B,$o),fun(B,$o),Uu),Uua)
        <=> ! [Y5: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),Y5)
             => aa(B,$o,Uu,Y5) ) ) ) ).

% ATP.lambda_409
tff(fact_8587_ATP_Olambda__410,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_akb(fun(B,$o),fun(set(B),$o),Uu),Uua)
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),Uua)
         => aa(B,$o,Uu,X3) ) ) ).

% ATP.lambda_410
tff(fact_8588_ATP_Olambda__411,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,option(B)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_us(fun(C,option(B)),fun(B,$o),Uu),Uua)
    <=> ? [A6: C] : aa(C,option(B),Uu,A6) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_411
tff(fact_8589_ATP_Olambda__412,axiom,
    ! [B: $tType,C: $tType,Uu: B,Uua: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aTP_Lamp_ul(B,fun(product_prod(B,C),$o),Uu),Uua)
    <=> ? [V6: C] : Uua = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),V6) ) ).

% ATP.lambda_412
tff(fact_8590_ATP_Olambda__413,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_vm(fun(C,B),fun(B,$o),Uu),Uua)
    <=> ? [X3: C] : Uua = aa(C,B,Uu,X3) ) ).

% ATP.lambda_413
tff(fact_8591_ATP_Olambda__414,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,$o),Uua: fun(B,C)] :
      ( aa(fun(B,C),$o,aTP_Lamp_ais(fun(C,$o),fun(fun(B,C),$o),Uu),Uua)
    <=> ! [X3: B] : aa(C,$o,Uu,aa(B,C,Uua,X3)) ) ).

% ATP.lambda_414
tff(fact_8592_ATP_Olambda__415,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: product_prod(product_prod($o,B),product_prod($o,C))] :
      ( aa(product_prod(product_prod($o,B),product_prod($o,C)),$o,aTP_Lamp_abt(set(product_prod(B,C)),fun(product_prod(product_prod($o,B),product_prod($o,C)),$o),Uu),Uua)
    <=> ? [X3: B,Y5: C] :
          ( ( Uua = aa(product_prod($o,C),product_prod(product_prod($o,B),product_prod($o,C)),aa(product_prod($o,B),fun(product_prod($o,C),product_prod(product_prod($o,B),product_prod($o,C))),product_Pair(product_prod($o,B),product_prod($o,C)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),X3)),aa(C,product_prod($o,C),aa($o,fun(C,product_prod($o,C)),product_Pair($o,C),$false),Y5)) )
          & aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y5)),Uu) ) ) ).

% ATP.lambda_415
tff(fact_8593_ATP_Olambda__416,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,option(C)),Uua: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aTP_Lamp_aoa(fun(B,option(C)),fun(product_prod(B,C),$o),Uu),Uua)
    <=> ? [A6: B,B9: C] :
          ( ( Uua = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A6),B9) )
          & ( aa(B,option(C),Uu,A6) = aa(C,option(C),some(C),B9) ) ) ) ).

% ATP.lambda_416
tff(fact_8594_ATP_Olambda__417,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: product_prod(set(B),set(B))] :
      ( aa(product_prod(set(B),set(B)),$o,aTP_Lamp_asg(set(product_prod(B,B)),fun(product_prod(set(B),set(B)),$o),Uu),Uua)
    <=> ? [X7: set(B),Y7: set(B)] :
          ( ( Uua = aa(set(B),product_prod(set(B),set(B)),aa(set(B),fun(set(B),product_prod(set(B),set(B))),product_Pair(set(B),set(B)),X7),Y7) )
          & ( X7 != bot_bot(set(B)) )
          & ! [X3: B] :
              ( aa(set(B),$o,member(B,X3),Y7)
             => ? [Xa4: B] :
                  ( aa(set(B),$o,member(B,Xa4),X7)
                  & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Xa4),X3)),Uu) ) ) ) ) ).

% ATP.lambda_417
tff(fact_8595_ATP_Olambda__418,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_vw(set(C),fun(B,set(C)),Uu),Uua) = aa(set(C),set(C),uminus_uminus(set(C)),Uu) ).

% ATP.lambda_418
tff(fact_8596_ATP_Olambda__419,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_adw(set(B),fun(B,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_419
tff(fact_8597_ATP_Olambda__420,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: array(B),Uua: array(B)] : aa(array(B),heap_Time_Heap(array(B)),aTP_Lamp_av(array(B),fun(array(B),heap_Time_Heap(array(B))),Uu),Uua) = aa(array(B),heap_Time_Heap(array(B)),heap_Time_ureturn(array(B)),Uu) ) ).

% ATP.lambda_420
tff(fact_8598_ATP_Olambda__421,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: B,Uua: product_unit] : aa(product_unit,heap_Time_Heap(B),aTP_Lamp_an(B,fun(product_unit,heap_Time_Heap(B)),Uu),Uua) = aa(B,heap_Time_Heap(B),heap_Time_return(B),Uu) ) ).

% ATP.lambda_421
tff(fact_8599_ATP_Olambda__422,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),option(B),aa(B,fun(list(B),option(B)),aTP_Lamp_rm(B,fun(list(B),option(B))),Uu),Uua) = aa(B,option(B),some(B),Uu) ).

% ATP.lambda_422
tff(fact_8600_ATP_Olambda__423,axiom,
    ! [B: $tType,C: $tType,Uu: list(C),Uua: B] : aa(B,set(C),aTP_Lamp_vy(list(C),fun(B,set(C)),Uu),Uua) = aa(list(C),set(C),set2(C),Uu) ).

% ATP.lambda_423
tff(fact_8601_ATP_Olambda__424,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,$o),Uua: B] : aa(B,set(C),aTP_Lamp_vr(fun(C,$o),fun(B,set(C)),Uu),Uua) = aa(fun(C,$o),set(C),collect(C),Uu) ).

% ATP.lambda_424
tff(fact_8602_ATP_Olambda__425,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),fun(set(B),set(B)),aa(B,fun(list(B),fun(set(B),set(B))),aTP_Lamp_ajr(B,fun(list(B),fun(set(B),set(B)))),Uu),Uua) = insert(B,Uu) ).

% ATP.lambda_425
tff(fact_8603_ATP_Olambda__426,axiom,
    ! [B: $tType,Uu: nat,Uua: B] : aa(B,nat,aa(nat,fun(B,nat),aTP_Lamp_ht(nat,fun(B,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).

% ATP.lambda_426
tff(fact_8604_ATP_Olambda__427,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_ln(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),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,one2))),Uua)),Uub)) ).

% ATP.lambda_427
tff(fact_8605_ATP_Olambda__428,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_lw(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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,one2))),Uua)),Uub)) ).

% ATP.lambda_428
tff(fact_8606_ATP_Olambda__429,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lx(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),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,one2))),Uua)),Uub)) ).

% ATP.lambda_429
tff(fact_8607_ATP_Olambda__430,axiom,
    ! [B: $tType] :
      ( unique1627219031080169319umeral(B)
     => ! [Uu: num,Uua: B,Uub: B] :
          aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),aTP_Lamp_ls(num,fun(B,fun(B,product_prod(B,B))),Uu),Uua),Uub) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(num,B,numeral_numeral(B),Uu)),Uub),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Uua)),one_one(B))),aa(B,B,aa(B,fun(B,B),minus_minus(B),Uub),aa(num,B,numeral_numeral(B),Uu))),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_430
tff(fact_8608_ATP_Olambda__431,axiom,
    ! [B: $tType,Uu: B,Uua: B,Uub: list(B)] :
      aa(list(B),list(B),aa(B,fun(list(B),list(B)),aTP_Lamp_aem(B,fun(B,fun(list(B),list(B))),Uu),Uua),Uub) = $ite(Uu = Uua,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uua),Uub),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uua),Uub))) ).

% ATP.lambda_431
tff(fact_8609_ATP_Olambda__432,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(B,C),Uua: set(B),Uub: B] :
          aa(B,C,aa(set(B),fun(B,C),aTP_Lamp_kr(fun(B,C),fun(set(B),fun(B,C)),Uu),Uua),Uub) = $ite(aa(set(B),$o,member(B,Uub),Uua),aa(B,C,Uu,Uub),zero_zero(C)) ) ).

% ATP.lambda_432
tff(fact_8610_ATP_Olambda__433,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(B,C),Uua: set(B),Uub: B] :
          aa(B,C,aa(set(B),fun(B,C),aTP_Lamp_ks(fun(B,C),fun(set(B),fun(B,C)),Uu),Uua),Uub) = $ite(aa(set(B),$o,member(B,Uub),Uua),aa(B,C,Uu,Uub),one_one(C)) ) ).

% ATP.lambda_433
tff(fact_8611_ATP_Olambda__434,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ev(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(B,C,Uua,Uub),zero_zero(C)) ) ).

% ATP.lambda_434
tff(fact_8612_ATP_Olambda__435,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ew(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(B,C,Uua,Uub),one_one(C)) ) ).

% ATP.lambda_435
tff(fact_8613_ATP_Olambda__436,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_eu(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(B,C,Uua,Uub),zero_zero(C)) ) ).

% ATP.lambda_436
tff(fact_8614_ATP_Olambda__437,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_ex(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(B,C,Uua,Uub),one_one(C)) ) ).

% ATP.lambda_437
tff(fact_8615_ATP_Olambda__438,axiom,
    ! [B: $tType,Uu: B,Uua: fun(B,nat),Uub: B] :
      aa(B,nat,aa(fun(B,nat),fun(B,nat),aa(B,fun(fun(B,nat),fun(B,nat)),aTP_Lamp_aia(B,fun(fun(B,nat),fun(B,nat))),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,nat,suc,aa(B,nat,Uua,Uub)),aa(B,nat,Uua,Uub)) ).

% ATP.lambda_438
tff(fact_8616_ATP_Olambda__439,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B,Uub: B] :
      aa(B,nat,aa(B,fun(B,nat),aTP_Lamp_aiu(fun(B,nat),fun(B,fun(B,nat)),Uu),Uua),Uub) = $ite(Uub = Uua,aa(nat,nat,suc,aa(B,nat,Uu,Uub)),aa(B,nat,Uu,Uub)) ).

% ATP.lambda_439
tff(fact_8617_ATP_Olambda__440,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: C,Uua: B,Uub: C] :
          aa(C,B,aa(B,fun(C,B),aTP_Lamp_ahq(C,fun(B,fun(C,B)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,zero_zero(B)) ) ).

% ATP.lambda_440
tff(fact_8618_ATP_Olambda__441,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: C,Uua: B,Uub: C] :
          aa(C,B,aa(B,fun(C,B),aTP_Lamp_aii(C,fun(B,fun(C,B)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,one_one(B)) ) ).

% ATP.lambda_441
tff(fact_8619_ATP_Olambda__442,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: C,Uua: B,Uub: C] :
          aa(C,B,aa(B,fun(C,B),aTP_Lamp_ahr(C,fun(B,fun(C,B)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,zero_zero(B)) ) ).

% ATP.lambda_442
tff(fact_8620_ATP_Olambda__443,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: C,Uua: B,Uub: C] :
          aa(C,B,aa(B,fun(C,B),aTP_Lamp_aih(C,fun(B,fun(C,B)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,one_one(B)) ) ).

% ATP.lambda_443
tff(fact_8621_ATP_Olambda__444,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_mm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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_444
tff(fact_8622_ATP_Olambda__445,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_mk(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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_445
tff(fact_8623_ATP_Olambda__446,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_ml(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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_446
tff(fact_8624_ATP_Olambda__447,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B,Uub: list(B)] :
      aa(list(B),list(B),aa(B,fun(list(B),list(B)),aTP_Lamp_pg(fun(B,$o),fun(B,fun(list(B),list(B))),Uu),Uua),Uub) = $ite(aa(B,$o,Uu,Uua),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uua),Uub),Uub) ).

% ATP.lambda_447
tff(fact_8625_ATP_Olambda__448,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B,Uub: set(B)] :
      aa(set(B),set(B),aa(B,fun(set(B),set(B)),aTP_Lamp_ace(fun(B,$o),fun(B,fun(set(B),set(B))),Uu),Uua),Uub) = $ite(aa(B,$o,Uu,Uua),aa(set(B),set(B),insert(B,Uua),Uub),Uub) ).

% ATP.lambda_448
tff(fact_8626_ATP_Olambda__449,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,nat),Uub: B] :
      aa(B,nat,aa(fun(B,nat),fun(B,nat),aa(fun(B,$o),fun(fun(B,nat),fun(B,nat)),aTP_Lamp_aib(fun(B,$o),fun(fun(B,nat),fun(B,nat))),Uu),Uua),Uub) = $ite(aa(B,$o,Uu,Uub),aa(B,nat,Uua,Uub),zero_zero(nat)) ).

% ATP.lambda_449
tff(fact_8627_ATP_Olambda__450,axiom,
    ! [C: $tType,B: $tType] :
      ( monoid_add(B)
     => ! [Uu: fun(C,B),Uua: fun(C,$o),Uub: C] :
          aa(C,B,aa(fun(C,$o),fun(C,B),aTP_Lamp_rb(fun(C,B),fun(fun(C,$o),fun(C,B)),Uu),Uua),Uub) = $ite(aa(C,$o,Uua,Uub),aa(C,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_450
tff(fact_8628_ATP_Olambda__451,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: fun(B,$o),Uub: B] :
      aa(B,nat,aa(fun(B,$o),fun(B,nat),aTP_Lamp_ait(fun(B,nat),fun(fun(B,$o),fun(B,nat)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,nat,Uu,Uub),zero_zero(nat)) ).

% ATP.lambda_451
tff(fact_8629_ATP_Olambda__452,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(B,C),Uua: fun(B,$o),Uub: B] :
          aa(B,C,aa(fun(B,$o),fun(B,C),aTP_Lamp_fs(fun(B,C),fun(fun(B,$o),fun(B,C)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,C,Uu,Uub),zero_zero(C)) ) ).

% ATP.lambda_452
tff(fact_8630_ATP_Olambda__453,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(B,C),Uua: fun(B,$o),Uub: B] :
          aa(B,C,aa(fun(B,$o),fun(B,C),aTP_Lamp_fv(fun(B,C),fun(fun(B,$o),fun(B,C)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,C,Uu,Uub),one_one(C)) ) ).

% ATP.lambda_453
tff(fact_8631_ATP_Olambda__454,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,fun(B,B)),Uua: fun(C,$o),Uub: C] :
      aa(C,fun(B,B),aa(fun(C,$o),fun(C,fun(B,B)),aTP_Lamp_rw(fun(C,fun(B,B)),fun(fun(C,$o),fun(C,fun(B,B))),Uu),Uua),Uub) = $ite(aa(C,$o,Uua,Uub),aa(C,fun(B,B),Uu,Uub),id(B)) ).

% ATP.lambda_454
tff(fact_8632_ATP_Olambda__455,axiom,
    ! [B: $tType,Uu: fun(heap_ext(product_unit),$o),Uua: fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),Uub: heap_ext(product_unit)] :
      aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_nn(fun(heap_ext(product_unit),$o),fun(fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = $ite(aa(heap_ext(product_unit),$o,Uu,Uub),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(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),Uua,Uub)),none(product_prod(B,product_prod(heap_ext(product_unit),nat)))) ).

% ATP.lambda_455
tff(fact_8633_ATP_Olambda__456,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,B),Uua: fun(C,$o),Uub: C] :
      aa(C,option(B),aa(fun(C,$o),fun(C,option(B)),aTP_Lamp_qg(fun(C,B),fun(fun(C,$o),fun(C,option(B))),Uu),Uua),Uub) = $ite(aa(C,$o,Uua,Uub),aa(B,option(B),some(B),aa(C,B,Uu,Uub)),none(B)) ).

% ATP.lambda_456
tff(fact_8634_ATP_Olambda__457,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B,Uub: set(product_prod(B,C))] : aa(set(product_prod(B,C)),set(product_prod(B,C)),aa(B,fun(set(product_prod(B,C)),set(product_prod(B,C))),aTP_Lamp_aca(set(C),fun(B,fun(set(product_prod(B,C)),set(product_prod(B,C)))),Uu),Uua),Uub) = finite_fold(C,set(product_prod(B,C)),aTP_Lamp_abz(B,fun(C,fun(set(product_prod(B,C)),set(product_prod(B,C)))),Uua),Uub,Uu) ).

% ATP.lambda_457
tff(fact_8635_ATP_Olambda__458,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C,Uub: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(C,fun(set(product_prod(C,B)),set(product_prod(C,B))),aTP_Lamp_acm(set(B),fun(C,fun(set(product_prod(C,B)),set(product_prod(C,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(C,B)),aTP_Lamp_abx(C,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),Uua),Uub,Uu) ).

% ATP.lambda_458
tff(fact_8636_ATP_Olambda__459,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: list(B),Uub: list(B)] : aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aTP_Lamp_vf(fun(B,fun(B,$o)),fun(list(B),fun(list(B),list(B))),Uu),Uua),Uub) = merges9089515139780605204_merge(B,Uu,aa(list(B),list(B),mergesort_by_rel(B,Uu),Uua),aa(list(B),list(B),mergesort_by_rel(B,Uu),Uub)) ).

% ATP.lambda_459
tff(fact_8637_ATP_Olambda__460,axiom,
    ! [B: $tType] :
      ( sup(B)
     => ! [Uu: option(B),Uua: option(B),Uub: B] : aa(B,option(B),aa(option(B),fun(B,option(B)),aTP_Lamp_abd(option(B),fun(option(B),fun(B,option(B))),Uu),Uua),Uub) = case_option(option(B),B,Uu,aTP_Lamp_abc(B,fun(B,option(B)),Uub),Uua) ) ).

% ATP.lambda_460
tff(fact_8638_ATP_Olambda__461,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,option(C)),Uua: B,Uub: C] : aa(C,fun(B,option(C)),aa(B,fun(C,fun(B,option(C))),aTP_Lamp_sr(fun(B,option(C)),fun(B,fun(C,fun(B,option(C)))),Uu),Uua),Uub) = fun_upd(B,option(C),Uu,Uua,aa(C,option(C),some(C),Uub)) ).

% ATP.lambda_461
tff(fact_8639_ATP_Olambda__462,axiom,
    ! [B: $tType,C: $tType,Uu: B,Uua: C,Uub: fun(B,option(C))] : aa(fun(B,option(C)),fun(B,option(C)),aa(C,fun(fun(B,option(C)),fun(B,option(C))),aa(B,fun(C,fun(fun(B,option(C)),fun(B,option(C)))),aTP_Lamp_ue(B,fun(C,fun(fun(B,option(C)),fun(B,option(C))))),Uu),Uua),Uub) = fun_upd(B,option(C),Uub,Uu,aa(C,option(C),some(C),Uua)) ).

% ATP.lambda_462
tff(fact_8640_ATP_Olambda__463,axiom,
    ! [B: $tType,Uu: fun(B,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_afn(fun(B,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Uub),Uu),Uua) ).

% ATP.lambda_463
tff(fact_8641_ATP_Olambda__464,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: $o] :
      aa($o,set(B),aa(set(B),fun($o,set(B)),aTP_Lamp_abs(set(B),fun(set(B),fun($o,set(B))),Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_464
tff(fact_8642_ATP_Olambda__465,axiom,
    ! [B: $tType,C: $tType,Uu: heap_Time_Heap(C),Uua: fun(C,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(C,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ng(heap_Time_Heap(C),fun(fun(C,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(C,product_prod(heap_ext(product_unit),nat)),none(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(C,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(C,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(C,product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_nb(fun(C,heap_Time_Heap(B)),fun(C,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(C,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(C,Uu),Uub)) ).

% ATP.lambda_465
tff(fact_8643_ATP_Olambda__466,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,Uu: fun(E,fun(C,D)),Uua: fun(B,E),Uub: product_prod(B,C)] : aa(product_prod(B,C),D,aa(fun(B,E),fun(product_prod(B,C),D),aTP_Lamp_tk(fun(E,fun(C,D)),fun(fun(B,E),fun(product_prod(B,C),D)),Uu),Uua),Uub) = aa(C,D,aa(E,fun(C,D),Uu,aa(B,E,Uua,aa(product_prod(B,C),B,product_fst(B,C),Uub))),aa(product_prod(B,C),C,product_snd(B,C),Uub)) ).

% ATP.lambda_466
tff(fact_8644_ATP_Olambda__467,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: fun(C,D),Uub: C] : aa(C,B,aa(fun(C,D),fun(C,B),aTP_Lamp_zr(fun(D,fun(C,B)),fun(fun(C,D),fun(C,B)),Uu),Uua),Uub) = aa(C,B,aa(D,fun(C,B),Uu,aa(C,D,Uua,Uub)),Uub) ) ).

% ATP.lambda_467
tff(fact_8645_ATP_Olambda__468,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_lh(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,aa(nat,fun(nat,B),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_468
tff(fact_8646_ATP_Olambda__469,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_lf(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,aa(nat,fun(nat,B),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_469
tff(fact_8647_ATP_Olambda__470,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(B,C),Uub: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_alo(fun(B,fun(C,$o)),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,Uub),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_470
tff(fact_8648_ATP_Olambda__471,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ea(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,aa(nat,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_471
tff(fact_8649_ATP_Olambda__472,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ek(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,aa(nat,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_472
tff(fact_8650_ATP_Olambda__473,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: C,Uub: D] : aa(D,B,aa(C,fun(D,B),aTP_Lamp_zp(fun(D,fun(C,B)),fun(C,fun(D,B)),Uu),Uua),Uub) = aa(C,B,aa(D,fun(C,B),Uu,Uub),Uua) ) ).

% ATP.lambda_473
tff(fact_8651_ATP_Olambda__474,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( complete_Sup(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: C,Uub: D] : aa(D,B,aa(C,fun(D,B),aTP_Lamp_xn(fun(D,fun(C,B)),fun(C,fun(D,B)),Uu),Uua),Uub) = aa(C,B,aa(D,fun(C,B),Uu,Uub),Uua) ) ).

% ATP.lambda_474
tff(fact_8652_ATP_Olambda__475,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( complete_Inf(B)
     => ! [Uu: fun(D,fun(C,B)),Uua: C,Uub: D] : aa(D,B,aa(C,fun(D,B),aTP_Lamp_zb(fun(D,fun(C,B)),fun(C,fun(D,B)),Uu),Uua),Uub) = aa(C,B,aa(D,fun(C,B),Uu,Uub),Uua) ) ).

% ATP.lambda_475
tff(fact_8653_ATP_Olambda__476,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: D,Uub: C] : aa(C,B,aa(D,fun(C,B),aTP_Lamp_xw(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uua),Uub) = aa(D,B,aa(C,fun(D,B),Uu,Uub),Uua) ) ).

% ATP.lambda_476
tff(fact_8654_ATP_Olambda__477,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: D,Uub: C] : aa(C,B,aa(D,fun(C,B),aTP_Lamp_ds(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uua),Uub) = aa(D,B,aa(C,fun(D,B),Uu,Uub),Uua) ) ).

% ATP.lambda_477
tff(fact_8655_ATP_Olambda__478,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: D,Uub: C] : aa(C,B,aa(D,fun(C,B),aTP_Lamp_bm(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uua),Uub) = aa(D,B,aa(C,fun(D,B),Uu,Uub),Uua) ) ).

% ATP.lambda_478
tff(fact_8656_ATP_Olambda__479,axiom,
    ! [C: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(C,fun(B,$o)),Uua: B,Uub: C] :
          ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_alz(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(C,fun(B,$o),Uu,Uub),Uua) ) ) ).

% ATP.lambda_479
tff(fact_8657_ATP_Olambda__480,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,fun(B,$o)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_ui(fun(C,fun(B,$o)),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(C,fun(B,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_480
tff(fact_8658_ATP_Olambda__481,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,fun(B,B)),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_kp(fun(C,fun(B,B)),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(C,fun(B,B),Uu,Uub),Uua) ).

% ATP.lambda_481
tff(fact_8659_ATP_Olambda__482,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,$o)),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_yp(fun(B,fun(C,$o)),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_482
tff(fact_8660_ATP_Olambda__483,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: fun(B,fun(C,D)),Uua: C,Uub: B] : aa(B,D,aa(C,fun(B,D),aTP_Lamp_fj(fun(B,fun(C,D)),fun(C,fun(B,D)),Uu),Uua),Uub) = aa(C,D,aa(B,fun(C,D),Uu,Uub),Uua) ) ).

% ATP.lambda_483
tff(fact_8661_ATP_Olambda__484,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: fun(B,fun(C,D)),Uua: C,Uub: B] : aa(B,D,aa(C,fun(B,D),aTP_Lamp_ff(fun(B,fun(C,D)),fun(C,fun(B,D)),Uu),Uua),Uub) = aa(C,D,aa(B,fun(C,D),Uu,Uub),Uua) ) ).

% ATP.lambda_484
tff(fact_8662_ATP_Olambda__485,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,C)),Uua: C,Uub: B] : aa(B,C,aa(C,fun(B,C),aTP_Lamp_ack(fun(B,fun(C,C)),fun(C,fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(B,fun(C,C),Uu,Uub),Uua) ).

% ATP.lambda_485
tff(fact_8663_ATP_Olambda__486,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(C,B)),Uua: C,Uub: B] : aa(B,B,aa(C,fun(B,B),aTP_Lamp_kq(fun(B,fun(C,B)),fun(C,fun(B,B)),Uu),Uua),Uub) = aa(C,B,aa(B,fun(C,B),Uu,Uub),Uua) ).

% ATP.lambda_486
tff(fact_8664_ATP_Olambda__487,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_kl(fun(B,fun(B,$o)),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(B,fun(B,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_487
tff(fact_8665_ATP_Olambda__488,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_mn(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),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))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_488
tff(fact_8666_ATP_Olambda__489,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_eb(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ea(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_489
tff(fact_8667_ATP_Olambda__490,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,fun(nat,B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_el(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(nat,fun(nat,B),aTP_Lamp_ek(fun(nat,fun(nat,B)),fun(nat,fun(nat,B)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_490
tff(fact_8668_ATP_Olambda__491,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_apv(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_apu(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_491
tff(fact_8669_ATP_Olambda__492,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_apo($o,fun(code_integer,fun($o,char)),(Uu)),Uua),(Uub)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_apn($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_492
tff(fact_8670_ATP_Olambda__493,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ov(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_ou(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_493
tff(fact_8671_ATP_Olambda__494,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ot(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_os(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_494
tff(fact_8672_ATP_Olambda__495,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_or(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_oq(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_495
tff(fact_8673_ATP_Olambda__496,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_op(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_oo(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_496
tff(fact_8674_ATP_Olambda__497,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_od(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_oc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_497
tff(fact_8675_ATP_Olambda__498,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_498
tff(fact_8676_ATP_Olambda__499,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(C),Uub: D] : aa(D,B,aa(set(C),fun(D,B),aTP_Lamp_dt(fun(C,fun(D,B)),fun(set(C),fun(D,B)),Uu),Uua),Uub) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(D,fun(C,B),aTP_Lamp_ds(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uub)),Uua) ) ).

% ATP.lambda_499
tff(fact_8677_ATP_Olambda__500,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(C),Uub: D] : aa(D,B,aa(set(C),fun(D,B),aTP_Lamp_bn(fun(C,fun(D,B)),fun(set(C),fun(D,B)),Uu),Uua),Uub) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(D,fun(C,B),aTP_Lamp_bm(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uub)),Uua) ) ).

% ATP.lambda_500
tff(fact_8678_ATP_Olambda__501,axiom,
    ! [E: $tType,F6: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(C,fun(D,fun(E,fun(F6,set(B))))),Uua: product_prod(C,D),Uub: product_prod(E,F6)] : aa(product_prod(E,F6),set(B),aa(product_prod(C,D),fun(product_prod(E,F6),set(B)),aTP_Lamp_yw(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(product_prod(C,D),fun(product_prod(E,F6),set(B))),Uu),Uua),Uub) = aa(product_prod(C,D),set(B),aa(fun(C,fun(D,set(B))),fun(product_prod(C,D),set(B)),product_case_prod(C,D,set(B)),aa(product_prod(E,F6),fun(C,fun(D,set(B))),aTP_Lamp_yv(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(product_prod(E,F6),fun(C,fun(D,set(B)))),Uu),Uub)),Uua) ).

% ATP.lambda_501
tff(fact_8679_ATP_Olambda__502,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: filter(C),Uub: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(filter(C),fun(fun(B,$o),$o),aTP_Lamp_aos(fun(C,B),fun(filter(C),fun(fun(B,$o),$o)),Uu),Uua),Uub)
    <=> eventually(C,aa(fun(B,$o),fun(C,$o),aTP_Lamp_aor(fun(C,B),fun(fun(B,$o),fun(C,$o)),Uu),Uub),Uua) ) ).

% ATP.lambda_502
tff(fact_8680_ATP_Olambda__503,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: array(B),Uua: nat,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aky(array(B),fun(nat,fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = 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)),aa(nat,B,nth(B,array_get(B,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_503
tff(fact_8681_ATP_Olambda__504,axiom,
    ! [B: $tType] :
      ( field(B)
     => ! [Uu: fun(nat,B),Uua: fun(nat,B),Uub: nat] : aa(nat,B,aa(fun(nat,B),fun(nat,B),aTP_Lamp_et(fun(nat,B),fun(fun(nat,B),fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),zero_zero(B)),Uub))),aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_504
tff(fact_8682_ATP_Olambda__505,axiom,
    ! [B: $tType,Uu: fun(set(B),set(B)),Uua: set(B),Uub: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),aTP_Lamp_ant(fun(set(B),set(B)),fun(set(B),fun(set(B),set(B))),Uu),Uua),Uub) = 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(set(B),set(B),Uu,Uub)),Uua)),complete_lattice_gfp(set(B),Uu)) ).

% ATP.lambda_505
tff(fact_8683_ATP_Olambda__506,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(set(B),set(B)),Uub: set(B)] : aa(set(B),set(B),aa(fun(set(B),set(B)),fun(set(B),set(B)),aTP_Lamp_ani(set(B),fun(fun(set(B),set(B)),fun(set(B),set(B))),Uu),Uua),Uub) = 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(set(B),set(B),Uua,Uub)),Uu)),complete_lattice_gfp(set(B),Uua)) ).

% ATP.lambda_506
tff(fact_8684_ATP_Olambda__507,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_re(nat,fun(nat,fun(list(nat),$o)),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_507
tff(fact_8685_ATP_Olambda__508,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,Uu: fun(B,fun(D,$o)),Uua: fun(C,fun(E,$o)),Uub: product_prod(product_prod(B,D),product_prod(C,E))] :
      ( aa(product_prod(product_prod(B,D),product_prod(C,E)),$o,aa(fun(C,fun(E,$o)),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o),aTP_Lamp_arg(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(product_prod(product_prod(B,D),product_prod(C,E)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(B,D)),$o,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),$o),ord_less_eq(set(product_prod(B,D))),basic_fsts(product_prod(B,D),product_prod(C,E),Uub)),aa(fun(product_prod(B,D),$o),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,$o)),fun(product_prod(B,D),$o),product_case_prod(B,D,$o),Uu)))
        & aa(set(product_prod(C,E)),$o,aa(set(product_prod(C,E)),fun(set(product_prod(C,E)),$o),ord_less_eq(set(product_prod(C,E))),basic_snds(product_prod(B,D),product_prod(C,E),Uub)),aa(fun(product_prod(C,E),$o),set(product_prod(C,E)),collect(product_prod(C,E)),aa(fun(C,fun(E,$o)),fun(product_prod(C,E),$o),product_case_prod(C,E,$o),Uua))) ) ) ).

% ATP.lambda_508
tff(fact_8686_ATP_Olambda__509,axiom,
    ! [B: $tType,C: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,fun(E,$o)),Uub: product_prod(product_prod(B,C),product_prod(D,E))] :
      ( aa(product_prod(product_prod(B,C),product_prod(D,E)),$o,aa(fun(D,fun(E,$o)),fun(product_prod(product_prod(B,C),product_prod(D,E)),$o),aTP_Lamp_aqu(fun(B,fun(C,$o)),fun(fun(D,fun(E,$o)),fun(product_prod(product_prod(B,C),product_prod(D,E)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),basic_fsts(product_prod(B,C),product_prod(D,E),Uub)),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Uu)))
        & aa(set(product_prod(D,E)),$o,aa(set(product_prod(D,E)),fun(set(product_prod(D,E)),$o),ord_less_eq(set(product_prod(D,E))),basic_snds(product_prod(B,C),product_prod(D,E),Uub)),aa(fun(product_prod(D,E),$o),set(product_prod(D,E)),collect(product_prod(D,E)),aa(fun(D,fun(E,$o)),fun(product_prod(D,E),$o),product_case_prod(D,E,$o),Uua))) ) ) ).

% ATP.lambda_509
tff(fact_8687_ATP_Olambda__510,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: list(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_ne(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(list(B),nat,size_size(list(B)),Uub))
        | ( ( aa(list(B),nat,size_size(list(B)),Uua) = aa(list(B),nat,size_size(list(B)),Uub) )
          & aa(set(product_prod(list(B),list(B))),$o,member(product_prod(list(B),list(B)),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)),Uua),Uub)),lex(B,Uu)) ) ) ) ).

% ATP.lambda_510
tff(fact_8688_ATP_Olambda__511,axiom,
    ! [B: $tType,Uu: nat,Uua: list(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_aao(nat,fun(list(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(B),nat,size_size(list(B)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_511
tff(fact_8689_ATP_Olambda__512,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: list(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_ux(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Uua) = aa(list(B),nat,size_size(list(B)),Uub) )
        & ? [Xys2: list(B),X3: B,Y5: B,Xs6: list(B),Ys6: list(B)] :
            ( ( Uua = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs6)) )
            & ( Uub = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys6)) )
            & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X3),Y5)),Uu) ) ) ) ).

% ATP.lambda_512
tff(fact_8690_ATP_Olambda__513,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_rf(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_513
tff(fact_8691_ATP_Olambda__514,axiom,
    ! [B: $tType,Uu: nat,Uua: set(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(set(B),fun(list(B),$o),aTP_Lamp_he(nat,fun(set(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Uub) = Uu )
        & distinct(B,Uub)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uub)),Uua) ) ) ).

% ATP.lambda_514
tff(fact_8692_ATP_Olambda__515,axiom,
    ! [B: $tType,Uu: set(B),Uua: nat,Uub: list(B)] :
      ( aa(list(B),$o,aa(nat,fun(list(B),$o),aTP_Lamp_hd(set(B),fun(nat,fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Uub) = Uua )
        & distinct(B,Uub)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uub)),Uu) ) ) ).

% ATP.lambda_515
tff(fact_8693_ATP_Olambda__516,axiom,
    ! [B: $tType,Uu: set(B),Uua: nat,Uub: list(B)] :
      ( aa(list(B),$o,aa(nat,fun(list(B),$o),aTP_Lamp_xh(set(B),fun(nat,fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uub)),Uu)
        & ( aa(list(B),nat,size_size(list(B)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_516
tff(fact_8694_ATP_Olambda__517,axiom,
    ! [B: $tType,Uu: nat,Uua: list(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_hu(nat,fun(list(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Uub) = Uu )
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uub)),aa(list(B),set(B),set2(B),Uua)) ) ) ).

% ATP.lambda_517
tff(fact_8695_ATP_Olambda__518,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(set(C),fun(product_prod(B,C),$o),aTP_Lamp_arh(set(B),fun(set(C),fun(product_prod(B,C),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),basic_fsts(B,C,Uub)),Uu)
        & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),basic_snds(B,C,Uub)),Uua) ) ) ).

% ATP.lambda_518
tff(fact_8696_ATP_Olambda__519,axiom,
    ! [B: $tType,Uu: set(B),Uua: nat,Uub: list(B)] :
      ( aa(list(B),$o,aa(nat,fun(list(B),$o),aTP_Lamp_gw(set(B),fun(nat,fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(B),nat,size_size(list(B)),Uub)),Uua) ) ) ).

% ATP.lambda_519
tff(fact_8697_ATP_Olambda__520,axiom,
    ! [B: $tType,Uu: set(B),Uua: nat,Uub: list(B)] :
      ( aa(list(B),$o,aa(nat,fun(list(B),$o),aTP_Lamp_ha(set(B),fun(nat,fun(list(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Uub)),Uu)
        & ( aa(list(B),nat,size_size(list(B)),Uub) = Uua ) ) ) ).

% ATP.lambda_520
tff(fact_8698_ATP_Olambda__521,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_rd(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_521
tff(fact_8699_ATP_Olambda__522,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: fun(B,option(C))] :
      ( aa(fun(B,option(C)),$o,aa(set(C),fun(fun(B,option(C)),$o),aTP_Lamp_anr(set(B),fun(set(C),fun(fun(B,option(C)),$o)),Uu),Uua),Uub)
    <=> ( ( dom(B,C,Uub) = Uu )
        & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),ran(B,C,Uub)),Uua) ) ) ).

% ATP.lambda_522
tff(fact_8700_ATP_Olambda__523,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gs(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_523
tff(fact_8701_ATP_Olambda__524,axiom,
    ! [B: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(B,nat)] :
      ( aa(product_prod(B,nat),$o,aa(nat,fun(product_prod(B,nat),$o),aTP_Lamp_tq(set(nat),fun(nat,fun(product_prod(B,nat),$o)),Uu),Uua),Uub)
    <=> aa(set(nat),$o,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(B,nat),nat,product_snd(B,nat),Uub)),Uua)),Uu) ) ).

% ATP.lambda_524
tff(fact_8702_ATP_Olambda__525,axiom,
    ! [B: $tType,Uu: set(list(B)),Uua: list(B),Uub: B] :
      ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_hx(set(list(B)),fun(list(B),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(list(B)),$o,member(list(B),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),Uub),nil(B)))),Uu) ) ).

% ATP.lambda_525
tff(fact_8703_ATP_Olambda__526,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_add(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(nat)),$o,member(set(nat),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_526
tff(fact_8704_ATP_Olambda__527,axiom,
    ! [B: $tType,Uu: list(B),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_sb(list(B),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(B),nat,size_size(list(B)),Uu))
        & aa(set(nat),$o,member(nat,Uub),Uua) ) ) ).

% ATP.lambda_527
tff(fact_8705_ATP_Olambda__528,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: list(B),Uub: nat] :
      ( aa(nat,$o,aa(list(B),fun(nat,$o),aTP_Lamp_pk(fun(B,$o),fun(list(B),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(B),nat,size_size(list(B)),Uua))
        & aa(B,$o,Uu,aa(nat,B,nth(B,Uua),Uub)) ) ) ).

% ATP.lambda_528
tff(fact_8706_ATP_Olambda__529,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: multiset(B),Uub: B] :
      ( aa(B,$o,aa(multiset(B),fun(B,$o),aTP_Lamp_aid(fun(B,$o),fun(multiset(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),aa(multiset(B),set(B),set_mset(B),Uua))
        & aa(B,$o,Uu,Uub) ) ) ).

% ATP.lambda_529
tff(fact_8707_ATP_Olambda__530,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: list(B),Uua: fun(B,$o),Uub: B] :
          ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_adf(list(B),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),aa(list(B),set(B),set2(B),Uu))
            & aa(B,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_530
tff(fact_8708_ATP_Olambda__531,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: list(B),Uub: B] :
      ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_pb(fun(B,$o),fun(list(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),aa(list(B),set(B),set2(B),Uua))
        & aa(B,$o,Uu,Uub) ) ) ).

% ATP.lambda_531
tff(fact_8709_ATP_Olambda__532,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,option(C)),Uua: B,Uub: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(B,fun(product_prod(B,C),$o),aTP_Lamp_aoc(fun(B,option(C)),fun(B,fun(product_prod(B,C),$o)),Uu),Uua),Uub)
    <=> ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),Uub),graph(B,C,Uu))
        & ( aa(product_prod(B,C),B,product_fst(B,C),Uub) != Uua ) ) ) ).

% ATP.lambda_532
tff(fact_8710_ATP_Olambda__533,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: B,Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ef(B,fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),Uu),aa(nat,B,semiring_1_of_nat(B),Uub))),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_533
tff(fact_8711_ATP_Olambda__534,axiom,
    ! [B: $tType,Uu: list(B),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_rz(list(B),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(set(nat),$o,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(B),nat,size_size(list(B)),Uu))),Uua) ) ).

% ATP.lambda_534
tff(fact_8712_ATP_Olambda__535,axiom,
    ! [B: $tType] :
      ( wellorder(B)
     => ! [Uu: set(B),Uua: nat,Uub: B] :
          ( aa(B,$o,aa(nat,fun(B,$o),aTP_Lamp_ahb(set(B),fun(nat,fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & aa(B,$o,aa(B,fun(B,$o),ord_less(B),infini527867602293511546merate(B,Uu,Uua)),Uub) ) ) ) ).

% ATP.lambda_535
tff(fact_8713_ATP_Olambda__536,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_ade(set(B),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & ( aa(B,C,Uua,Uub) = aa(set(C),C,lattic643756798350308766er_Min(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_536
tff(fact_8714_ATP_Olambda__537,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_arw(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(set(nat),$o,member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_arv(set(nat),fun(nat,fun(nat,$o)),Uu),Uub)))),Uua) ) ) ).

% ATP.lambda_537
tff(fact_8715_ATP_Olambda__538,axiom,
    ! [B: $tType,Uu: set(B),Uua: nat,Uub: set(B)] :
      ( aa(set(B),$o,aa(nat,fun(set(B),$o),aTP_Lamp_gp(set(B),fun(nat,fun(set(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),Uu)
        & ( aa(set(B),nat,finite_card(B),Uub) = Uua ) ) ) ).

% ATP.lambda_538
tff(fact_8716_ATP_Olambda__539,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gt(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_539
tff(fact_8717_ATP_Olambda__540,axiom,
    ! [B: $tType,Uu: multiset(B),Uua: multiset(B),Uub: B] :
      ( aa(B,$o,aa(multiset(B),fun(B,$o),aTP_Lamp_ahl(multiset(B),fun(multiset(B),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uua),Uub)),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uu),Uub)) ) ).

% ATP.lambda_540
tff(fact_8718_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cl(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_541
tff(fact_8719_ATP_Olambda__542,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_abb(set(B),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uu)
        | aa(set(B),$o,member(B,Uub),Uua) ) ) ).

% ATP.lambda_542
tff(fact_8720_ATP_Olambda__543,axiom,
    ! [B: $tType,Uu: B,Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_hi(B,fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa(set(B),$o,member(B,Uub),Uua) ) ) ).

% ATP.lambda_543
tff(fact_8721_ATP_Olambda__544,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_lk(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_544
tff(fact_8722_ATP_Olambda__545,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_gf(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_545
tff(fact_8723_ATP_Olambda__546,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ll(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_546
tff(fact_8724_ATP_Olambda__547,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_gi(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_547
tff(fact_8725_ATP_Olambda__548,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_gh(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_548
tff(fact_8726_ATP_Olambda__549,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_gg(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_549
tff(fact_8727_ATP_Olambda__550,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),aTP_Lamp_aom(set(B),fun(set(B),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),Uua),Uub)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),Uu) ) ) ).

% ATP.lambda_550
tff(fact_8728_ATP_Olambda__551,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),aTP_Lamp_aol(set(B),fun(set(B),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),Uub),Uua)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uua),Uu) ) ) ).

% ATP.lambda_551
tff(fact_8729_ATP_Olambda__552,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aam(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu) ) ) ).

% ATP.lambda_552
tff(fact_8730_ATP_Olambda__553,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_aat(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uua)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uu) ) ) ).

% ATP.lambda_553
tff(fact_8731_ATP_Olambda__554,axiom,
    ! [B: $tType,Uu: filter(B),Uua: filter(B),Uub: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(filter(B),fun(fun(B,$o),$o),aTP_Lamp_amt(filter(B),fun(filter(B),fun(fun(B,$o),$o)),Uu),Uua),Uub)
    <=> ( eventually(B,Uub,Uu)
        & eventually(B,Uub,Uua) ) ) ).

% ATP.lambda_554
tff(fact_8732_ATP_Olambda__555,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_arv(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_555
tff(fact_8733_ATP_Olambda__556,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_kk(set(B),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uu)
        & aa(set(B),$o,member(B,Uub),Uua) ) ) ).

% ATP.lambda_556
tff(fact_8734_ATP_Olambda__557,axiom,
    ! [B: $tType,Uu: B,Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aqs(B,fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uu )
        & ( Uub = Uu ) ) ) ).

% ATP.lambda_557
tff(fact_8735_ATP_Olambda__558,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),aTP_Lamp_ez(set(B),fun(set(B),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uub),Uu) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uua),Uu) ) ) ).

% ATP.lambda_558
tff(fact_8736_ATP_Olambda__559,axiom,
    ! [B: $tType,Uu: list(B),Uua: fun(B,nat),Uub: B] : aa(B,nat,aa(fun(B,nat),fun(B,nat),aTP_Lamp_rh(list(B),fun(fun(B,nat),fun(B,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(B,nat,count_list(B,Uu),Uub)),aa(B,nat,Uua,Uub)) ).

% ATP.lambda_559
tff(fact_8737_ATP_Olambda__560,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: list(B),Uub: B] : aa(B,nat,aa(list(B),fun(B,nat),aTP_Lamp_rg(fun(B,nat),fun(list(B),fun(B,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(B,nat,count_list(B,Uua),Uub)),aa(B,nat,Uu,Uub)) ).

% ATP.lambda_560
tff(fact_8738_ATP_Olambda__561,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_aln(fun(B,$o),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uua)
       => aa(B,$o,Uu,Uub) ) ) ).

% ATP.lambda_561
tff(fact_8739_ATP_Olambda__562,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,fun(B,B)),Uua: multiset(C),Uub: C] : aa(C,fun(B,B),aa(multiset(C),fun(C,fun(B,B)),aTP_Lamp_aie(fun(C,fun(B,B)),fun(multiset(C),fun(C,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(C,nat,aa(multiset(C),fun(C,nat),count(C),Uua),Uub)),aa(C,fun(B,B),Uu,Uub)) ).

% ATP.lambda_562
tff(fact_8740_ATP_Olambda__563,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: set(B),Uua: fun(B,$o),Uub: B] :
          ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ahd(set(B),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & aa(B,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_563
tff(fact_8741_ATP_Olambda__564,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ah(set(B),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uu)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_564
tff(fact_8742_ATP_Olambda__565,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_aci(fun(B,$o),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uua)
        & aa(B,$o,Uu,Uub) ) ) ).

% ATP.lambda_565
tff(fact_8743_ATP_Olambda__566,axiom,
    ! [B: $tType,Uu: B,Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_hm(B,fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_566
tff(fact_8744_ATP_Olambda__567,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aah(fun(B,$o),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(B,$o,Uu,Uua) ) ) ).

% ATP.lambda_567
tff(fact_8745_ATP_Olambda__568,axiom,
    ! [B: $tType,Uu: B,Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_hl(B,fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_568
tff(fact_8746_ATP_Olambda__569,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,set(C)),Uub: B] :
      ( aa(B,$o,aa(fun(B,set(C)),fun(B,$o),aTP_Lamp_wz(set(B),fun(fun(B,set(C)),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uu)
        & ( aa(B,set(C),Uua,Uub) != bot_bot(set(C)) ) ) ) ).

% ATP.lambda_569
tff(fact_8747_ATP_Olambda__570,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_fn(set(B),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & ( aa(B,C,Uua,Uub) != zero_zero(C) ) ) ) ) ).

% ATP.lambda_570
tff(fact_8748_ATP_Olambda__571,axiom,
    ! [B: $tType,C: $tType] :
      ( ab_group_add(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_hn(set(B),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & ( aa(B,C,Uua,Uub) != zero_zero(C) ) ) ) ) ).

% ATP.lambda_571
tff(fact_8749_ATP_Olambda__572,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_fp(set(B),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & ( aa(B,C,Uua,Uub) != one_one(C) ) ) ) ) ).

% ATP.lambda_572
tff(fact_8750_ATP_Olambda__573,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,B),Uua: set(C),Uub: C] :
          ( aa(C,$o,aa(set(C),fun(C,$o),aTP_Lamp_ho(fun(C,B),fun(set(C),fun(C,$o)),Uu),Uua),Uub)
        <=> ( aa(set(C),$o,member(C,Uub),Uua)
            & ( aa(C,B,Uu,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_573
tff(fact_8751_ATP_Olambda__574,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,B),Uua: set(C),Uub: C] :
          ( aa(C,$o,aa(set(C),fun(C,$o),aTP_Lamp_hv(fun(C,B),fun(set(C),fun(C,$o)),Uu),Uua),Uub)
        <=> ( aa(set(C),$o,member(C,Uub),Uua)
            & ( aa(C,B,Uu,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_574
tff(fact_8752_ATP_Olambda__575,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_parity(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_gm(set(B),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uu)
            & ~ aa(C,$o,aa(C,fun(C,$o),dvd_dvd(C),aa(num,C,numeral_numeral(C),aa(num,num,bit0,one2))),aa(B,C,Uua,Uub)) ) ) ) ).

% ATP.lambda_575
tff(fact_8753_ATP_Olambda__576,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_hk(set(B),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,member(B,Uub),Uu)
        & ~ aa(set(B),$o,member(B,Uub),Uua) ) ) ).

% ATP.lambda_576
tff(fact_8754_ATP_Olambda__577,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: multiset(B),Uub: B] : aa(B,nat,aa(multiset(B),fun(B,nat),aTP_Lamp_ahs(fun(B,nat),fun(multiset(B),fun(B,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Uua),Uub)),aa(nat,nat,suc,aa(B,nat,Uu,Uub))) ).

% ATP.lambda_577
tff(fact_8755_ATP_Olambda__578,axiom,
    ! [B: $tType,E: $tType,D: $tType,Uu: fun(D,fun(E,$o)),Uua: fun(D,B),Uub: set(B)] : aa(set(B),set(D),aa(fun(D,B),fun(set(B),set(D)),aTP_Lamp_amr(fun(D,fun(E,$o)),fun(fun(D,B),fun(set(B),set(D))),Uu),Uua),Uub) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),inf_inf(set(D)),aa(set(B),set(D),aa(fun(D,B),fun(set(B),set(D)),vimage(D,B),Uua),Uub)),aa(fun(D,$o),set(D),collect(D),aa(fun(D,fun(E,$o)),fun(D,$o),domainp(D,E),Uu))) ).

% ATP.lambda_578
tff(fact_8756_ATP_Olambda__579,axiom,
    ! [B: $tType,C: $tType,Uu: list(product_prod(B,C)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_td(list(product_prod(B,C)),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> ( aa(B,option(C),map_of(B,C,Uu),Uua) = aa(C,option(C),some(C),Uub) ) ) ).

% ATP.lambda_579
tff(fact_8757_ATP_Olambda__580,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aal(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_580
tff(fact_8758_ATP_Olambda__581,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_le(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_581
tff(fact_8759_ATP_Olambda__582,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_lj(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_582
tff(fact_8760_ATP_Olambda__583,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Uu: B,Uua: B,Uub: B] : aa(B,B,aa(B,fun(B,B),aTP_Lamp_jg(B,fun(B,fun(B,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),Uub),Uu)),Uua) ) ).

% ATP.lambda_583
tff(fact_8761_ATP_Olambda__584,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Uu: B,Uua: B,Uub: B] : aa(B,B,aa(B,fun(B,B),aTP_Lamp_je(B,fun(B,fun(B,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Uu),Uub)),Uua) ) ).

% ATP.lambda_584
tff(fact_8762_ATP_Olambda__585,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Uu: B,Uua: B,Uub: B] : aa(B,B,aa(B,fun(B,B),aTP_Lamp_jf(B,fun(B,fun(B,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),Uub),Uu)),Uua) ) ).

% ATP.lambda_585
tff(fact_8763_ATP_Olambda__586,axiom,
    ! [B: $tType] :
      ( linordered_field(B)
     => ! [Uu: B,Uua: B,Uub: B] : aa(B,B,aa(B,fun(B,B),aTP_Lamp_jd(B,fun(B,fun(B,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Uu),Uub)),Uua) ) ).

% ATP.lambda_586
tff(fact_8764_ATP_Olambda__587,axiom,
    ! [C: $tType,D: $tType,Uu: set(product_prod(D,C)),Uua: D,Uub: C] :
      ( aa(C,$o,aa(D,fun(C,$o),aTP_Lamp_arn(set(product_prod(D,C)),fun(D,fun(C,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(D,C)),$o,member(product_prod(D,C),aa(C,product_prod(D,C),aa(D,fun(C,product_prod(D,C)),product_Pair(D,C),Uua),Uub)),Uu) ) ).

% ATP.lambda_587
tff(fact_8765_ATP_Olambda__588,axiom,
    ! [D: $tType,C: $tType,Uu: set(product_prod(C,D)),Uua: C,Uub: D] :
      ( aa(D,$o,aa(C,fun(D,$o),aTP_Lamp_arp(set(product_prod(C,D)),fun(C,fun(D,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(C,D)),$o,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),Uua),Uub)),Uu) ) ).

% ATP.lambda_588
tff(fact_8766_ATP_Olambda__589,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_mh(set(product_prod(C,B)),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uua),Uub)),Uu) ) ).

% ATP.lambda_589
tff(fact_8767_ATP_Olambda__590,axiom,
    ! [D: $tType,B: $tType,Uu: set(product_prod(B,D)),Uua: B,Uub: D] :
      ( aa(D,$o,aa(B,fun(D,$o),aTP_Lamp_arm(set(product_prod(B,D)),fun(B,fun(D,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,D)),$o,member(product_prod(B,D),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),Uua),Uub)),Uu) ) ).

% ATP.lambda_590
tff(fact_8768_ATP_Olambda__591,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(product_prod(B,C)),fun(B,fun(C,$o)),aTP_Lamp_lp(set(product_prod(B,C)),fun(B,fun(C,$o))),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,C)),$o,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_591
tff(fact_8769_ATP_Olambda__592,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aab(set(product_prod(B,B)),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uua),Uub)),Uu) ) ).

% ATP.lambda_592
tff(fact_8770_ATP_Olambda__593,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_ajf(set(product_prod(C,B)),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uub),Uua)),Uu) ) ).

% ATP.lambda_593
tff(fact_8771_ATP_Olambda__594,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aoi(set(product_prod(B,B)),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uub),Uua)),Uu) ) ).

% ATP.lambda_594
tff(fact_8772_ATP_Olambda__595,axiom,
    ! [B: $tType,Uu: set(list(B)),Uua: B,Uub: list(B)] :
      ( aa(list(B),$o,aa(B,fun(list(B),$o),aTP_Lamp_if(set(list(B)),fun(B,fun(list(B),$o)),Uu),Uua),Uub)
    <=> aa(set(list(B)),$o,member(list(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uua),Uub)),Uu) ) ).

% ATP.lambda_595
tff(fact_8773_ATP_Olambda__596,axiom,
    ! [B: $tType,Uu: list(list(B)),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_qw(list(list(B)),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,nth(B,aa(nat,list(B),nth(list(B),Uu),Uub)),Uua) ).

% ATP.lambda_596
tff(fact_8774_ATP_Olambda__597,axiom,
    ! [B: $tType] :
      ( archim2362893244070406136eiling(B)
     => ! [Uu: B,Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_fx(B,fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),ring_1_Ints(B))
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),Uub)
            & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uub),Uua) ) ) ) ).

% ATP.lambda_597
tff(fact_8775_ATP_Olambda__598,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: list(B),Uub: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_oz(fun(B,C),fun(list(B),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,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,one2)))))),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_598
tff(fact_8776_ATP_Olambda__599,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(C,B),Uub: B] :
      ( aa(B,$o,aa(fun(C,B),fun(B,$o),aTP_Lamp_aop(fun(B,C),fun(fun(C,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(C,B,Uua,aa(B,C,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_599
tff(fact_8777_ATP_Olambda__600,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(C,$o),Uub: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(fun(C,$o),fun(product_prod(B,C),$o),aTP_Lamp_aoy(fun(B,$o),fun(fun(C,$o),fun(product_prod(B,C),$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,aa(product_prod(B,C),B,product_fst(B,C),Uub))
        & aa(C,$o,Uua,aa(product_prod(B,C),C,product_snd(B,C),Uub)) ) ) ).

% ATP.lambda_600
tff(fact_8778_ATP_Olambda__601,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fb(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_601
tff(fact_8779_ATP_Olambda__602,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_amn(fun(B,fun(C,$o)),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uua,Uub)
        & aa(B,$o,aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),Uu),Uub) ) ) ).

% ATP.lambda_602
tff(fact_8780_ATP_Olambda__603,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: list(B),Uub: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_ox(fun(B,C),fun(list(B),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,Uu,Uub)),aa(B,C,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,one2)))))) ) ) ).

% ATP.lambda_603
tff(fact_8781_ATP_Olambda__604,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: list(B),Uub: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_oy(fun(B,C),fun(list(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(B,C,Uu,Uub) = aa(B,C,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,one2))))) ) ) ) ).

% ATP.lambda_604
tff(fact_8782_ATP_Olambda__605,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,D)),Uub: B] : aa(B,D,aa(fun(B,fun(C,D)),fun(B,D),aTP_Lamp_xe(fun(B,set(C)),fun(fun(B,fun(C,D)),fun(B,D)),Uu),Uua),Uub) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),aa(B,fun(C,D),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_605
tff(fact_8783_ATP_Olambda__606,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,D)),Uub: B] : aa(B,D,aa(fun(B,fun(C,D)),fun(B,D),aTP_Lamp_xd(fun(B,set(C)),fun(fun(B,fun(C,D)),fun(B,D)),Uu),Uua),Uub) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),aa(B,fun(C,D),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_606
tff(fact_8784_ATP_Olambda__607,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [Uu: fun(C,B),Uua: C,Uub: C] :
          ( aa(C,$o,aa(C,fun(C,$o),aTP_Lamp_qb(fun(C,B),fun(C,fun(C,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,Uu,Uua)),aa(C,B,Uu,Uub)) ) ) ).

% ATP.lambda_607
tff(fact_8785_ATP_Olambda__608,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_ame(fun(B,C),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,Uu,Uub)),aa(B,C,Uua,Uub)) ) ) ).

% ATP.lambda_608
tff(fact_8786_ATP_Olambda__609,axiom,
    ! [B: $tType,C: $tType] :
      ( field(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_dv(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_609
tff(fact_8787_ATP_Olambda__610,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,fun(D,C)),Uua: fun(B,option(D)),Uub: B] : aa(B,option(C),aa(fun(B,option(D)),fun(B,option(C)),aTP_Lamp_anq(fun(B,fun(D,C)),fun(fun(B,option(D)),fun(B,option(C))),Uu),Uua),Uub) = aa(option(D),option(C),map_option(D,C,aa(B,fun(D,C),Uu,Uub)),aa(B,option(D),Uua,Uub)) ).

% ATP.lambda_610
tff(fact_8788_ATP_Olambda__611,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_du(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_611
tff(fact_8789_ATP_Olambda__612,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(B,C),Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_hw(fun(B,C),fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,Uu,Uub)),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_612
tff(fact_8790_ATP_Olambda__613,axiom,
    ! [C: $tType,B: $tType] :
      ( linordered_idom(C)
     => ! [Uu: fun(B,C),Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_bi(fun(B,C),fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,Uua,Uub)),aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_613
tff(fact_8791_ATP_Olambda__614,axiom,
    ! [B: $tType,C: $tType] :
      ( ab_group_add(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bt(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_614
tff(fact_8792_ATP_Olambda__615,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(C),aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_ww(fun(B,set(C)),fun(fun(B,set(C)),fun(B,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),aa(B,set(C),Uu,Uub)),aa(B,set(C),Uua,Uub)) ).

% ATP.lambda_615
tff(fact_8793_ATP_Olambda__616,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: fun(B,nat),Uub: B] : aa(B,nat,aa(fun(B,nat),fun(B,nat),aa(fun(B,nat),fun(fun(B,nat),fun(B,nat)),aTP_Lamp_ahz(fun(B,nat),fun(fun(B,nat),fun(B,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(B,nat,Uu,Uub)),aa(B,nat,Uua,Uub)) ).

% ATP.lambda_616
tff(fact_8794_ATP_Olambda__617,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: fun(B,nat),Uua: fun(B,nat),Uub: B] : aa(B,nat,aa(fun(B,nat),fun(B,nat),aTP_Lamp_id(fun(B,nat),fun(fun(B,nat),fun(B,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(B,nat,Uua,Uub)),aa(B,nat,Uu,Uub)) ) ).

% ATP.lambda_617
tff(fact_8795_ATP_Olambda__618,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: fun(B,nat),Uub: B] : aa(B,nat,aa(fun(B,nat),fun(B,nat),aTP_Lamp_bw(fun(B,nat),fun(fun(B,nat),fun(B,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(B,nat,Uua,Uub)),aa(B,nat,Uu,Uub)) ).

% ATP.lambda_618
tff(fact_8796_ATP_Olambda__619,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_abk(fun(C,set(B)),fun(fun(C,set(B)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(C,set(B),Uu,Uub)),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_619
tff(fact_8797_ATP_Olambda__620,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_abe(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_620
tff(fact_8798_ATP_Olambda__621,axiom,
    ! [B: $tType,C: $tType] :
      ( lattice(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_anz(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_621
tff(fact_8799_ATP_Olambda__622,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(C),aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_aaz(fun(B,set(C)),fun(fun(B,set(C)),fun(B,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),aa(B,set(C),Uu,Uub)),aa(B,set(C),Uua,Uub)) ).

% ATP.lambda_622
tff(fact_8800_ATP_Olambda__623,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(B,C),Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aob(fun(B,C),fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(B,C,Uu,Uub)),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_623
tff(fact_8801_ATP_Olambda__624,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_zj(fun(C,set(B)),fun(fun(C,set(B)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(C,set(B),Uu,Uub)),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_624
tff(fact_8802_ATP_Olambda__625,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_zi(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_625
tff(fact_8803_ATP_Olambda__626,axiom,
    ! [B: $tType,C: $tType] :
      ( lattice(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_aoh(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_626
tff(fact_8804_ATP_Olambda__627,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(C),aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_wt(fun(B,set(C)),fun(fun(B,set(C)),fun(B,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),Uu,Uub)),aa(B,set(C),Uua,Uub)) ).

% ATP.lambda_627
tff(fact_8805_ATP_Olambda__628,axiom,
    ! [C: $tType,B: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(B,C),Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_aok(fun(B,C),fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(B,C,Uu,Uub)),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_628
tff(fact_8806_ATP_Olambda__629,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,B),Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bo(fun(C,B),fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(C,B,Uu,Uub)),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_629
tff(fact_8807_ATP_Olambda__630,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: fun(B,nat),Uub: B] : aa(B,nat,aa(fun(B,nat),fun(B,nat),aa(fun(B,nat),fun(fun(B,nat),fun(B,nat)),aTP_Lamp_ahy(fun(B,nat),fun(fun(B,nat),fun(B,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,Uu,Uub)),aa(B,nat,Uua,Uub)) ).

% ATP.lambda_630
tff(fact_8808_ATP_Olambda__631,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(B,C),Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_hp(fun(B,C),fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,Uu,Uub)),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_631
tff(fact_8809_ATP_Olambda__632,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,multiset(B)),Uua: fun(C,multiset(B)),Uub: C] : aa(C,multiset(B),aa(fun(C,multiset(B)),fun(C,multiset(B)),aTP_Lamp_anh(fun(C,multiset(B)),fun(fun(C,multiset(B)),fun(C,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(C,multiset(B),Uu,Uub)),aa(C,multiset(B),Uua,Uub)) ).

% ATP.lambda_632
tff(fact_8810_ATP_Olambda__633,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,multiset(C)),Uua: fun(B,multiset(C)),Uub: B] : aa(B,multiset(C),aa(fun(B,multiset(C)),fun(B,multiset(C)),aTP_Lamp_ang(fun(B,multiset(C)),fun(fun(B,multiset(C)),fun(B,multiset(C))),Uu),Uua),Uub) = aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),union_mset(C),aa(B,multiset(C),Uu,Uub)),aa(B,multiset(C),Uua,Uub)) ).

% ATP.lambda_633
tff(fact_8811_ATP_Olambda__634,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,multiset(B)),Uua: fun(C,multiset(B)),Uub: C] : aa(C,multiset(B),aa(fun(C,multiset(B)),fun(C,multiset(B)),aTP_Lamp_amz(fun(C,multiset(B)),fun(fun(C,multiset(B)),fun(C,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(C,multiset(B),Uu,Uub)),aa(C,multiset(B),Uua,Uub)) ).

% ATP.lambda_634
tff(fact_8812_ATP_Olambda__635,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,multiset(C)),Uua: fun(B,multiset(C)),Uub: B] : aa(B,multiset(C),aa(fun(B,multiset(C)),fun(B,multiset(C)),aTP_Lamp_amy(fun(B,multiset(C)),fun(fun(B,multiset(C)),fun(B,multiset(C))),Uu),Uua),Uub) = aa(multiset(C),multiset(C),aa(multiset(C),fun(multiset(C),multiset(C)),inter_mset(C),aa(B,multiset(C),Uu,Uub)),aa(B,multiset(C),Uua,Uub)) ).

% ATP.lambda_635
tff(fact_8813_ATP_Olambda__636,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(D,B),Uua: fun(D,C),Uub: D] : aa(D,product_prod(B,C),aa(fun(D,C),fun(D,product_prod(B,C)),aTP_Lamp_adn(fun(D,B),fun(fun(D,C),fun(D,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(D,B,Uu,Uub)),aa(D,C,Uua,Uub)) ).

% ATP.lambda_636
tff(fact_8814_ATP_Olambda__637,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,C),Uua: fun(B,D),Uub: B] : aa(B,product_prod(C,D),aa(fun(B,D),fun(B,product_prod(C,D)),aTP_Lamp_aov(fun(B,C),fun(fun(B,D),fun(B,product_prod(C,D))),Uu),Uua),Uub) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(B,C,Uu,Uub)),aa(B,D,Uua,Uub)) ).

% ATP.lambda_637
tff(fact_8815_ATP_Olambda__638,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aba(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
       => aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_638
tff(fact_8816_ATP_Olambda__639,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,fun(C,C)),Uua: fun(B,nat),Uub: B] : aa(B,fun(C,C),aa(fun(B,nat),fun(B,fun(C,C)),aTP_Lamp_ach(fun(B,fun(C,C)),fun(fun(B,nat),fun(B,fun(C,C))),Uu),Uua),Uub) = aa(fun(C,C),fun(C,C),aa(nat,fun(fun(C,C),fun(C,C)),compow(fun(C,C)),aa(B,nat,Uua,Uub)),aa(B,fun(C,C),Uu,Uub)) ).

% ATP.lambda_639
tff(fact_8817_ATP_Olambda__640,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
          ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_anb(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(B,$o,Uu,Uub)
            | aa(B,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_640
tff(fact_8818_ATP_Olambda__641,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ge(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        | aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_641
tff(fact_8819_ATP_Olambda__642,axiom,
    ! [C: $tType,Uu: fun(C,$o),Uua: fun(C,$o),Uub: C] :
      ( aa(C,$o,aa(fun(C,$o),fun(C,$o),aTP_Lamp_kf(fun(C,$o),fun(fun(C,$o),fun(C,$o)),Uu),Uua),Uub)
    <=> ( aa(C,$o,Uu,Uub)
        & aa(C,$o,Uua,Uub) ) ) ).

% ATP.lambda_642
tff(fact_8820_ATP_Olambda__643,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_gd(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_643
tff(fact_8821_ATP_Olambda__644,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_pa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uua,Uub)
        & aa(B,$o,Uu,Uub) ) ) ).

% ATP.lambda_644
tff(fact_8822_ATP_Olambda__645,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_qf(fun(B,C),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(B,C,Uu,Uua) = aa(B,C,Uu,Uub) ) ) ) ).

% ATP.lambda_645
tff(fact_8823_ATP_Olambda__646,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_afk(fun(B,C),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,C,Uu,Uua) = aa(B,C,Uu,Uub) ) ) ).

% ATP.lambda_646
tff(fact_8824_ATP_Olambda__647,axiom,
    ! [E: $tType,C: $tType,Uu: fun(C,E),Uua: fun(C,E),Uub: C] :
      ( aa(C,$o,aa(fun(C,E),fun(C,$o),aTP_Lamp_are(fun(C,E),fun(fun(C,E),fun(C,$o)),Uu),Uua),Uub)
    <=> ( aa(C,E,Uu,Uub) = aa(C,E,Uua,Uub) ) ) ).

% ATP.lambda_647
tff(fact_8825_ATP_Olambda__648,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(C,D),Uub: C] :
      ( aa(C,$o,aa(fun(C,D),fun(C,$o),aTP_Lamp_all(fun(C,D),fun(fun(C,D),fun(C,$o)),Uu),Uua),Uub)
    <=> ( aa(C,D,Uu,Uub) = aa(C,D,Uua,Uub) ) ) ).

% ATP.lambda_648
tff(fact_8826_ATP_Olambda__649,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(C,B),Uub: C] :
      ( aa(C,$o,aa(fun(C,B),fun(C,$o),aTP_Lamp_alr(fun(C,B),fun(fun(C,B),fun(C,$o)),Uu),Uua),Uub)
    <=> ( aa(C,B,Uu,Uub) = aa(C,B,Uua,Uub) ) ) ).

% ATP.lambda_649
tff(fact_8827_ATP_Olambda__650,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_alv(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
      <=> aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_650
tff(fact_8828_ATP_Olambda__651,axiom,
    ! [D: $tType,B: $tType,Uu: fun(B,D),Uua: fun(B,D),Uub: B] :
      ( aa(B,$o,aa(fun(B,D),fun(B,$o),aTP_Lamp_ard(fun(B,D),fun(fun(B,D),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,D,Uu,Uub) = aa(B,D,Uua,Uub) ) ) ).

% ATP.lambda_651
tff(fact_8829_ATP_Olambda__652,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(B,C),Uub: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_alq(fun(B,C),fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,C,Uu,Uub) = aa(B,C,Uua,Uub) ) ) ).

% ATP.lambda_652
tff(fact_8830_ATP_Olambda__653,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_qh(B,fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(B,C,Uua,Uu) = aa(B,C,Uua,Uub) ) ) ) ).

% ATP.lambda_653
tff(fact_8831_ATP_Olambda__654,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(B,C),Uua: fun(B,$o),Uub: B] : aa(B,C,aa(fun(B,$o),fun(B,C),aTP_Lamp_ki(fun(B,C),fun(fun(B,$o),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,Uu,Uub)),aa($o,C,zero_neq_one_of_bool(C),aa(B,$o,Uua,Uub))) ) ).

% ATP.lambda_654
tff(fact_8832_ATP_Olambda__655,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aip(fun(B,fun(B,$o)),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uua,Uub)
        & ! [Y5: B] :
            ( aa(B,$o,Uua,Y5)
           => aa(B,$o,aa(B,fun(B,$o),Uu,Uub),Y5) ) ) ) ).

% ATP.lambda_655
tff(fact_8833_ATP_Olambda__656,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,option(C)),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_ud(fun(B,option(C)),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> ( aa(B,option(C),Uu,Uua) = aa(C,option(C),some(C),Uub) ) ) ).

% ATP.lambda_656
tff(fact_8834_ATP_Olambda__657,axiom,
    ! [Uu: fun(b,b),Uua: ref(b),Uub: b] :
      aa(b,heap_Time_Heap(b),aa(ref(b),fun(b,heap_Time_Heap(b)),aTP_Lamp_ao(fun(b,b),fun(ref(b),fun(b,heap_Time_Heap(b))),Uu),Uua),Uub) = $let(
        y: b,
        y:= aa(b,b,Uu,Uub),
        heap_Time_bind(product_unit,b,ref_update(b,Uua,y),aTP_Lamp_an(b,fun(product_unit,heap_Time_Heap(b)),y)) ) ).

% ATP.lambda_657
tff(fact_8835_ATP_Olambda__658,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(D),Uub: C] : aa(C,B,aa(set(D),fun(C,B),aTP_Lamp_dr(fun(C,fun(D,B)),fun(set(D),fun(C,B)),Uu),Uua),Uub) = aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7121269368397514597t_prod(D,B),aa(C,fun(D,B),Uu,Uub)),Uua) ) ).

% ATP.lambda_658
tff(fact_8836_ATP_Olambda__659,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(D),Uub: C] : aa(C,B,aa(set(D),fun(C,B),aTP_Lamp_bl(fun(C,fun(D,B)),fun(set(D),fun(C,B)),Uu),Uua),Uub) = aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7311177749621191930dd_sum(D,B),aa(C,fun(D,B),Uu,Uub)),Uua) ) ).

% ATP.lambda_659
tff(fact_8837_ATP_Olambda__660,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fc(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_660
tff(fact_8838_ATP_Olambda__661,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,set(B)),Uua: set(B),Uub: C] :
      ( aa(C,$o,aa(set(B),fun(C,$o),aTP_Lamp_wx(fun(C,set(B)),fun(set(B),fun(C,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),Uu,Uub)),Uua) ) ).

% ATP.lambda_661
tff(fact_8839_ATP_Olambda__662,axiom,
    ! [B: $tType,C: $tType] :
      ( unboun7993243217541854897norder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_amj(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_662
tff(fact_8840_ATP_Olambda__663,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_amg(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_663
tff(fact_8841_ATP_Olambda__664,axiom,
    ! [C: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_bf(fun(C,B),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),modulo_modulo(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_664
tff(fact_8842_ATP_Olambda__665,axiom,
    ! [C: $tType,B: $tType] :
      ( field(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_bv(fun(C,B),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_665
tff(fact_8843_ATP_Olambda__666,axiom,
    ! [B: $tType,C: $tType] :
      ( euclid4440199948858584721cancel(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] : aa(B,C,aa(C,fun(B,C),aTP_Lamp_kw(fun(B,C),fun(C,fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),divide_divide(C),aa(B,C,Uu,Uub)),Uua) ) ).

% ATP.lambda_666
tff(fact_8844_ATP_Olambda__667,axiom,
    ! [B: $tType,C: $tType] :
      ( ( dense_linorder(C)
        & no_bot(C) )
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_amh(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_667
tff(fact_8845_ATP_Olambda__668,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_pl(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less(C),aa(B,C,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_668
tff(fact_8846_ATP_Olambda__669,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_bq(fun(C,B),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_669
tff(fact_8847_ATP_Olambda__670,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,set(B)),Uua: set(B),Uub: C] : aa(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_ym(fun(C,set(B)),fun(set(B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(C,set(B),Uu,Uub)),Uua) ).

% ATP.lambda_670
tff(fact_8848_ATP_Olambda__671,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(C,B),Uua: nat,Uub: C] : aa(C,B,aa(nat,fun(C,B),aTP_Lamp_dw(fun(C,B),fun(nat,fun(C,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_671
tff(fact_8849_ATP_Olambda__672,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,multiset(B)),Uua: B,Uub: C] : aa(C,nat,aa(B,fun(C,nat),aTP_Lamp_ahj(fun(C,multiset(B)),fun(B,fun(C,nat)),Uu),Uua),Uub) = aa(B,nat,aa(multiset(B),fun(B,nat),count(B),aa(C,multiset(B),Uu,Uub)),Uua) ).

% ATP.lambda_672
tff(fact_8850_ATP_Olambda__673,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,set(B)),Uua: set(B),Uub: C] : aa(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_aaw(fun(C,set(B)),fun(set(B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(C,set(B),Uu,Uub)),Uua) ).

% ATP.lambda_673
tff(fact_8851_ATP_Olambda__674,axiom,
    ! [C: $tType,B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_abj(fun(C,B),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_674
tff(fact_8852_ATP_Olambda__675,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,set(B)),Uua: set(B),Uub: C] : aa(C,set(B),aa(set(B),fun(C,set(B)),aTP_Lamp_yl(fun(C,set(B)),fun(set(B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(C,set(B),Uu,Uub)),Uua) ).

% ATP.lambda_675
tff(fact_8853_ATP_Olambda__676,axiom,
    ! [C: $tType,B: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_wr(fun(C,B),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_676
tff(fact_8854_ATP_Olambda__677,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] : aa(B,C,aa(C,fun(B,C),aTP_Lamp_zm(fun(B,C),fun(C,fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),inf_inf(C),aa(B,C,Uu,Uub)),Uua) ) ).

% ATP.lambda_677
tff(fact_8855_ATP_Olambda__678,axiom,
    ! [B: $tType,C: $tType] :
      ( linord4140545234300271783up_add(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] : aa(B,C,aa(C,fun(B,C),aTP_Lamp_aak(fun(B,C),fun(C,fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,Uu,Uub)),Uua) ) ).

% ATP.lambda_678
tff(fact_8856_ATP_Olambda__679,axiom,
    ! [D: $tType,B: $tType,C: $tType,Uu: fun(D,heap_Time_Heap(C)),Uua: fun(C,heap_Time_Heap(B)),Uub: D] : aa(D,heap_Time_Heap(B),aa(fun(C,heap_Time_Heap(B)),fun(D,heap_Time_Heap(B)),aTP_Lamp_aa(fun(D,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(D,heap_Time_Heap(B))),Uu),Uua),Uub) = heap_Time_bind(C,B,aa(D,heap_Time_Heap(C),Uu,Uub),Uua) ).

% ATP.lambda_679
tff(fact_8857_ATP_Olambda__680,axiom,
    ! [F6: $tType,G4: $tType,C: $tType,E: $tType,D: $tType,Uu: fun(F6,fun(G4,product_prod(D,E))),Uua: fun(D,fun(E,C)),Uub: F6] : aa(F6,fun(G4,C),aa(fun(D,fun(E,C)),fun(F6,fun(G4,C)),aTP_Lamp_apz(fun(F6,fun(G4,product_prod(D,E))),fun(fun(D,fun(E,C)),fun(F6,fun(G4,C))),Uu),Uua),Uub) = product_scomp(G4,D,E,C,aa(F6,fun(G4,product_prod(D,E)),Uu,Uub),Uua) ).

% ATP.lambda_680
tff(fact_8858_ATP_Olambda__681,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,filter(C)),Uua: filter(D),Uub: B] : aa(B,filter(product_prod(C,D)),aa(filter(D),fun(B,filter(product_prod(C,D))),aTP_Lamp_apd(fun(B,filter(C)),fun(filter(D),fun(B,filter(product_prod(C,D)))),Uu),Uua),Uub) = prod_filter(C,D,aa(B,filter(C),Uu,Uub),Uua) ).

% ATP.lambda_681
tff(fact_8859_ATP_Olambda__682,axiom,
    ! [E: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(E,fun(B,fun(D,$o))),Uua: fun(D,fun(C,$o)),Uub: E] : aa(E,fun(B,fun(C,$o)),aa(fun(D,fun(C,$o)),fun(E,fun(B,fun(C,$o))),aTP_Lamp_arl(fun(E,fun(B,fun(D,$o))),fun(fun(D,fun(C,$o)),fun(E,fun(B,fun(C,$o)))),Uu),Uua),Uub) = aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),aa(E,fun(B,fun(D,$o)),Uu,Uub)),Uua) ).

% ATP.lambda_682
tff(fact_8860_ATP_Olambda__683,axiom,
    ! [E: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(E,set(product_prod(B,D))),Uua: set(product_prod(D,C)),Uub: E] : aa(E,set(product_prod(B,C)),aa(set(product_prod(D,C)),fun(E,set(product_prod(B,C))),aTP_Lamp_wg(fun(E,set(product_prod(B,D))),fun(set(product_prod(D,C)),fun(E,set(product_prod(B,C)))),Uu),Uua),Uub) = relcomp(B,D,C,aa(E,set(product_prod(B,D)),Uu,Uub),Uua) ).

% ATP.lambda_683
tff(fact_8861_ATP_Olambda__684,axiom,
    ! [D: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(C,B))),Uua: set(C),Uub: D] : aa(D,set(B),aa(set(C),fun(D,set(B)),aTP_Lamp_agc(fun(D,set(product_prod(C,B))),fun(set(C),fun(D,set(B))),Uu),Uua),Uub) = aa(set(C),set(B),image(C,B,aa(D,set(product_prod(C,B)),Uu,Uub)),Uua) ).

% ATP.lambda_684
tff(fact_8862_ATP_Olambda__685,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,C),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_ads(fun(B,C),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(C),$o,member(C,aa(B,C,Uu,Uub)),Uua) ) ).

% ATP.lambda_685
tff(fact_8863_ATP_Olambda__686,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(C,B),Uub: C] :
      ( aa(C,$o,aa(fun(C,B),fun(C,$o),aTP_Lamp_aff(set(B),fun(fun(C,B),fun(C,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,aa(C,B,Uua,Uub)),Uu) ) ).

% ATP.lambda_686
tff(fact_8864_ATP_Olambda__687,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: $o,Uub: B] :
      ( aa(B,$o,aa($o,fun(B,$o),aTP_Lamp_alw(fun(B,$o),fun($o,fun(B,$o)),Uu),(Uua)),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_687
tff(fact_8865_ATP_Olambda__688,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_pc(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(B,C,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_688
tff(fact_8866_ATP_Olambda__689,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,C),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_afr(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,C,Uu,Uub) = Uua ) ) ).

% ATP.lambda_689
tff(fact_8867_ATP_Olambda__690,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: fun(nat,B),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(nat,B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_qo(nat,fun(fun(nat,B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(B),heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mu(nat,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(B,aa(list(nat),list(B),map(nat,B,Uua),upt(zero_zero(nat),Uu)),Uub)) ) ).

% ATP.lambda_690
tff(fact_8868_ATP_Olambda__691,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: B,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(B,fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mv(nat,fun(B,fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(B),heap_ext(product_unit)),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(B),heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mu(nat,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(B,replicate(B,Uu,Uua),Uub)) ) ).

% ATP.lambda_691
tff(fact_8869_ATP_Olambda__692,axiom,
    ! [B: $tType,Uu: B,Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_hj(B,fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_692
tff(fact_8870_ATP_Olambda__693,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_eg(nat,fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_693
tff(fact_8871_ATP_Olambda__694,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(B,$o),Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_kj(fun(B,$o),fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),times_times(C),aa($o,C,zero_neq_one_of_bool(C),aa(B,$o,Uu,Uub))),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_694
tff(fact_8872_ATP_Olambda__695,axiom,
    ! [B: $tType] :
      ( comple9053668089753744459l_ccpo(B)
     => ! [Uu: fun(B,B),Uua: fun(B,$o),Uub: B] :
          ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_akj(fun(B,B),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
        <=> ( ? [X3: B] :
                ( ( Uub = aa(B,B,Uu,X3) )
                & aa(B,$o,Uua,X3) )
            | ? [M10: set(B)] :
                ( ( Uub = aa(set(B),B,complete_Sup_Sup(B),M10) )
                & comple1602240252501008431_chain(B,ord_less_eq(B),M10)
                & ! [X3: B] :
                    ( aa(set(B),$o,member(B,X3),M10)
                   => aa(B,$o,Uua,X3) ) ) ) ) ) ).

% ATP.lambda_695
tff(fact_8873_ATP_Olambda__696,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: fun(B,C)] :
      ( aa(fun(B,C),$o,aa(set(C),fun(fun(B,C),$o),aTP_Lamp_akh(set(B),fun(set(C),fun(fun(B,C),$o)),Uu),Uua),Uub)
    <=> ( ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),Uu)
           => aa(set(C),$o,member(C,aa(B,C,Uub,X3)),Uua) )
        & ! [A6: B] :
            ( ~ aa(set(B),$o,member(B,A6),Uu)
           => ( aa(B,C,Uub,A6) = undefined(C) ) ) ) ) ).

% ATP.lambda_696
tff(fact_8874_ATP_Olambda__697,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: fun(list(B),fun(list(B),$o)),Uua: list(B),Uub: list(B)] :
          ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o)),aTP_Lamp_aeq(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),Uu),Uua),Uub)
        <=> ( ? [Y5: B,Ys4: list(B)] :
                ( ( Uua = nil(B) )
                & ( Uub = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) ) )
            | ? [X3: B,Y5: B,Xs3: list(B),Ys4: list(B)] :
                ( ( Uua = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs3) )
                & ( Uub = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y5) )
            | ? [X3: B,Y5: B,Xs3: list(B),Ys4: list(B)] :
                ( ( Uua = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs3) )
                & ( Uub = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
                & ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y5)
                & ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y5),X3)
                & aa(list(B),$o,aa(list(B),fun(list(B),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_697
tff(fact_8875_ATP_Olambda__698,axiom,
    ! [B: $tType,Uu: fun(list(multiset(B)),fun(list(multiset(B)),$o)),Uua: list(multiset(B)),Uub: list(multiset(B))] :
      ( aa(list(multiset(B)),$o,aa(list(multiset(B)),fun(list(multiset(B)),$o),aa(fun(list(multiset(B)),fun(list(multiset(B)),$o)),fun(list(multiset(B)),fun(list(multiset(B)),$o)),aTP_Lamp_aqj(fun(list(multiset(B)),fun(list(multiset(B)),$o)),fun(list(multiset(B)),fun(list(multiset(B)),$o))),Uu),Uua),Uub)
    <=> ( ? [Y5: multiset(B),Ys4: list(multiset(B))] :
            ( ( Uua = nil(multiset(B)) )
            & ( Uub = aa(list(multiset(B)),list(multiset(B)),aa(multiset(B),fun(list(multiset(B)),list(multiset(B))),cons(multiset(B)),Y5),Ys4) ) )
        | ? [X3: multiset(B),Y5: multiset(B),Xs3: list(multiset(B)),Ys4: list(multiset(B))] :
            ( ( Uua = aa(list(multiset(B)),list(multiset(B)),aa(multiset(B),fun(list(multiset(B)),list(multiset(B))),cons(multiset(B)),X3),Xs3) )
            & ( Uub = aa(list(multiset(B)),list(multiset(B)),aa(multiset(B),fun(list(multiset(B)),list(multiset(B))),cons(multiset(B)),Y5),Ys4) )
            & aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),X3),Y5) )
        | ? [X3: multiset(B),Y5: multiset(B),Xs3: list(multiset(B)),Ys4: list(multiset(B))] :
            ( ( Uua = aa(list(multiset(B)),list(multiset(B)),aa(multiset(B),fun(list(multiset(B)),list(multiset(B))),cons(multiset(B)),X3),Xs3) )
            & ( Uub = aa(list(multiset(B)),list(multiset(B)),aa(multiset(B),fun(list(multiset(B)),list(multiset(B))),cons(multiset(B)),Y5),Ys4) )
            & ~ aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),X3),Y5)
            & ~ aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),Y5),X3)
            & aa(list(multiset(B)),$o,aa(list(multiset(B)),fun(list(multiset(B)),$o),Uu,Xs3),Ys4) ) ) ) ).

% ATP.lambda_698
tff(fact_8876_ATP_Olambda__699,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),aTP_Lamp_ajo(set(B),fun(set(B),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,finite_finite2(B),Uub)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uua),Uub)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),Uu) ) ) ).

% ATP.lambda_699
tff(fact_8877_ATP_Olambda__700,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: set(B),Uub: set(B)] :
      ( aa(set(B),$o,aa(set(B),fun(set(B),$o),aTP_Lamp_ash(set(product_prod(B,B)),fun(set(B),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ( aa(set(B),$o,finite_finite2(B),Uua)
        & aa(set(B),$o,finite_finite2(B),Uub)
        & ( Uub != bot_bot(set(B)) )
        & ! [X3: B] :
            ( aa(set(B),$o,member(B,X3),Uua)
           => ? [Xa4: B] :
                ( aa(set(B),$o,member(B,Xa4),Uub)
                & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X3),Xa4)),Uu) ) ) ) ) ).

% ATP.lambda_700
tff(fact_8878_ATP_Olambda__701,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B,Uub: B] : aa(B,set(B),aa(B,fun(B,set(B)),aTP_Lamp_aof(set(product_prod(B,B)),fun(B,fun(B,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),image(B,B,converse(B,B,Uu)),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B)))) ).

% ATP.lambda_701
tff(fact_8879_ATP_Olambda__702,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: list(B),Uub: B] : aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_oi(fun(B,C),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_oh(fun(B,C),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_702
tff(fact_8880_ATP_Olambda__703,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ig(fun(nat,B),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),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_703
tff(fact_8881_ATP_Olambda__704,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ie(fun(nat,B),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),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_704
tff(fact_8882_ATP_Olambda__705,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,B),Uua: list(B),Uub: C] : aa(C,list(B),aa(list(B),fun(C,list(B)),aTP_Lamp_qe(fun(C,B),fun(list(B),fun(C,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(C,B,Uu,Uub)),nil(B))) ).

% ATP.lambda_705
tff(fact_8883_ATP_Olambda__706,axiom,
    ! [B: $tType] :
      ( euclid5411537665997757685th_nat(B)
     => ! [Uu: B,Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_di(B,fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uub)),Uua)) ) ).

% ATP.lambda_706
tff(fact_8884_ATP_Olambda__707,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: B,Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_de(B,fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uub)),Uua)) ) ).

% ATP.lambda_707
tff(fact_8885_ATP_Olambda__708,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: array(B),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mu(nat,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_Pair(array(B),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_708
tff(fact_8886_ATP_Olambda__709,axiom,
    ! [B: $tType,Uu: B,Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ale(B,fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,Uub),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uua),aa(set(B),set(B),insert(B,Uu),bot_bot(set(B))))) ) ).

% ATP.lambda_709
tff(fact_8887_ATP_Olambda__710,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: list(B),Uua: array(B),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ms(list(B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_Pair(array(B),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(B),nat,size_size(list(B)),Uu)))) ) ).

% ATP.lambda_710
tff(fact_8888_ATP_Olambda__711,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [Uu: B,Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_hz(B,fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_711
tff(fact_8889_ATP_Olambda__712,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,option(C)),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_anv(fun(B,option(C)),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,Uub),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uua),dom(B,C,Uu))) ) ).

% ATP.lambda_712
tff(fact_8890_ATP_Olambda__713,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,option(C)),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_anw(fun(B,option(C)),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,Uub),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uua),dom(B,C,Uu))) ) ).

% ATP.lambda_713
tff(fact_8891_ATP_Olambda__714,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: array(B),Uub: heap_ext(product_unit)] :
          ( aa(heap_ext(product_unit),$o,aa(array(B),fun(heap_ext(product_unit),$o),aTP_Lamp_akz(nat,fun(array(B),fun(heap_ext(product_unit),$o)),Uu),Uua),Uub)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),array_length(B,Uub,Uua)) ) ) ).

% ATP.lambda_714
tff(fact_8892_ATP_Olambda__715,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: array(B),Uua: nat,Uub: heap_ext(product_unit)] :
          ( aa(heap_ext(product_unit),$o,aa(nat,fun(heap_ext(product_unit),$o),aTP_Lamp_akx(array(B),fun(nat,fun(heap_ext(product_unit),$o)),Uu),Uua),Uub)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),array_length(B,Uub,Uu)) ) ) ).

% ATP.lambda_715
tff(fact_8893_ATP_Olambda__716,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [Uu: B,Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_bg(B,fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_716
tff(fact_8894_ATP_Olambda__717,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(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_717
tff(fact_8895_ATP_Olambda__718,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: B,Uua: C,Uub: D] : aa(D,product_prod(B,product_prod(C,D)),aa(C,fun(D,product_prod(B,product_prod(C,D))),aa(B,fun(C,fun(D,product_prod(B,product_prod(C,D)))),aTP_Lamp_sd(B,fun(C,fun(D,product_prod(B,product_prod(C,D))))),Uu),Uua),Uub) = 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)),Uu),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),Uua),Uub)) ).

% ATP.lambda_718
tff(fact_8896_ATP_Olambda__719,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: C,Uua: B,Uub: D] : aa(D,product_prod(B,product_prod(C,D)),aa(B,fun(D,product_prod(B,product_prod(C,D))),aTP_Lamp_sg(C,fun(B,fun(D,product_prod(B,product_prod(C,D)))),Uu),Uua),Uub) = 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)),Uua),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),Uu),Uub)) ).

% ATP.lambda_719
tff(fact_8897_ATP_Olambda__720,axiom,
    ! [B: $tType,Uu: B,Uua: list(B),Uub: nat] : aa(nat,list(B),aa(list(B),fun(nat,list(B)),aTP_Lamp_kt(B,fun(list(B),fun(nat,list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),take(B,Uub,Uua)) ).

% ATP.lambda_720
tff(fact_8898_ATP_Olambda__721,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,option(B)),Uua: list(C),Uub: B] : aa(B,list(B),aa(list(C),fun(B,list(B)),aTP_Lamp_qm(fun(C,option(B)),fun(list(C),fun(B,list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),map_filter(C,B,Uu,Uua)) ).

% ATP.lambda_721
tff(fact_8899_ATP_Olambda__722,axiom,
    ! [C: $tType,B: $tType] :
      ( ( order(B)
        & order(C) )
     => ! [Uu: fun(B,C),Uua: set(B),Uub: C] :
          ( aa(C,$o,aa(set(B),fun(C,$o),aTP_Lamp_ahf(fun(B,C),fun(set(B),fun(C,$o)),Uu),Uua),Uub)
        <=> aa(set(C),$o,member(C,Uub),aa(set(B),set(C),image2(B,C,Uu),Uua)) ) ) ).

% ATP.lambda_722
tff(fact_8900_ATP_Olambda__723,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_xg(fun(B,set(C)),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_723
tff(fact_8901_ATP_Olambda__724,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_xf(fun(B,set(C)),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_724
tff(fact_8902_ATP_Olambda__725,axiom,
    ! [C: $tType,B: $tType] :
      ( unboun7993243217541854897norder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_ami(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Uua),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_725
tff(fact_8903_ATP_Olambda__726,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_amd(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),Uua),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_726
tff(fact_8904_ATP_Olambda__727,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C] :
          ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_rj(fun(C,B),fun(B,fun(C,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(C,B,Uu,Uub)) ) ) ).

% ATP.lambda_727
tff(fact_8905_ATP_Olambda__728,axiom,
    ! [C: $tType,B: $tType] :
      ( unboun7993243217541854897norder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_amf(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less(C),Uua),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_728
tff(fact_8906_ATP_Olambda__729,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_pm(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),ord_less(C),Uua),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_729
tff(fact_8907_ATP_Olambda__730,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(D,B),Uua: fun(C,option(D)),Uub: C] : aa(C,option(B),aa(fun(C,option(D)),fun(C,option(B)),aTP_Lamp_ka(fun(D,B),fun(fun(C,option(D)),fun(C,option(B))),Uu),Uua),Uub) = aa(option(D),option(B),map_option(D,B,Uu),aa(C,option(D),Uua,Uub)) ).

% ATP.lambda_730
tff(fact_8908_ATP_Olambda__731,axiom,
    ! [B: $tType,Uu: nat,Uua: fun(B,nat),Uub: B] : aa(B,nat,aa(fun(B,nat),fun(B,nat),aa(nat,fun(fun(B,nat),fun(B,nat)),aTP_Lamp_aic(nat,fun(fun(B,nat),fun(B,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(B,nat,Uua,Uub)) ).

% ATP.lambda_731
tff(fact_8909_ATP_Olambda__732,axiom,
    ! [B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: B,Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_bp(B,fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_732
tff(fact_8910_ATP_Olambda__733,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: nat,Uub: B] : aa(B,nat,aa(nat,fun(B,nat),aTP_Lamp_air(fun(B,nat),fun(nat,fun(B,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(B,nat,Uu,Uub)) ).

% ATP.lambda_733
tff(fact_8911_ATP_Olambda__734,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_zd(set(B),fun(fun(C,set(B)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_734
tff(fact_8912_ATP_Olambda__735,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B,Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_ee(B,fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uu),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_735
tff(fact_8913_ATP_Olambda__736,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_aax(set(B),fun(fun(C,set(B)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_736
tff(fact_8914_ATP_Olambda__737,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: B,Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_abh(B,fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),Uu),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_737
tff(fact_8915_ATP_Olambda__738,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_yk(set(B),fun(fun(C,set(B)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_738
tff(fact_8916_ATP_Olambda__739,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [Uu: C,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_zl(C,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(C,C,aa(C,fun(C,C),inf_inf(C),Uu),aa(B,C,Uua,Uub)) ) ).

% ATP.lambda_739
tff(fact_8917_ATP_Olambda__740,axiom,
    ! [B: $tType,C: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: B,Uua: fun(C,B),Uub: C] : aa(C,B,aa(fun(C,B),fun(C,B),aTP_Lamp_wp(B,fun(fun(C,B),fun(C,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_740
tff(fact_8918_ATP_Olambda__741,axiom,
    ! [C: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_kx(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(C,$o,aa(C,fun(C,$o),dvd_dvd(C),Uua),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_741
tff(fact_8919_ATP_Olambda__742,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: filter(C),Uua: fun(B,filter(D)),Uub: B] : aa(B,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(B,filter(product_prod(C,D))),aTP_Lamp_ape(filter(C),fun(fun(B,filter(D)),fun(B,filter(product_prod(C,D)))),Uu),Uua),Uub) = prod_filter(C,D,Uu,aa(B,filter(D),Uua,Uub)) ).

% ATP.lambda_742
tff(fact_8920_ATP_Olambda__743,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(B,D),Uua: fun(C,filter(D)),Uub: C] : aa(C,filter(B),aa(fun(C,filter(D)),fun(C,filter(B)),aTP_Lamp_anf(fun(B,D),fun(fun(C,filter(D)),fun(C,filter(B))),Uu),Uua),Uub) = filtercomap(B,D,Uu,aa(C,filter(D),Uua,Uub)) ).

% ATP.lambda_743
tff(fact_8921_ATP_Olambda__744,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,C),Uua: fun(D,filter(C)),Uub: D] : aa(D,filter(B),aa(fun(D,filter(C)),fun(D,filter(B)),aTP_Lamp_ane(fun(B,C),fun(fun(D,filter(C)),fun(D,filter(B))),Uu),Uua),Uub) = filtercomap(B,C,Uu,aa(D,filter(C),Uua,Uub)) ).

% ATP.lambda_744
tff(fact_8922_ATP_Olambda__745,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType,Uu: fun(B,fun(D,$o)),Uua: fun(E,fun(D,fun(C,$o))),Uub: E] : aa(E,fun(B,fun(C,$o)),aa(fun(E,fun(D,fun(C,$o))),fun(E,fun(B,fun(C,$o))),aTP_Lamp_ark(fun(B,fun(D,$o)),fun(fun(E,fun(D,fun(C,$o))),fun(E,fun(B,fun(C,$o)))),Uu),Uua),Uub) = aa(fun(D,fun(C,$o)),fun(B,fun(C,$o)),aa(fun(B,fun(D,$o)),fun(fun(D,fun(C,$o)),fun(B,fun(C,$o))),relcompp(B,D,C),Uu),aa(E,fun(D,fun(C,$o)),Uua,Uub)) ).

% ATP.lambda_745
tff(fact_8923_ATP_Olambda__746,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(D,C),Uua: B,Uub: D] : aa(D,product_prod(B,C),aa(B,fun(D,product_prod(B,C)),aTP_Lamp_sc(fun(D,C),fun(B,fun(D,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),Uua),aa(D,C,Uu,Uub)) ).

% ATP.lambda_746
tff(fact_8924_ATP_Olambda__747,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType,Uu: set(product_prod(B,D)),Uua: fun(E,set(product_prod(D,C))),Uub: E] : aa(E,set(product_prod(B,C)),aa(fun(E,set(product_prod(D,C))),fun(E,set(product_prod(B,C))),aTP_Lamp_wf(set(product_prod(B,D)),fun(fun(E,set(product_prod(D,C))),fun(E,set(product_prod(B,C)))),Uu),Uua),Uub) = relcomp(B,D,C,Uu,aa(E,set(product_prod(D,C)),Uua,Uub)) ).

% ATP.lambda_747
tff(fact_8925_ATP_Olambda__748,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(C,B),Uua: fun(D,filter(C)),Uub: D] : aa(D,filter(B),aa(fun(D,filter(C)),fun(D,filter(B)),aTP_Lamp_aoq(fun(C,B),fun(fun(D,filter(C)),fun(D,filter(B))),Uu),Uua),Uub) = filtermap(C,B,Uu,aa(D,filter(C),Uua,Uub)) ).

% ATP.lambda_748
tff(fact_8926_ATP_Olambda__749,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: set(product_prod(C,B)),Uua: fun(D,set(C)),Uub: D] : aa(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_aga(set(product_prod(C,B)),fun(fun(D,set(C)),fun(D,set(B))),Uu),Uua),Uub) = aa(set(C),set(B),image(C,B,Uu),aa(D,set(C),Uua,Uub)) ).

% ATP.lambda_749
tff(fact_8927_ATP_Olambda__750,axiom,
    ! [B: $tType,Uu: $o,Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_aly($o,fun(fun(B,$o),fun(B,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_750
tff(fact_8928_ATP_Olambda__751,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,C),Uua: fun(D,set(C)),Uub: D] : aa(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_adr(fun(B,C),fun(fun(D,set(C)),fun(D,set(B))),Uu),Uua),Uub) = aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),Uu),aa(D,set(C),Uua,Uub)) ).

% ATP.lambda_751
tff(fact_8929_ATP_Olambda__752,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_xt(fun(C,set(B)),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,Uub),aa(C,set(B),Uu,Uua)) ) ).

% ATP.lambda_752
tff(fact_8930_ATP_Olambda__753,axiom,
    ! [C: $tType,B: $tType,Uu: C,Uua: fun(B,set(C)),Uub: B] : aa(B,set(C),aa(fun(B,set(C)),fun(B,set(C)),aTP_Lamp_yh(C,fun(fun(B,set(C)),fun(B,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),insert(C,Uu),aa(B,set(C),Uua,Uub)) ).

% ATP.lambda_753
tff(fact_8931_ATP_Olambda__754,axiom,
    ! [B: $tType,C: $tType,Uu: B,Uua: fun(C,set(B)),Uub: C] : aa(C,set(B),aa(fun(C,set(B)),fun(C,set(B)),aTP_Lamp_xk(B,fun(fun(C,set(B)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_754
tff(fact_8932_ATP_Olambda__755,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(C,B),Uua: fun(D,set(C)),Uub: D] : aa(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_yf(fun(C,B),fun(fun(D,set(C)),fun(D,set(B))),Uu),Uua),Uub) = aa(set(C),set(B),image2(C,B,Uu),aa(D,set(C),Uua,Uub)) ).

% ATP.lambda_755
tff(fact_8933_ATP_Olambda__756,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,C),Uua: fun(D,set(B)),Uub: D] : aa(D,set(C),aa(fun(D,set(B)),fun(D,set(C)),aTP_Lamp_afm(fun(B,C),fun(fun(D,set(B)),fun(D,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image2(B,C,Uu),aa(D,set(B),Uua,Uub)) ).

% ATP.lambda_756
tff(fact_8934_ATP_Olambda__757,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( condit1219197933456340205attice(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,D),Uub: B] : aa(B,set(D),aa(fun(C,D),fun(B,set(D)),aTP_Lamp_anx(fun(B,set(C)),fun(fun(C,D),fun(B,set(D))),Uu),Uua),Uub) = aa(set(C),set(D),image2(C,D,Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_757
tff(fact_8935_ATP_Olambda__758,axiom,
    ! [B: $tType,Uu: $o,Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_alx($o,fun(fun(B,$o),fun(B,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_758
tff(fact_8936_ATP_Olambda__759,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B,Uub: $o] :
      ( aa($o,$o,aa(B,fun($o,$o),aTP_Lamp_arx(fun(B,$o),fun(B,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        | aa(B,$o,Uu,Uua) ) ) ).

% ATP.lambda_759
tff(fact_8937_ATP_Olambda__760,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B,Uub: $o] :
      ( aa($o,$o,aa(B,fun($o,$o),aTP_Lamp_aka(fun(B,$o),fun(B,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(B,$o,Uu,Uua) ) ) ).

% ATP.lambda_760
tff(fact_8938_ATP_Olambda__761,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(list(B),B),Uua: list(B),Uub: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_pf(fun(list(B),B),fun(list(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(B),B,Uu,Uua) ) ) ) ).

% ATP.lambda_761
tff(fact_8939_ATP_Olambda__762,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_afi(fun(B,C),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(B,C,Uu,Uua) ) ) ).

% ATP.lambda_762
tff(fact_8940_ATP_Olambda__763,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: B,Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_en(B,fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_763
tff(fact_8941_ATP_Olambda__764,axiom,
    ! [B: $tType,Uu: list(B),Uua: set(nat),Uub: B] :
      ( aa(B,$o,aa(set(nat),fun(B,$o),aTP_Lamp_sa(list(B),fun(set(nat),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,Uub),aa(list(B),set(B),set2(B),nths(B,Uu,Uua))) ) ).

% ATP.lambda_764
tff(fact_8942_ATP_Olambda__765,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(B),Uub: B] : aa(B,set(B),aa(set(B),fun(B,set(B)),aTP_Lamp_adv(set(B),fun(set(B),fun(B,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uu),Uua) ).

% ATP.lambda_765
tff(fact_8943_ATP_Olambda__766,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: set(C),Uub: B] : aa(B,set(C),aa(set(C),fun(B,set(C)),aTP_Lamp_ws(set(C),fun(set(C),fun(B,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),Uu),Uua) ).

% ATP.lambda_766
tff(fact_8944_ATP_Olambda__767,axiom,
    ! [B: $tType,Uu: list(B),Uua: B,Uub: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),aTP_Lamp_ra(list(B),fun(B,fun(list(B),list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uub),Uu) ).

% ATP.lambda_767
tff(fact_8945_ATP_Olambda__768,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: set(C),Uub: B] : aa(B,set(C),aa(set(C),fun(B,set(C)),aTP_Lamp_aay(C,fun(set(C),fun(B,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),insert(C,Uu),Uua) ).

% ATP.lambda_768
tff(fact_8946_ATP_Olambda__769,axiom,
    ! [B: $tType,C: $tType,E: $tType,Uu: fun(E,C),Uua: set(E),Uub: B] : aa(B,set(C),aa(set(E),fun(B,set(C)),aTP_Lamp_xc(fun(E,C),fun(set(E),fun(B,set(C))),Uu),Uua),Uub) = aa(set(E),set(C),image2(E,C,Uu),Uua) ).

% ATP.lambda_769
tff(fact_8947_ATP_Olambda__770,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: set(C),Uub: B] : aa(B,set(B),aa(set(C),fun(B,set(B)),aTP_Lamp_wa(fun(C,B),fun(set(C),fun(B,set(B))),Uu),Uua),Uub) = aa(set(C),set(B),image2(C,B,Uu),Uua) ).

% ATP.lambda_770
tff(fact_8948_ATP_Olambda__771,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_fy(fun(B,nat),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(Uub = Uua,aa(nat,nat,suc,aa(B,nat,Uu,Uub)),aa(B,nat,Uu,Uub))) ) ).

% ATP.lambda_771
tff(fact_8949_ATP_Olambda__772,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_fu(fun(B,nat),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(aa(B,$o,Uua,Uub),aa(B,nat,Uu,Uub),zero_zero(nat))) ) ).

% ATP.lambda_772
tff(fact_8950_ATP_Olambda__773,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: ref(B),Uua: B,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(B,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_apt(ref(B),fun(B,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(B,Uu,Uua,Uub)),one_one(nat))) ) ).

% ATP.lambda_773
tff(fact_8951_ATP_Olambda__774,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: fun(B,nat),Uub: B] :
      ( aa(B,$o,aa(fun(B,nat),fun(B,$o),aTP_Lamp_fz(fun(B,nat),fun(fun(B,nat),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(B,nat,Uu,Uub)),aa(B,nat,Uua,Uub))) ) ).

% ATP.lambda_774
tff(fact_8952_ATP_Olambda__775,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ic(fun(nat,B),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_775
tff(fact_8953_ATP_Olambda__776,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_ib(fun(nat,B),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_776
tff(fact_8954_ATP_Olambda__777,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_alm(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_777
tff(fact_8955_ATP_Olambda__778,axiom,
    ! [B: $tType,Uu: fun(nat,set(B)),Uua: nat,Uub: nat] : aa(nat,set(B),aa(nat,fun(nat,set(B)),aTP_Lamp_xa(fun(nat,set(B)),fun(nat,fun(nat,set(B))),Uu),Uua),Uub) = aa(nat,set(B),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_778
tff(fact_8956_ATP_Olambda__779,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_dp(fun(nat,B),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_779
tff(fact_8957_ATP_Olambda__780,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat] : aa(nat,B,aa(nat,fun(nat,B),aTP_Lamp_cc(fun(nat,B),fun(nat,fun(nat,B)),Uu),Uua),Uub) = aa(nat,B,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_780
tff(fact_8958_ATP_Olambda__781,axiom,
    ! [B: $tType,C: $tType,Uu: fun(product_prod(B,C),$o),Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_lq(fun(product_prod(B,C),$o),fun(B,fun(C,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(B,C),$o,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ) ).

% ATP.lambda_781
tff(fact_8959_ATP_Olambda__782,axiom,
    ! [D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),D),Uua: B,Uub: C] : aa(C,D,aa(B,fun(C,D),aTP_Lamp_ly(fun(product_prod(B,C),D),fun(B,fun(C,D)),Uu),Uua),Uub) = aa(product_prod(B,C),D,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_782
tff(fact_8960_ATP_Olambda__783,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: list(B),Uub: nat] :
      ( aa(nat,$o,aa(list(B),fun(nat,$o),aTP_Lamp_qs(fun(B,$o),fun(list(B),fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,aa(nat,B,nth(B,Uua),Uub)) ) ).

% ATP.lambda_783
tff(fact_8961_ATP_Olambda__784,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(multiset(B),C),Uua: fun(D,multiset(B)),Uub: D] : aa(D,C,aa(fun(D,multiset(B)),fun(D,C),aTP_Lamp_ahv(fun(multiset(B),C),fun(fun(D,multiset(B)),fun(D,C)),Uu),Uua),Uub) = aa(multiset(B),C,Uu,aa(D,multiset(B),Uua,Uub)) ) ).

% ATP.lambda_784
tff(fact_8962_ATP_Olambda__785,axiom,
    ! [C: $tType,D: $tType,E: $tType,B: $tType,Uu: fun(E,fun(C,D)),Uua: fun(B,E),Uub: B] : aa(B,fun(C,D),aa(fun(B,E),fun(B,fun(C,D)),aTP_Lamp_tl(fun(E,fun(C,D)),fun(fun(B,E),fun(B,fun(C,D))),Uu),Uua),Uub) = aa(E,fun(C,D),Uu,aa(B,E,Uua,Uub)) ).

% ATP.lambda_785
tff(fact_8963_ATP_Olambda__786,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(D,set(B)),Uua: fun(C,D),Uub: C] : aa(C,set(B),aa(fun(C,D),fun(C,set(B)),aTP_Lamp_ye(fun(D,set(B)),fun(fun(C,D),fun(C,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,aa(C,D,Uua,Uub)) ).

% ATP.lambda_786
tff(fact_8964_ATP_Olambda__787,axiom,
    ! [E: $tType,D: $tType,G4: $tType,Uu: fun(D,fun(E,$o)),Uua: fun(G4,D),Uub: G4] : aa(G4,fun(E,$o),aa(fun(G4,D),fun(G4,fun(E,$o)),aTP_Lamp_aqx(fun(D,fun(E,$o)),fun(fun(G4,D),fun(G4,fun(E,$o))),Uu),Uua),Uub) = aa(D,fun(E,$o),Uu,aa(G4,D,Uua,Uub)) ).

% ATP.lambda_787
tff(fact_8965_ATP_Olambda__788,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(D,B),Uua: fun(C,D),Uub: C] : aa(C,B,aa(fun(C,D),fun(C,B),aTP_Lamp_vk(fun(D,B),fun(fun(C,D),fun(C,B)),Uu),Uua),Uub) = aa(D,B,Uu,aa(C,D,Uua,Uub)) ).

% ATP.lambda_788
tff(fact_8966_ATP_Olambda__789,axiom,
    ! [D: $tType,C: $tType,E: $tType,Uu: fun(C,fun(D,$o)),Uua: fun(E,C),Uub: E] : aa(E,fun(D,$o),aa(fun(E,C),fun(E,fun(D,$o)),aTP_Lamp_age(fun(C,fun(D,$o)),fun(fun(E,C),fun(E,fun(D,$o))),Uu),Uua),Uub) = aa(C,fun(D,$o),Uu,aa(E,C,Uua,Uub)) ).

% ATP.lambda_789
tff(fact_8967_ATP_Olambda__790,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(num,C),Uub: num] : aa(num,B,aa(fun(num,C),fun(num,B),aTP_Lamp_mf(fun(C,B),fun(fun(num,C),fun(num,B)),Uu),Uua),Uub) = aa(C,B,Uu,aa(num,C,Uua,Uub)) ).

% ATP.lambda_790
tff(fact_8968_ATP_Olambda__791,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(nat,C),Uub: nat] : aa(nat,B,aa(fun(nat,C),fun(nat,B),aTP_Lamp_fa(fun(C,B),fun(fun(nat,C),fun(nat,B)),Uu),Uua),Uub) = aa(C,B,Uu,aa(nat,C,Uua,Uub)) ).

% ATP.lambda_791
tff(fact_8969_ATP_Olambda__792,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(C,B),Uua: fun(D,C),Uub: D] : aa(D,B,aa(fun(D,C),fun(D,B),aTP_Lamp_ip(fun(C,B),fun(fun(D,C),fun(D,B)),Uu),Uua),Uub) = aa(C,B,Uu,aa(D,C,Uua,Uub)) ).

% ATP.lambda_792
tff(fact_8970_ATP_Olambda__793,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(C) )
     => ! [Uu: fun(B,C),Uua: fun(D,B),Uub: D] : aa(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aoe(fun(B,C),fun(fun(D,B),fun(D,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_793
tff(fact_8971_ATP_Olambda__794,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(C) )
     => ! [Uu: fun(B,C),Uua: fun(D,B),Uub: D] : aa(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aef(fun(B,C),fun(fun(D,B),fun(D,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_794
tff(fact_8972_ATP_Olambda__795,axiom,
    ! [B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(C)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(B,C),Uua: fun(C,B),Uub: C] : aa(C,C,aa(fun(C,B),fun(C,C),aTP_Lamp_aeb(fun(B,C),fun(fun(C,B),fun(C,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_795
tff(fact_8973_ATP_Olambda__796,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,$o),Uua: fun(C,B),Uub: C] :
      ( aa(C,$o,aa(fun(C,B),fun(C,$o),aTP_Lamp_als(fun(B,$o),fun(fun(C,B),fun(C,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,aa(C,B,Uua,Uub)) ) ).

% ATP.lambda_796
tff(fact_8974_ATP_Olambda__797,axiom,
    ! [C: $tType,B: $tType,F6: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(F6,B),Uub: F6] : aa(F6,fun(C,$o),aa(fun(F6,B),fun(F6,fun(C,$o)),aTP_Lamp_aqw(fun(B,fun(C,$o)),fun(fun(F6,B),fun(F6,fun(C,$o))),Uu),Uua),Uub) = aa(B,fun(C,$o),Uu,aa(F6,B,Uua,Uub)) ).

% ATP.lambda_797
tff(fact_8975_ATP_Olambda__798,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,B),Uub: D] : aa(D,fun(C,$o),aa(fun(D,B),fun(D,fun(C,$o)),aTP_Lamp_agv(fun(B,fun(C,$o)),fun(fun(D,B),fun(D,fun(C,$o))),Uu),Uua),Uub) = aa(B,fun(C,$o),Uu,aa(D,B,Uua,Uub)) ).

% ATP.lambda_798
tff(fact_8976_ATP_Olambda__799,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,C),Uua: fun(D,B),Uub: D] : aa(D,C,aa(fun(D,B),fun(D,C),aTP_Lamp_aji(fun(B,C),fun(fun(D,B),fun(D,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(D,B,Uua,Uub)) ).

% ATP.lambda_799
tff(fact_8977_ATP_Olambda__800,axiom,
    ! [E: $tType,D: $tType,Uu: fun(D,E),Uua: fun(E,$o),Uub: D] :
      ( aa(D,$o,aa(fun(E,$o),fun(D,$o),aTP_Lamp_arq(fun(D,E),fun(fun(E,$o),fun(D,$o)),Uu),Uua),Uub)
    <=> aa(E,$o,Uua,aa(D,E,Uu,Uub)) ) ).

% ATP.lambda_800
tff(fact_8978_ATP_Olambda__801,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(B,$o),Uub: C] :
      ( aa(C,$o,aa(fun(B,$o),fun(C,$o),aTP_Lamp_aor(fun(C,B),fun(fun(B,$o),fun(C,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(C,B,Uu,Uub)) ) ).

% ATP.lambda_801
tff(fact_8979_ATP_Olambda__802,axiom,
    ! [D: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(B,D),Uub: C] : aa(C,D,aa(fun(B,D),fun(C,D),aTP_Lamp_vl(fun(C,B),fun(fun(B,D),fun(C,D)),Uu),Uua),Uub) = aa(B,D,Uua,aa(C,B,Uu,Uub)) ).

% ATP.lambda_802
tff(fact_8980_ATP_Olambda__803,axiom,
    ! [C: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(C) )
     => ! [Uu: fun(B,C),Uua: fun(C,B),Uub: B] : aa(B,B,aa(fun(C,B),fun(B,B),aTP_Lamp_aea(fun(B,C),fun(fun(C,B),fun(B,B)),Uu),Uua),Uub) = aa(C,B,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_803
tff(fact_8981_ATP_Olambda__804,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(C,$o),Uub: B] :
      ( aa(B,$o,aa(fun(C,$o),fun(B,$o),aTP_Lamp_adp(fun(B,C),fun(fun(C,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(C,$o,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_804
tff(fact_8982_ATP_Olambda__805,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: fun(B,C),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_ajt(fun(B,C),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_805
tff(fact_8983_ATP_Olambda__806,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: fun(B,C),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_ajs(fun(B,C),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_806
tff(fact_8984_ATP_Olambda__807,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( semiring_1(D)
     => ! [Uu: fun(B,C),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_jh(fun(B,C),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_807
tff(fact_8985_ATP_Olambda__808,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_and(fun(B,C),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(B,C,Uu,Uub)) ).

% ATP.lambda_808
tff(fact_8986_ATP_Olambda__809,axiom,
    ! [B: $tType,Uu: fun(code_natural,B),Uua: code_natural,Uub: B] : aa(B,B,aa(code_natural,fun(B,B),aTP_Lamp_aqa(fun(code_natural,B),fun(code_natural,fun(B,B)),Uu),Uua),Uub) = aa(code_natural,B,Uu,Uua) ).

% ATP.lambda_809
tff(fact_8987_ATP_Olambda__810,axiom,
    ! [B: $tType,C: $tType,E: $tType,Uu: fun(E,set(C)),Uua: E,Uub: B] : aa(B,set(C),aa(E,fun(B,set(C)),aTP_Lamp_wh(fun(E,set(C)),fun(E,fun(B,set(C))),Uu),Uua),Uub) = aa(E,set(C),Uu,Uua) ).

% ATP.lambda_810
tff(fact_8988_ATP_Olambda__811,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(D,fun(B,D)),Uua: D,Uub: C] : aa(C,fun(B,D),aa(D,fun(C,fun(B,D)),aTP_Lamp_st(fun(D,fun(B,D)),fun(D,fun(C,fun(B,D))),Uu),Uua),Uub) = aa(D,fun(B,D),Uu,Uua) ).

% ATP.lambda_811
tff(fact_8989_ATP_Olambda__812,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,$o),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_apa(fun(C,$o),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(C,$o,Uu,Uua) ) ).

% ATP.lambda_812
tff(fact_8990_ATP_Olambda__813,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,D),Uua: B,Uub: C] : aa(C,D,aa(B,fun(C,D),aTP_Lamp_tb(fun(B,D),fun(B,fun(C,D)),Uu),Uua),Uub) = aa(B,D,Uu,Uua) ).

% ATP.lambda_813
tff(fact_8991_ATP_Olambda__814,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: B,Uub: D] : aa(D,C,aa(B,fun(D,C),aTP_Lamp_afe(fun(B,C),fun(B,fun(D,C)),Uu),Uua),Uub) = aa(B,C,Uu,Uua) ).

% ATP.lambda_814
tff(fact_8992_ATP_Olambda__815,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: fun(C,heap_Time_Heap(B)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(B),aa(fun(C,heap_Time_Heap(B)),fun(product_unit,heap_Time_Heap(B)),aTP_Lamp_ai(C,fun(fun(C,heap_Time_Heap(B)),fun(product_unit,heap_Time_Heap(B))),Uu),Uua),Uub) = aa(C,heap_Time_Heap(B),Uua,Uu) ).

% ATP.lambda_815
tff(fact_8993_ATP_Olambda__816,axiom,
    ! [B: $tType,Uu: heap_Time_Heap(B),Uua: heap_ext(product_unit),Uub: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),$o),aa(heap_ext(product_unit),fun(B,fun(product_prod(heap_ext(product_unit),nat),$o)),aTP_Lamp_nd(heap_Time_Heap(B),fun(heap_ext(product_unit),fun(B,fun(product_prod(heap_ext(product_unit),nat),$o))),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),fun(nat,$o)),fun(product_prod(heap_ext(product_unit),nat),$o),product_case_prod(heap_ext(product_unit),nat,$o),aa(B,fun(heap_ext(product_unit),fun(nat,$o)),aa(heap_ext(product_unit),fun(B,fun(heap_ext(product_unit),fun(nat,$o))),aTP_Lamp_nc(heap_Time_Heap(B),fun(heap_ext(product_unit),fun(B,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_816
tff(fact_8994_ATP_Olambda__817,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: multiset(C),Uub: D] : aa(D,B,aa(multiset(C),fun(D,B),aTP_Lamp_aig(fun(C,fun(D,B)),fun(multiset(C),fun(D,B)),Uu),Uua),Uub) = comm_m9189036328036947845d_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(D,fun(C,B),aTP_Lamp_ds(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_817
tff(fact_8995_ATP_Olambda__818,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: multiset(C),Uub: D] : aa(D,B,aa(multiset(C),fun(D,B),aTP_Lamp_agm(fun(C,fun(D,B)),fun(multiset(C),fun(D,B)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(D,fun(C,B),aTP_Lamp_bm(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_818
tff(fact_8996_ATP_Olambda__819,axiom,
    ! [C: $tType,D: $tType,B: $tType,F6: $tType,E: $tType,Uu: fun(C,fun(D,fun(E,fun(F6,set(B))))),Uua: set(product_prod(E,F6)),Uub: product_prod(C,D)] : aa(product_prod(C,D),set(B),aa(set(product_prod(E,F6)),fun(product_prod(C,D),set(B)),aTP_Lamp_yx(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(set(product_prod(E,F6)),fun(product_prod(C,D),set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(product_prod(E,F6)),set(set(B)),image2(product_prod(E,F6),set(B),aa(product_prod(C,D),fun(product_prod(E,F6),set(B)),aTP_Lamp_yw(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(product_prod(C,D),fun(product_prod(E,F6),set(B))),Uu),Uub)),Uua)) ).

% ATP.lambda_819
tff(fact_8997_ATP_Olambda__820,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(C),Uub: D] : aa(D,B,aa(set(C),fun(D,B),aTP_Lamp_xx(fun(C,fun(D,B)),fun(set(C),fun(D,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(D,fun(C,B),aTP_Lamp_xw(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_820
tff(fact_8998_ATP_Olambda__821,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(C),Uub: D] : aa(D,B,aa(set(C),fun(D,B),aTP_Lamp_zf(fun(C,fun(D,B)),fun(set(C),fun(D,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(D,fun(C,B),aTP_Lamp_xw(fun(C,fun(D,B)),fun(D,fun(C,B)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_821
tff(fact_8999_ATP_Olambda__822,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: multiset(D),Uub: C] : aa(C,B,aa(multiset(D),fun(C,B),aTP_Lamp_aif(fun(C,fun(D,B)),fun(multiset(D),fun(C,B)),Uu),Uua),Uub) = comm_m9189036328036947845d_mset(B,aa(multiset(D),multiset(B),image_mset(D,B,aa(C,fun(D,B),Uu,Uub)),Uua)) ) ).

% ATP.lambda_822
tff(fact_9000_ATP_Olambda__823,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: multiset(D),Uub: C] : aa(C,B,aa(multiset(D),fun(C,B),aTP_Lamp_agl(fun(C,fun(D,B)),fun(multiset(D),fun(C,B)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(B,aa(multiset(D),multiset(B),image_mset(D,B,aa(C,fun(D,B),Uu,Uub)),Uua)) ) ).

% ATP.lambda_823
tff(fact_9001_ATP_Olambda__824,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(D),Uub: C] : aa(C,B,aa(set(D),fun(C,B),aTP_Lamp_xv(fun(C,fun(D,B)),fun(set(D),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),Uu,Uub)),Uua)) ) ).

% ATP.lambda_824
tff(fact_9002_ATP_Olambda__825,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(B,fun(C,set(D))),Uua: set(C),Uub: B] : aa(B,set(D),aa(set(C),fun(B,set(D)),aTP_Lamp_agq(fun(B,fun(C,set(D))),fun(set(C),fun(B,set(D))),Uu),Uua),Uub) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(B,fun(C,set(D)),Uu,Uub)),Uua)) ).

% ATP.lambda_825
tff(fact_9003_ATP_Olambda__826,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,fun(D,B)),Uua: set(D),Uub: C] : aa(C,B,aa(set(D),fun(C,B),aTP_Lamp_ze(fun(C,fun(D,B)),fun(set(D),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),Uu,Uub)),Uua)) ) ).

% ATP.lambda_826
tff(fact_9004_ATP_Olambda__827,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,C),Uua: C,Uub: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_rn(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,C,Uu,Uub) != Uua ) ) ).

% ATP.lambda_827
tff(fact_9005_ATP_Olambda__828,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: fun(B,C),Uub: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aTP_Lamp_ro(C,fun(fun(B,C),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,C,Uua,Uub) != Uu ) ) ).

% ATP.lambda_828
tff(fact_9006_ATP_Olambda__829,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(D,set(B)),Uua: fun(C,set(D)),Uub: C] : aa(C,set(B),aa(fun(C,set(D)),fun(C,set(B)),aTP_Lamp_yd(fun(D,set(B)),fun(fun(C,set(D)),fun(C,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),aa(C,set(D),Uua,Uub))) ).

% ATP.lambda_829
tff(fact_9007_ATP_Olambda__830,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(C,set(B)),Uua: fun(D,set(C)),Uub: D] : aa(D,set(B),aa(fun(D,set(C)),fun(D,set(B)),aTP_Lamp_yb(fun(C,set(B)),fun(fun(D,set(C)),fun(D,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),Uu),aa(D,set(C),Uua,Uub))) ).

% ATP.lambda_830
tff(fact_9008_ATP_Olambda__831,axiom,
    ! [B: $tType,C: $tType,D: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(C,B),Uua: fun(D,set(C)),Uub: D] : aa(D,B,aa(fun(D,set(C)),fun(D,B),aTP_Lamp_yo(fun(C,B),fun(fun(D,set(C)),fun(D,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,Uu),aa(D,set(C),Uua,Uub))) ) ).

% ATP.lambda_831
tff(fact_9009_ATP_Olambda__832,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( condit1219197933456340205attice(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_any(fun(B,set(C)),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(set(D),D,complete_Sup_Sup(D),aa(set(C),set(D),image2(C,D,Uua),aa(B,set(C),Uu,Uub))) ) ).

% ATP.lambda_832
tff(fact_9010_ATP_Olambda__833,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(D,set(B)),Uua: fun(C,set(D)),Uub: C] : aa(C,set(B),aa(fun(C,set(D)),fun(C,set(B)),aTP_Lamp_zk(fun(D,set(B)),fun(fun(C,set(D)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),aa(C,set(D),Uua,Uub))) ).

% ATP.lambda_833
tff(fact_9011_ATP_Olambda__834,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( condit1219197933456340205attice(D)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,D),Uub: B] : aa(B,D,aa(fun(C,D),fun(B,D),aTP_Lamp_aog(fun(B,set(C)),fun(fun(C,D),fun(B,D)),Uu),Uua),Uub) = aa(set(D),D,complete_Inf_Inf(D),aa(set(C),set(D),image2(C,D,Uua),aa(B,set(C),Uu,Uub))) ) ).

% ATP.lambda_834
tff(fact_9012_ATP_Olambda__835,axiom,
    ! [C: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(C)
     => ! [Uu: fun(B,C),Uua: C,Uub: B] :
          ( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_ky(fun(B,C),fun(C,fun(B,$o)),Uu),Uua),Uub)
        <=> ~ aa(C,$o,aa(C,fun(C,$o),dvd_dvd(C),Uua),aa(B,C,Uu,Uub)) ) ) ).

% ATP.lambda_835
tff(fact_9013_ATP_Olambda__836,axiom,
    ! [B: $tType,C: $tType,E: $tType,Uu: fun(E,set(C)),Uua: set(E),Uub: B] : aa(B,set(C),aa(set(E),fun(B,set(C)),aTP_Lamp_wk(fun(E,set(C)),fun(set(E),fun(B,set(C))),Uu),Uua),Uub) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(E),set(set(C)),image2(E,set(C),Uu),Uua)) ).

% ATP.lambda_836
tff(fact_9014_ATP_Olambda__837,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: C,Uub: list(C)] : aa(list(C),fun(list(B),list(B)),aa(C,fun(list(C),fun(list(B),list(B))),aTP_Lamp_ajm(fun(C,B),fun(C,fun(list(C),fun(list(B),list(B)))),Uu),Uua),Uub) = aa(B,fun(list(B),list(B)),cons(B),aa(C,B,Uu,Uua)) ).

% ATP.lambda_837
tff(fact_9015_ATP_Olambda__838,axiom,
    ! [C: $tType,B: $tType,Uu: set(C),Uua: fun(B,fun(C,$o)),Uub: B] : aa(B,nat,aa(fun(B,fun(C,$o)),fun(B,nat),aTP_Lamp_gq(set(C),fun(fun(B,fun(C,$o)),fun(B,nat)),Uu),Uua),Uub) = aa(set(C),nat,finite_card(C),aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_fd(set(C),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_838
tff(fact_9016_ATP_Olambda__839,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,D)),Uua: set(D),Uub: B] :
      ( aa(B,$o,aa(set(D),fun(B,$o),aTP_Lamp_un(fun(B,fun(C,D)),fun(set(D),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [B9: C] : aa(set(D),$o,member(D,aa(C,D,aa(B,fun(C,D),Uu,Uub),B9)),Uua) ) ).

% ATP.lambda_839
tff(fact_9017_ATP_Olambda__840,axiom,
    ! [B: $tType,C: $tType,Uu: list(B),Uua: list(C),Uub: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(list(C),fun(product_prod(B,C),$o),aTP_Lamp_ug(list(B),fun(list(C),fun(product_prod(B,C),$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(nat,B,nth(B,Uu),I3)),aa(nat,C,nth(C,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(list(C),nat,size_size(list(C)),Uua))) ) ) ).

% ATP.lambda_840
tff(fact_9018_ATP_Olambda__841,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(B,C),Uub: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(fun(B,C),fun(product_prod(B,C),$o),aTP_Lamp_adk(set(B),fun(fun(B,C),fun(product_prod(B,C),$o)),Uu),Uua),Uub)
    <=> ? [A6: B] :
          ( ( Uub = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A6),aa(B,C,Uua,A6)) )
          & aa(set(B),$o,member(B,A6),Uu) ) ) ).

% ATP.lambda_841
tff(fact_9019_ATP_Olambda__842,axiom,
    ! [B: $tType,Uu: list(B),Uua: set(nat),Uub: B] :
      ( aa(B,$o,aa(set(nat),fun(B,$o),aTP_Lamp_vb(list(B),fun(set(nat),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,B,nth(B,Uu),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Uu))
          & aa(set(nat),$o,member(nat,I3),Uua) ) ) ).

% ATP.lambda_842
tff(fact_9020_ATP_Olambda__843,axiom,
    ! [B: $tType,Uu: nat,Uua: list(B),Uub: B] :
      ( aa(B,$o,aa(list(B),fun(B,$o),aTP_Lamp_vc(nat,fun(list(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,B,nth(B,Uua),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uu),I3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Uua)) ) ) ).

% ATP.lambda_843
tff(fact_9021_ATP_Olambda__844,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [Uu: set(B),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_acz(set(B),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> ? [A6: B] :
              ( ( Uub = aa(B,B,aa(B,fun(B,B),sup_sup(B),Uua),A6) )
              & aa(set(B),$o,member(B,A6),Uu) ) ) ) ).

% ATP.lambda_844
tff(fact_9022_ATP_Olambda__845,axiom,
    ! [B: $tType] :
      ( finite8700451911770168679attice(B)
     => ! [Uu: B,Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_zw(B,fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ? [B9: B] :
              ( ( Uub = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),B9) )
              & aa(set(B),$o,member(B,B9),Uua) ) ) ) ).

% ATP.lambda_845
tff(fact_9023_ATP_Olambda__846,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [Uu: set(B),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_acx(set(B),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> ? [A6: B] :
              ( ( Uub = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uua),A6) )
              & aa(set(B),$o,member(B,A6),Uu) ) ) ) ).

% ATP.lambda_846
tff(fact_9024_ATP_Olambda__847,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: multiset(B),Uub: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aqp(set(multiset(B)),fun(multiset(B),fun(multiset(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: multiset(B)] :
          ( ( Uub = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),Uua),A6) )
          & aa(set(multiset(B)),$o,member(multiset(B),A6),Uu) ) ) ).

% ATP.lambda_847
tff(fact_9025_ATP_Olambda__848,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: multiset(B),Uub: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aql(set(multiset(B)),fun(multiset(B),fun(multiset(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: multiset(B)] :
          ( ( Uub = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),Uua),A6) )
          & aa(set(multiset(B)),$o,member(multiset(B),A6),Uu) ) ) ).

% ATP.lambda_848
tff(fact_9026_ATP_Olambda__849,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: list(C),Uub: set(B)] :
      ( aa(set(B),$o,aa(list(C),fun(set(B),$o),aTP_Lamp_aai(fun(C,set(B)),fun(list(C),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(C,set(B),Uu,aa(nat,C,nth(C,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(C),nat,size_size(list(C)),Uua)) ) ) ).

% ATP.lambda_849
tff(fact_9027_ATP_Olambda__850,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: list(C),Uub: set(B)] :
      ( aa(set(B),$o,aa(list(C),fun(set(B),$o),aTP_Lamp_aan(fun(C,set(B)),fun(list(C),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: C] :
          ( ( Uub = aa(C,set(B),Uu,A6) )
          & aa(set(C),$o,member(C,A6),aa(list(C),set(C),set2(C),Uua)) ) ) ).

% ATP.lambda_850
tff(fact_9028_ATP_Olambda__851,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_uk(fun(C,B),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [L5: C] :
          ( ( Uub = aa(C,B,Uu,L5) )
          & aa(set(C),$o,member(C,L5),Uua) ) ) ).

% ATP.lambda_851
tff(fact_9029_ATP_Olambda__852,axiom,
    ! [C: $tType,D: $tType,Uu: fun(C,set(D)),Uua: fun(D,$o),Uub: C] :
      ( aa(C,$o,aa(fun(D,$o),fun(C,$o),aTP_Lamp_akc(fun(C,set(D)),fun(fun(D,$o),fun(C,$o)),Uu),Uua),Uub)
    <=> ! [X3: D] :
          ( aa(set(D),$o,member(D,X3),aa(C,set(D),Uu,Uub))
         => aa(D,$o,Uua,X3) ) ) ).

% ATP.lambda_852
tff(fact_9030_ATP_Olambda__853,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: fun(C,$o),Uub: set(B)] :
      ( aa(set(B),$o,aa(fun(C,$o),fun(set(B),$o),aTP_Lamp_xy(fun(C,set(B)),fun(fun(C,$o),fun(set(B),$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( ( Uub = aa(C,set(B),Uu,X3) )
          & aa(C,$o,Uua,X3) ) ) ).

% ATP.lambda_853
tff(fact_9031_ATP_Olambda__854,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(C,$o),Uub: B] :
      ( aa(B,$o,aa(fun(C,$o),fun(B,$o),aTP_Lamp_ur(fun(C,B),fun(fun(C,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( ( Uub = aa(C,B,Uu,X3) )
          & aa(C,$o,Uua,X3) ) ) ).

% ATP.lambda_854
tff(fact_9032_ATP_Olambda__855,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,C),Uub: C] :
      ( aa(C,$o,aa(fun(B,C),fun(C,$o),aTP_Lamp_up(fun(B,$o),fun(fun(B,C),fun(C,$o)),Uu),Uua),Uub)
    <=> ? [X3: B] :
          ( ( Uub = aa(B,C,Uua,X3) )
          & aa(B,$o,Uu,X3) ) ) ).

% ATP.lambda_855
tff(fact_9033_ATP_Olambda__856,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,fun(B,$o)),Uub: D] :
      ( aa(D,$o,aa(fun(D,fun(B,$o)),fun(D,$o),aTP_Lamp_ase(fun(B,fun(C,$o)),fun(fun(D,fun(B,$o)),fun(D,$o)),Uu),Uua),Uub)
    <=> ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),Uu)))
          & aa(B,$o,aa(D,fun(B,$o),Uua,Uub),X3) ) ) ).

% ATP.lambda_856
tff(fact_9034_ATP_Olambda__857,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(B,D),Uub: fun(B,D)] :
      ( aa(fun(B,D),$o,aa(fun(B,D),fun(fun(B,D),$o),aTP_Lamp_amq(fun(B,fun(C,$o)),fun(fun(B,D),fun(fun(B,D),$o)),Uu),Uua),Uub)
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),Uu)))
         => ( aa(B,D,Uua,X3) = aa(B,D,Uub,X3) ) ) ) ).

% ATP.lambda_857
tff(fact_9035_ATP_Olambda__858,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,option(B)),Uua: list(C),Uub: B] :
      ( aa(B,$o,aa(list(C),fun(B,$o),aTP_Lamp_uz(fun(C,option(B)),fun(list(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),aa(list(C),set(C),set2(C),Uua))
          & ( aa(C,option(B),Uu,X3) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_858
tff(fact_9036_ATP_Olambda__859,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(C,$o)),fun(B,$o),aTP_Lamp_ajv(set(C),fun(fun(B,fun(C,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uu)
         => aa(C,$o,aa(B,fun(C,$o),Uua,Uub),X3) ) ) ).

% ATP.lambda_859
tff(fact_9037_ATP_Olambda__860,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(C,fun(B,$o)),Uub: C] :
      ( aa(C,$o,aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_alp(set(B),fun(fun(C,fun(B,$o)),fun(C,$o)),Uu),Uua),Uub)
    <=> ! [X3: B] :
          ( aa(set(B),$o,member(B,X3),Uu)
         => aa(B,$o,aa(C,fun(B,$o),Uua,Uub),X3) ) ) ).

% ATP.lambda_860
tff(fact_9038_ATP_Olambda__861,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(C,$o)),fun(B,$o),aTP_Lamp_asa(set(C),fun(fun(B,fun(C,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uu)
          & aa(C,$o,aa(B,fun(C,$o),Uua,Uub),X3) ) ) ).

% ATP.lambda_861
tff(fact_9039_ATP_Olambda__862,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(C,fun(B,$o)),Uub: C] :
      ( aa(C,$o,aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_ars(set(B),fun(fun(C,fun(B,$o)),fun(C,$o)),Uu),Uua),Uub)
    <=> ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),Uu)
          & aa(B,$o,aa(C,fun(B,$o),Uua,Uub),X3) ) ) ).

% ATP.lambda_862
tff(fact_9040_ATP_Olambda__863,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,fun(B,$o)),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_art(fun(C,fun(B,$o)),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uua)
          & aa(B,$o,aa(C,fun(B,$o),Uu,X3),Uub) ) ) ).

% ATP.lambda_863
tff(fact_9041_ATP_Olambda__864,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_akg(set(product_prod(B,B)),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X3: product_prod(B,B)] :
          ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),X3),Uu)
         => aa(product_prod(B,B),$o,aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),aa(B,fun(B,fun(B,$o)),aa(B,fun(B,fun(B,fun(B,$o))),aTP_Lamp_akf(set(product_prod(B,B)),fun(B,fun(B,fun(B,fun(B,$o)))),Uu),Uua),Uub)),X3) ) ) ).

% ATP.lambda_864
tff(fact_9042_ATP_Olambda__865,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,D)),Uua: set(D),Uub: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(set(D),fun(product_prod(B,C),$o),aTP_Lamp_um(fun(B,fun(C,D)),fun(set(D),fun(product_prod(B,C),$o)),Uu),Uua),Uub)
    <=> ? [P6: product_prod(B,C)] :
          ( ( Uub = P6 )
          & aa(set(D),$o,member(D,aa(C,D,aa(B,fun(C,D),Uu,aa(product_prod(B,C),B,product_fst(B,C),P6)),aa(product_prod(B,C),C,product_snd(B,C),P6))),Uua) ) ) ).

% ATP.lambda_865
tff(fact_9043_ATP_Olambda__866,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_asd(set(product_prod(C,B)),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uua)
          & aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Uub)),Uu) ) ) ).

% ATP.lambda_866
tff(fact_9044_ATP_Olambda__867,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: set(B),Uub: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_asb(fun(B,C),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),Uua)
          & ( aa(B,C,Uu,X3) = aa(B,C,Uu,Uub) ) ) ) ).

% ATP.lambda_867
tff(fact_9045_ATP_Olambda__868,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,option(B)),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_asf(fun(C,option(B)),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uua)
          & ( aa(C,option(B),Uu,X3) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_868
tff(fact_9046_ATP_Olambda__869,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_ajw(fun(C,set(B)),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uua)
         => aa(set(B),$o,member(B,Uub),aa(C,set(B),Uu,X3)) ) ) ).

% ATP.lambda_869
tff(fact_9047_ATP_Olambda__870,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_arz(fun(C,set(B)),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uua)
          & aa(set(B),$o,member(B,Uub),aa(C,set(B),Uu,X3)) ) ) ).

% ATP.lambda_870
tff(fact_9048_ATP_Olambda__871,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: set(C),Uub: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aTP_Lamp_asc(fun(C,B),fun(set(C),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(set(C),$o,member(C,X3),Uua)
          & ( Uub = aa(C,B,Uu,X3) ) ) ) ).

% ATP.lambda_871
tff(fact_9049_ATP_Olambda__872,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: filter(C),Uub: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(filter(C),fun(fun(B,$o),$o),aTP_Lamp_anc(fun(B,C),fun(filter(C),fun(fun(B,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(C,$o)] :
          ( eventually(C,Q7,Uua)
          & ! [X3: B] :
              ( aa(C,$o,Q7,aa(B,C,Uu,X3))
             => aa(B,$o,Uub,X3) ) ) ) ).

% ATP.lambda_872
tff(fact_9050_ATP_Olambda__873,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(C,fun(B,$o)),Uub: C] :
      ( aa(C,$o,aa(fun(C,fun(B,$o)),fun(C,$o),aTP_Lamp_uh(fun(B,$o),fun(fun(C,fun(B,$o)),fun(C,$o)),Uu),Uua),Uub)
    <=> ? [Y5: B] :
          ( aa(B,$o,Uu,Y5)
          & aa(B,$o,aa(C,fun(B,$o),Uua,Uub),Y5) ) ) ).

% ATP.lambda_873
tff(fact_9051_ATP_Olambda__874,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,set(B)),Uua: fun(C,$o),Uub: B] :
      ( aa(B,$o,aa(fun(C,$o),fun(B,$o),aTP_Lamp_xz(fun(C,set(B)),fun(fun(C,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X3: C] :
          ( aa(C,$o,Uua,X3)
          & aa(set(B),$o,member(B,Uub),aa(C,set(B),Uu,X3)) ) ) ).

% ATP.lambda_874
tff(fact_9052_ATP_Olambda__875,axiom,
    ! [B: $tType,Uu: fun(B,B),Uua: B,Uub: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_afq(fun(B,B),fun(B,fun(B,$o)),Uu),Uua),Uub)
    <=> ? [N4: nat] : Uub = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),N4),Uu),Uua) ) ).

% ATP.lambda_875
tff(fact_9053_ATP_Olambda__876,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: fun(C,fun(C,$o)),Uub: product_prod(B,B)] :
      ( aa(product_prod(B,B),$o,aa(fun(C,fun(C,$o)),fun(product_prod(B,B),$o),aTP_Lamp_ve(fun(C,B),fun(fun(C,fun(C,$o)),fun(product_prod(B,B),$o)),Uu),Uua),Uub)
    <=> ? [A6: C,B9: C] :
          ( ( Uub = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(C,B,Uu,A6)),aa(C,B,Uu,B9)) )
          & aa(C,$o,aa(C,fun(C,$o),Uua,A6),B9) ) ) ).

% ATP.lambda_876
tff(fact_9054_ATP_Olambda__877,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(C,C)),Uua: fun(C,B),Uub: product_prod(B,B)] :
      ( aa(product_prod(B,B),$o,aa(fun(C,B),fun(product_prod(B,B),$o),aTP_Lamp_agy(set(product_prod(C,C)),fun(fun(C,B),fun(product_prod(B,B),$o)),Uu),Uua),Uub)
    <=> ? [A15: C,A26: C] :
          ( ( Uub = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(C,B,Uua,A15)),aa(C,B,Uua,A26)) )
          & aa(set(product_prod(C,C)),$o,member(product_prod(C,C),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),A15),A26)),Uu) ) ) ).

% ATP.lambda_877
tff(fact_9055_ATP_Olambda__878,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: list(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_uw(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: B,V5: list(B)] :
          ( ( 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),A6),V5)) )
          | ? [U6: list(B),Aa3: B,B9: B,Va3: list(B),W3: list(B)] :
              ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Aa3),B9)),Uu)
              & ( Uua = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),U6),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Aa3),Va3)) )
              & ( Uub = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),U6),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B9),W3)) ) ) ) ) ).

% ATP.lambda_878
tff(fact_9056_ATP_Olambda__879,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [Uu: set(B),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_acy(set(B),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ? [A6: B,B9: B] :
              ( ( Uub = aa(B,B,aa(B,fun(B,B),sup_sup(B),A6),B9) )
              & aa(set(B),$o,member(B,A6),Uu)
              & aa(set(B),$o,member(B,B9),Uua) ) ) ) ).

% ATP.lambda_879
tff(fact_9057_ATP_Olambda__880,axiom,
    ! [B: $tType] :
      ( distrib_lattice(B)
     => ! [Uu: set(B),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_acw(set(B),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ? [A6: B,B9: B] :
              ( ( Uub = aa(B,B,aa(B,fun(B,B),inf_inf(B),A6),B9) )
              & aa(set(B),$o,member(B,A6),Uu)
              & aa(set(B),$o,member(B,B9),Uua) ) ) ) ).

% ATP.lambda_880
tff(fact_9058_ATP_Olambda__881,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: set(multiset(B)),Uub: multiset(B)] :
      ( aa(multiset(B),$o,aa(set(multiset(B)),fun(multiset(B),$o),aTP_Lamp_aqo(set(multiset(B)),fun(set(multiset(B)),fun(multiset(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: multiset(B),B9: multiset(B)] :
          ( ( Uub = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A6),B9) )
          & aa(set(multiset(B)),$o,member(multiset(B),A6),Uu)
          & aa(set(multiset(B)),$o,member(multiset(B),B9),Uua) ) ) ).

% ATP.lambda_881
tff(fact_9059_ATP_Olambda__882,axiom,
    ! [B: $tType,Uu: set(multiset(B)),Uua: set(multiset(B)),Uub: multiset(B)] :
      ( aa(multiset(B),$o,aa(set(multiset(B)),fun(multiset(B),$o),aTP_Lamp_aqm(set(multiset(B)),fun(set(multiset(B)),fun(multiset(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: multiset(B),B9: multiset(B)] :
          ( ( Uub = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A6),B9) )
          & aa(set(multiset(B)),$o,member(multiset(B),A6),Uu)
          & aa(set(multiset(B)),$o,member(multiset(B),B9),Uua) ) ) ).

% ATP.lambda_882
tff(fact_9060_ATP_Olambda__883,axiom,
    ! [B: $tType,Uu: set(B),Uua: set(list(B)),Uub: list(B)] :
      ( aa(list(B),$o,aa(set(list(B)),fun(list(B),$o),aTP_Lamp_uo(set(B),fun(set(list(B)),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ? [X3: B,Xs3: list(B)] :
          ( ( Uub = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs3) )
          & aa(set(B),$o,member(B,X3),Uu)
          & aa(set(list(B)),$o,member(list(B),Xs3),Uua) ) ) ).

% ATP.lambda_883
tff(fact_9061_ATP_Olambda__884,axiom,
    ! [B: $tType,Uu: filter(B),Uua: filter(B),Uub: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(filter(B),fun(fun(B,$o),$o),aTP_Lamp_amx(filter(B),fun(filter(B),fun(fun(B,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(B,$o),R8: fun(B,$o)] :
          ( eventually(B,Q7,Uu)
          & eventually(B,R8,Uua)
          & ! [X3: B] :
              ( ( aa(B,$o,Q7,X3)
                & aa(B,$o,R8,X3) )
             => aa(B,$o,Uub,X3) ) ) ) ).

% ATP.lambda_884
tff(fact_9062_ATP_Olambda__885,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: multiset(B),Uub: multiset(B)] :
      ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),aTP_Lamp_aik(set(product_prod(B,B)),fun(multiset(B),fun(multiset(B),$o)),Uu),Uua),Uub)
    <=> ? [A6: B,M03: multiset(B),K8: multiset(B)] :
          ( ( Uub = aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A6),M03) )
          & ( Uua = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M03),K8) )
          & ! [B9: B] :
              ( aa(set(B),$o,member(B,B9),aa(multiset(B),set(B),set_mset(B),K8))
             => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B9),A6)),Uu) ) ) ) ).

% ATP.lambda_885
tff(fact_9063_ATP_Olambda__886,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: list(B),Uub: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aTP_Lamp_uv(set(product_prod(B,B)),fun(list(B),fun(list(B),$o)),Uu),Uua),Uub)
    <=> ? [Us2: list(B),Z5: B,Z9: B,Vs2: list(B)] :
          ( ( Uua = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Vs2)) )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Z5),Z9)),Uu)
          & ( Uub = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z9),Vs2)) ) ) ) ).

% ATP.lambda_886
tff(fact_9064_ATP_Olambda__887,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,Uu: fun(C,fun(D,fun(E,B))),Uua: E,Uub: C,Uuc: D] : aa(D,B,aa(C,fun(D,B),aa(E,fun(C,fun(D,B)),aTP_Lamp_la(fun(C,fun(D,fun(E,B))),fun(E,fun(C,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(E,B,aa(D,fun(E,B),aa(C,fun(D,fun(E,B)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_887
tff(fact_9065_ATP_Olambda__888,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: product_prod(D,B),Uua: B,Uub: C,Uuc: set(product_prod(D,C))] :
      aa(set(product_prod(D,C)),set(product_prod(D,C)),aa(C,fun(set(product_prod(D,C)),set(product_prod(D,C))),aa(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aTP_Lamp_abv(product_prod(D,B),fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(D,B),B,product_snd(D,B),Uu) = Uua,aa(set(product_prod(D,C)),set(product_prod(D,C)),insert(product_prod(D,C),aa(C,product_prod(D,C),aa(D,fun(C,product_prod(D,C)),product_Pair(D,C),aa(product_prod(D,B),D,product_fst(D,B),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_888
tff(fact_9066_ATP_Olambda__889,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: set(B),Uub: B,Uuc: C] :
      aa(C,B,aa(B,fun(C,B),aa(set(B),fun(B,fun(C,B)),aTP_Lamp_afw(fun(B,C),fun(set(B),fun(B,fun(C,B))),Uu),Uua),Uub),Uuc) = $ite(aa(set(C),$o,member(C,Uuc),aa(set(B),set(C),image2(B,C,Uu),Uua)),the_inv_into(B,C,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_889
tff(fact_9067_ATP_Olambda__890,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(B,C),Uub: B,Uuc: C] :
      aa(C,B,aa(B,fun(C,B),aa(fun(B,C),fun(B,fun(C,B)),aTP_Lamp_afu(set(B),fun(fun(B,C),fun(B,fun(C,B))),Uu),Uua),Uub),Uuc) = $ite(aa(set(C),$o,member(C,Uuc),aa(set(B),set(C),image2(B,C,Uua),Uu)),the_inv_into(B,C,Uu,Uua,Uuc),Uub) ).

% ATP.lambda_890
tff(fact_9068_ATP_Olambda__891,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: B,Uub: C,Uuc: set(C)] :
      aa(set(C),set(C),aa(C,fun(set(C),set(C)),aa(B,fun(C,fun(set(C),set(C))),aTP_Lamp_acl(set(B),fun(B,fun(C,fun(set(C),set(C)))),Uu),Uua),Uub),Uuc) = $ite(aa(set(B),$o,member(B,Uua),Uu),aa(set(C),set(C),insert(C,Uub),Uuc),Uuc) ).

% ATP.lambda_891
tff(fact_9069_ATP_Olambda__892,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: nat,Uua: fun(nat,B),Uub: fun(nat,B),Uuc: nat] :
          aa(nat,B,aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_cv(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,B,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(B),aa(nat,B,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_892
tff(fact_9070_ATP_Olambda__893,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: nat,Uua: fun(nat,B),Uub: fun(nat,B),Uuc: nat] :
          aa(nat,B,aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_er(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,B,Uua,Uuc),
            $ite(Uuc = Uu,one_one(B),aa(nat,B,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_893
tff(fact_9071_ATP_Olambda__894,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: nat,Uua: fun(nat,B),Uub: fun(nat,B),Uuc: nat] :
          aa(nat,B,aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_es(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,B,Uua,Uuc),aa(nat,B,Uub,Uuc)) ) ).

% ATP.lambda_894
tff(fact_9072_ATP_Olambda__895,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: nat,Uua: fun(nat,B),Uub: fun(nat,B),Uuc: nat] :
          aa(nat,B,aa(fun(nat,B),fun(nat,B),aa(fun(nat,B),fun(fun(nat,B),fun(nat,B)),aTP_Lamp_cw(nat,fun(fun(nat,B),fun(fun(nat,B),fun(nat,B))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,B,Uua,Uuc),aa(nat,B,Uub,Uuc)) ) ).

% ATP.lambda_895
tff(fact_9073_ATP_Olambda__896,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: set(B),Uub: fun(B,C),Uuc: B] :
      aa(B,C,aa(fun(B,C),fun(B,C),aa(set(B),fun(fun(B,C),fun(B,C)),aTP_Lamp_afl(fun(B,C),fun(set(B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(aa(set(B),$o,member(B,Uuc),Uua),aa(B,C,Uu,Uuc),aa(B,C,Uub,Uuc)) ).

% ATP.lambda_896
tff(fact_9074_ATP_Olambda__897,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: fun(B,C),Uuc: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_hc(B,fun(fun(B,C),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,C,Uua,Uuc),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_897
tff(fact_9075_ATP_Olambda__898,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: fun(B,C),Uuc: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_hb(B,fun(fun(B,C),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,C,Uua,Uuc),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_898
tff(fact_9076_ATP_Olambda__899,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: B,Uua: fun(B,C),Uub: C,Uuc: B] :
          aa(B,C,aa(C,fun(B,C),aa(fun(B,C),fun(C,fun(B,C)),aTP_Lamp_gv(B,fun(fun(B,C),fun(C,fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,C,Uua,Uuc),Uub) ) ).

% ATP.lambda_899
tff(fact_9077_ATP_Olambda__900,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C,Uub: fun(B,option(C)),Uuc: B] :
      aa(B,option(C),aa(fun(B,option(C)),fun(B,option(C)),aa(C,fun(fun(B,option(C)),fun(B,option(C))),aTP_Lamp_anj(B,fun(C,fun(fun(B,option(C)),fun(B,option(C)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(C,option(C),some(C),Uua),aa(B,option(C),Uub,Uuc)) ).

% ATP.lambda_900
tff(fact_9078_ATP_Olambda__901,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,option(B)),Uua: C,Uub: B,Uuc: C] :
      aa(C,option(B),aa(B,fun(C,option(B)),aa(C,fun(B,fun(C,option(B))),aTP_Lamp_jx(fun(C,option(B)),fun(C,fun(B,fun(C,option(B)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uua,aa(B,option(B),some(B),Uub),aa(C,option(B),Uu,Uuc)) ).

% ATP.lambda_901
tff(fact_9079_ATP_Olambda__902,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: C,Uub: C,Uuc: B] :
      aa(B,C,aa(C,fun(B,C),aa(C,fun(C,fun(B,C)),aTP_Lamp_adq(set(B),fun(C,fun(C,fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(aa(set(B),$o,member(B,Uuc),Uu),Uua,Uub) ).

% ATP.lambda_902
tff(fact_9080_ATP_Olambda__903,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,$o),Uua: fun(C,B),Uub: fun(C,B),Uuc: C] :
      aa(C,B,aa(fun(C,B),fun(C,B),aa(fun(C,B),fun(fun(C,B),fun(C,B)),aTP_Lamp_aau(fun(C,$o),fun(fun(C,B),fun(fun(C,B),fun(C,B))),Uu),Uua),Uub),Uuc) = $ite(aa(C,$o,Uu,Uuc),aa(C,B,Uua,Uuc),aa(C,B,Uub,Uuc)) ).

% ATP.lambda_903
tff(fact_9081_ATP_Olambda__904,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,option(C)),Uub: fun(B,option(C)),Uuc: B] :
      aa(B,option(C),aa(fun(B,option(C)),fun(B,option(C)),aa(fun(B,option(C)),fun(fun(B,option(C)),fun(B,option(C))),aTP_Lamp_anm(fun(B,$o),fun(fun(B,option(C)),fun(fun(B,option(C)),fun(B,option(C)))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,option(C),Uua,Uuc),aa(B,option(C),Uub,Uuc)) ).

% ATP.lambda_904
tff(fact_9082_ATP_Olambda__905,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(B,$o),Uua: fun(B,C),Uub: fun(B,C),Uuc: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_kv(fun(B,$o),fun(fun(B,C),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,C,Uua,Uuc),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_905
tff(fact_9083_ATP_Olambda__906,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(B,$o),Uua: fun(B,C),Uub: fun(B,C),Uuc: B] :
          aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(B,C),fun(fun(B,C),fun(B,C)),aTP_Lamp_ku(fun(B,$o),fun(fun(B,C),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,C,Uua,Uuc),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_906
tff(fact_9084_ATP_Olambda__907,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(B,$o),Uub: fun(B,C),Uuc: B] :
      aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(B,$o),fun(fun(B,C),fun(B,C)),aTP_Lamp_ajj(fun(B,C),fun(fun(B,$o),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uua,Uuc),aa(B,C,Uu,Uuc),aa(B,C,Uub,Uuc)) ).

% ATP.lambda_907
tff(fact_9085_ATP_Olambda__908,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: D,Uub: B,Uuc: set(product_prod(D,C))] : aa(set(product_prod(D,C)),set(product_prod(D,C)),aa(B,fun(set(product_prod(D,C)),set(product_prod(D,C))),aa(D,fun(B,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aTP_Lamp_acg(set(product_prod(B,C)),fun(D,fun(B,fun(set(product_prod(D,C)),set(product_prod(D,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(D,C)),aa(fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),fun(product_prod(B,C),fun(set(product_prod(D,C)),set(product_prod(D,C)))),product_case_prod(B,C,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),aTP_Lamp_acf(D,fun(B,fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_908
tff(fact_9086_ATP_Olambda__909,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: set(product_prod(C,D)),Uua: B,Uub: C,Uuc: set(product_prod(B,D))] : aa(set(product_prod(B,D)),set(product_prod(B,D)),aa(C,fun(set(product_prod(B,D)),set(product_prod(B,D))),aa(B,fun(C,fun(set(product_prod(B,D)),set(product_prod(B,D)))),aTP_Lamp_acc(set(product_prod(C,D)),fun(B,fun(C,fun(set(product_prod(B,D)),set(product_prod(B,D))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(C,D),set(product_prod(B,D)),aa(fun(C,fun(D,fun(set(product_prod(B,D)),set(product_prod(B,D))))),fun(product_prod(C,D),fun(set(product_prod(B,D)),set(product_prod(B,D)))),product_case_prod(C,D,fun(set(product_prod(B,D)),set(product_prod(B,D)))),aa(C,fun(C,fun(D,fun(set(product_prod(B,D)),set(product_prod(B,D))))),aTP_Lamp_acb(B,fun(C,fun(C,fun(D,fun(set(product_prod(B,D)),set(product_prod(B,D)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_909
tff(fact_9087_ATP_Olambda__910,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_semiring_0(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: list(C),Uuc: nat] : aa(nat,B,aa(list(C),fun(nat,B),aa(B,fun(list(C),fun(nat,B)),aTP_Lamp_gx(fun(C,B),fun(B,fun(list(C),fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Uuc),aa(B,fun(B,B),times_times(B),Uua)),aa(C,B,Uu,aa(nat,C,nth(C,Uub),Uuc))) ) ).

% ATP.lambda_910
tff(fact_9088_ATP_Olambda__911,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(D,fun(list(D),fun(C,C))),Uua: fun(B,D),Uub: B,Uuc: list(B)] : aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),aa(fun(B,D),fun(B,fun(list(B),fun(C,C))),aTP_Lamp_ajn(fun(D,fun(list(D),fun(C,C))),fun(fun(B,D),fun(B,fun(list(B),fun(C,C)))),Uu),Uua),Uub),Uuc) = aa(list(D),fun(C,C),aa(D,fun(list(D),fun(C,C)),Uu,aa(B,D,Uua,Uub)),aa(list(B),list(D),map(B,D,Uua),Uuc)) ).

% ATP.lambda_911
tff(fact_9089_ATP_Olambda__912,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,fun(B,$o)),Uua: fun(C,B),Uub: C,Uuc: C] :
      ( aa(C,$o,aa(C,fun(C,$o),aa(fun(C,B),fun(C,fun(C,$o)),aTP_Lamp_py(fun(B,fun(B,$o)),fun(fun(C,B),fun(C,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(B,$o,aa(B,fun(B,$o),Uu,aa(C,B,Uua,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_912
tff(fact_9090_ATP_Olambda__913,axiom,
    ! [C: $tType,B: $tType,D: $tType,E: $tType,Uu: fun(C,fun(D,B)),Uua: fun(E,C),Uub: fun(E,D),Uuc: E] : aa(E,B,aa(fun(E,D),fun(E,B),aa(fun(E,C),fun(fun(E,D),fun(E,B)),aTP_Lamp_si(fun(C,fun(D,B)),fun(fun(E,C),fun(fun(E,D),fun(E,B))),Uu),Uua),Uub),Uuc) = aa(D,B,aa(C,fun(D,B),Uu,aa(E,C,Uua,Uuc)),aa(E,D,Uub,Uuc)) ).

% ATP.lambda_913
tff(fact_9091_ATP_Olambda__914,axiom,
    ! [D: $tType,E: $tType,G4: $tType,Uu: fun(D,fun(E,$o)),Uua: fun(G4,E),Uub: D,Uuc: G4] :
      ( aa(G4,$o,aa(D,fun(G4,$o),aa(fun(G4,E),fun(D,fun(G4,$o)),aTP_Lamp_aqz(fun(D,fun(E,$o)),fun(fun(G4,E),fun(D,fun(G4,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(E,$o,aa(D,fun(E,$o),Uu,Uub),aa(G4,E,Uua,Uuc)) ) ).

% ATP.lambda_914
tff(fact_9092_ATP_Olambda__915,axiom,
    ! [C: $tType,D: $tType,E: $tType,Uu: fun(C,fun(D,$o)),Uua: fun(E,D),Uub: C,Uuc: E] :
      ( aa(E,$o,aa(C,fun(E,$o),aa(fun(E,D),fun(C,fun(E,$o)),aTP_Lamp_agf(fun(C,fun(D,$o)),fun(fun(E,D),fun(C,fun(E,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(D,$o,aa(C,fun(D,$o),Uu,Uub),aa(E,D,Uua,Uuc)) ) ).

% ATP.lambda_915
tff(fact_9093_ATP_Olambda__916,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType] :
      ( ( order(D)
        & order(B) )
     => ! [Uu: fun(B,fun(C,D)),Uua: fun(E,C),Uub: B,Uuc: E] : aa(E,D,aa(B,fun(E,D),aa(fun(E,C),fun(B,fun(E,D)),aTP_Lamp_aet(fun(B,fun(C,D)),fun(fun(E,C),fun(B,fun(E,D))),Uu),Uua),Uub),Uuc) = aa(C,D,aa(B,fun(C,D),Uu,Uub),aa(E,C,Uua,Uuc)) ) ).

% ATP.lambda_916
tff(fact_9094_ATP_Olambda__917,axiom,
    ! [B: $tType,C: $tType,F6: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(F6,C),Uub: B,Uuc: F6] :
      ( aa(F6,$o,aa(B,fun(F6,$o),aa(fun(F6,C),fun(B,fun(F6,$o)),aTP_Lamp_aqy(fun(B,fun(C,$o)),fun(fun(F6,C),fun(B,fun(F6,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,Uub),aa(F6,C,Uua,Uuc)) ) ).

% ATP.lambda_917
tff(fact_9095_ATP_Olambda__918,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,C),Uub: B,Uuc: D] :
      ( aa(D,$o,aa(B,fun(D,$o),aa(fun(D,C),fun(B,fun(D,$o)),aTP_Lamp_agu(fun(B,fun(C,$o)),fun(fun(D,C),fun(B,fun(D,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(C,$o,aa(B,fun(C,$o),Uu,Uub),aa(D,C,Uua,Uuc)) ) ).

% ATP.lambda_918
tff(fact_9096_ATP_Olambda__919,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,B)),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,B,aa(B,fun(D,B),aa(fun(D,C),fun(B,fun(D,B)),aTP_Lamp_pz(fun(B,fun(C,B)),fun(fun(D,C),fun(B,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(C,B,aa(B,fun(C,B),Uu,Uub),aa(D,C,Uua,Uuc)) ).

% ATP.lambda_919
tff(fact_9097_ATP_Olambda__920,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_apn($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uua)),Uub),(Uuc)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_apm($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_920
tff(fact_9098_ATP_Olambda__921,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: set(D),Uuc: C] : aa(C,B,aa(set(D),fun(C,B),aa(fun(D,B),fun(set(D),fun(C,B)),aTP_Lamp_bs(fun(C,B),fun(fun(D,B),fun(set(D),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(D),B,aa(fun(D,B),fun(set(D),B),groups7311177749621191930dd_sum(D,B),aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_br(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_921
tff(fact_9099_ATP_Olambda__922,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(C,B),Uub: filter(C),Uuc: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(filter(C),fun(fun(B,$o),$o),aa(fun(C,B),fun(filter(C),fun(fun(B,$o),$o)),aTP_Lamp_amw(set(C),fun(fun(C,B),fun(filter(C),fun(fun(B,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> eventually(C,aa(fun(B,$o),fun(C,$o),aa(fun(C,B),fun(fun(B,$o),fun(C,$o)),aTP_Lamp_amv(set(C),fun(fun(C,B),fun(fun(B,$o),fun(C,$o))),Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_922
tff(fact_9100_ATP_Olambda__923,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),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_oh(fun(B,C),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub),Uuc) = case_list(list(B),B,
            $ite(aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,Uu,Uub)),aa(B,C,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_og(fun(B,C),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua)) ) ).

% ATP.lambda_923
tff(fact_9101_ATP_Olambda__924,axiom,
    ! [D: $tType,B: $tType,C: $tType,F6: $tType,E: $tType,Uu: fun(C,fun(D,fun(E,fun(F6,set(B))))),Uua: product_prod(E,F6),Uub: C,Uuc: D] : aa(D,set(B),aa(C,fun(D,set(B)),aa(product_prod(E,F6),fun(C,fun(D,set(B))),aTP_Lamp_yv(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(product_prod(E,F6),fun(C,fun(D,set(B)))),Uu),Uua),Uub),Uuc) = aa(product_prod(E,F6),set(B),aa(fun(E,fun(F6,set(B))),fun(product_prod(E,F6),set(B)),product_case_prod(E,F6,set(B)),aa(D,fun(E,fun(F6,set(B))),aa(C,fun(D,fun(E,fun(F6,set(B)))),Uu,Uub),Uuc)),Uua) ).

% ATP.lambda_924
tff(fact_9102_ATP_Olambda__925,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: set(B),Uua: fun(B,fun(C,D)),Uub: fun(B,fun(C,$o)),Uuc: C] : aa(C,D,aa(fun(B,fun(C,$o)),fun(C,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D)),aTP_Lamp_fk(set(B),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D))),Uu),Uua),Uub),Uuc) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),aa(C,fun(B,D),aTP_Lamp_fj(fun(B,fun(C,D)),fun(C,fun(B,D)),Uua),Uuc)),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,fun(C,$o)),fun(C,fun(B,$o)),aTP_Lamp_fg(set(B),fun(fun(B,fun(C,$o)),fun(C,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_925
tff(fact_9103_ATP_Olambda__926,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: set(B),Uua: fun(B,fun(C,D)),Uub: fun(B,fun(C,$o)),Uuc: C] : aa(C,D,aa(fun(B,fun(C,$o)),fun(C,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D)),aTP_Lamp_fh(set(B),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(C,D))),Uu),Uua),Uub),Uuc) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),aa(C,fun(B,D),aTP_Lamp_ff(fun(B,fun(C,D)),fun(C,fun(B,D)),Uua),Uuc)),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,fun(C,$o)),fun(C,fun(B,$o)),aTP_Lamp_fg(set(B),fun(fun(B,fun(C,$o)),fun(C,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_926
tff(fact_9104_ATP_Olambda__927,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_apu(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,one2))))))))))))))))))))))))))))))),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,one2))))))))))))))))))))))))))))))),Uua))) ).

% ATP.lambda_927
tff(fact_9105_ATP_Olambda__928,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: set(B),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(set(B),fun(B,fun(B,$o)),aTP_Lamp_aoo(fun(B,fun(B,$o)),fun(set(B),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uub),Uuc)),product_Sigma(B,B,Uua,aTP_Lamp_zv(set(B),fun(B,set(B)),Uua)))
        & aa(B,$o,aa(B,fun(B,$o),Uu,Uub),Uuc) ) ) ).

% ATP.lambda_928
tff(fact_9106_ATP_Olambda__929,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_no(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_929
tff(fact_9107_ATP_Olambda__930,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: B,Uub: array(B),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(B,fun(array(B),fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ala(nat,fun(B,fun(array(B),fun(heap_ext(product_unit),product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = 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)),aa(nat,B,nth(B,array_get(B,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(B,Uub,Uu,Uua,Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_930
tff(fact_9108_ATP_Olambda__931,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: set(B),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(set(B),fun(B,fun(B,$o)),aTP_Lamp_adt(set(product_prod(B,B)),fun(set(B),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uub),Uuc)),Uu)
        & ~ aa(set(B),$o,member(B,Uub),Uua)
        & ~ aa(set(B),$o,member(B,Uuc),Uua) ) ) ).

% ATP.lambda_931
tff(fact_9109_ATP_Olambda__932,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: set(product_prod(B,B)),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aTP_Lamp_aaa(fun(B,nat),fun(set(product_prod(B,B)),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(B,nat,Uu,Uub)),aa(B,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(B,nat,Uu,Uub)),aa(B,nat,Uu,Uuc))
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uub),Uuc)),Uua) ) ) ) ).

% ATP.lambda_932
tff(fact_9110_ATP_Olambda__933,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,option(C)),Uua: fun(product_prod(B,C),$o),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(product_prod(B,C),$o),fun(B,fun(C,$o)),aTP_Lamp_lr(fun(B,option(C)),fun(fun(product_prod(B,C),$o),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(B,option(C),Uu,Uub) = aa(C,option(C),some(C),Uuc) )
        & aa(product_prod(B,C),$o,Uua,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),Uuc)) ) ) ).

% ATP.lambda_933
tff(fact_9111_ATP_Olambda__934,axiom,
    ! [B: $tType,C: $tType] :
      ( order(B)
     => ! [Uu: fun(B,set(C)),Uua: set(C),Uub: set(C),Uuc: B] : aa(B,set(C),aa(set(C),fun(B,set(C)),aa(set(C),fun(set(C),fun(B,set(C))),aTP_Lamp_aej(fun(B,set(C)),fun(set(C),fun(set(C),fun(B,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),aa(B,set(C),Uu,Uuc)),Uua)),Uub) ) ).

% ATP.lambda_934
tff(fact_9112_ATP_Olambda__935,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(set(B),set(B)),Uub: set(B),Uuc: set(B)] : aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),aa(fun(set(B),set(B)),fun(set(B),fun(set(B),set(B))),aTP_Lamp_anu(set(B),fun(fun(set(B),set(B)),fun(set(B),fun(set(B),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(set(B),set(B),Uua,Uuc)),Uub)),Uu) ).

% ATP.lambda_935
tff(fact_9113_ATP_Olambda__936,axiom,
    ! [B: $tType] :
      ( comm_ring_1(B)
     => ! [Uu: B,Uua: B,Uub: nat,Uuc: nat] : aa(nat,B,aa(nat,fun(nat,B),aa(B,fun(nat,fun(nat,B)),aTP_Lamp_ct(B,fun(B,fun(nat,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(B,Uu,Uuc))),comm_s3205402744901411588hammer(B,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_936
tff(fact_9114_ATP_Olambda__937,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_cp(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_937
tff(fact_9115_ATP_Olambda__938,axiom,
    ! [B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: B,Uua: B,Uub: nat,Uuc: nat] : aa(nat,B,aa(nat,fun(nat,B),aa(B,fun(nat,fun(nat,B)),aTP_Lamp_cs(B,fun(B,fun(nat,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uu),Uuc))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_938
tff(fact_9116_ATP_Olambda__939,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: nat,Uub: list(B),Uuc: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(nat,fun(list(B),fun(list(B),$o)),aTP_Lamp_vd(set(product_prod(B,B)),fun(nat,fun(list(B),fun(list(B),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(list(B),nat,size_size(list(B)),Uub) = Uua )
        & ( aa(list(B),nat,size_size(list(B)),Uuc) = Uua )
        & ? [Xys2: list(B),X3: B,Y5: B,Xs6: list(B),Ys6: list(B)] :
            ( ( Uub = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs6)) )
            & ( Uuc = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys6)) )
            & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),X3),Y5)),Uu) ) ) ) ).

% ATP.lambda_939
tff(fact_9117_ATP_Olambda__940,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [Uu: B,Uua: nat,Uub: B,Uuc: nat] : aa(nat,B,aa(B,fun(nat,B),aa(nat,fun(B,fun(nat,B)),aTP_Lamp_ij(B,fun(nat,fun(B,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uu),Uuc)) ) ).

% ATP.lambda_940
tff(fact_9118_ATP_Olambda__941,axiom,
    ! [B: $tType,Uu: $o,Uua: B,Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(B,fun(B,fun(B,$o)),aTP_Lamp_rs($o,fun(B,fun(B,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_941
tff(fact_9119_ATP_Olambda__942,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,B),Uua: set(C),Uub: fun(B,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aa(set(C),fun(fun(B,$o),fun(B,$o)),aTP_Lamp_iq(fun(C,B),fun(set(C),fun(fun(B,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),aa(set(C),set(B),image2(C,B,Uu),Uua))
        & aa(B,$o,Uub,Uuc) ) ) ).

% ATP.lambda_942
tff(fact_9120_ATP_Olambda__943,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: multiset(C),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(multiset(C),fun(B,fun(C,$o)),aTP_Lamp_ahm(fun(C,B),fun(multiset(C),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(C),$o,member(C,Uuc),aa(multiset(C),set(C),set_mset(C),Uua))
        & ( Uub = aa(C,B,Uu,Uuc) ) ) ) ).

% ATP.lambda_943
tff(fact_9121_ATP_Olambda__944,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,$o)),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_fd(set(C),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(C),$o,member(C,Uuc),Uu)
        & aa(C,$o,aa(B,fun(C,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_944
tff(fact_9122_ATP_Olambda__945,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,fun(C,$o)),Uub: C,Uuc: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aa(fun(B,fun(C,$o)),fun(C,fun(B,$o)),aTP_Lamp_fg(set(B),fun(fun(B,fun(C,$o)),fun(C,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & aa(C,$o,aa(B,fun(C,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_945
tff(fact_9123_ATP_Olambda__946,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(B,C),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,C),fun(B,fun(B,$o)),aTP_Lamp_is(set(B),fun(fun(B,C),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & ( aa(B,C,Uua,Uuc) = aa(B,C,Uua,Uub) ) ) ) ).

% ATP.lambda_946
tff(fact_9124_ATP_Olambda__947,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,C),Uua: set(B),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(set(B),fun(B,fun(B,$o)),aTP_Lamp_aqe(fun(B,C),fun(set(B),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uua)
        & ( aa(B,C,Uu,Uub) = aa(B,C,Uu,Uuc) ) ) ) ).

% ATP.lambda_947
tff(fact_9125_ATP_Olambda__948,axiom,
    ! [C: $tType,B: $tType,Uu: set(C),Uua: fun(C,B),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(C,B),fun(B,fun(C,$o)),aTP_Lamp_afv(set(C),fun(fun(C,B),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(C),$o,member(C,Uuc),Uu)
        & ( aa(C,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_948
tff(fact_9126_ATP_Olambda__949,axiom,
    ! [B: $tType,D: $tType,Uu: set(B),Uua: fun(B,D),Uub: D,Uuc: B] :
      ( aa(B,$o,aa(D,fun(B,$o),aa(fun(B,D),fun(D,fun(B,$o)),aTP_Lamp_iv(set(B),fun(fun(B,D),fun(D,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & ( aa(B,D,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_949
tff(fact_9127_ATP_Olambda__950,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aa(fun(B,C),fun(C,fun(B,$o)),aTP_Lamp_iy(set(B),fun(fun(B,C),fun(C,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_950
tff(fact_9128_ATP_Olambda__951,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,C),Uua: set(B),Uub: C,Uuc: B] :
      ( aa(B,$o,aa(C,fun(B,$o),aa(set(B),fun(C,fun(B,$o)),aTP_Lamp_afs(fun(B,C),fun(set(B),fun(C,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uua)
        & ( aa(B,C,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_951
tff(fact_9129_ATP_Olambda__952,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(C,B),Uub: set(C),Uuc: C] :
      ( aa(C,$o,aa(set(C),fun(C,$o),aa(fun(C,B),fun(set(C),fun(C,$o)),aTP_Lamp_afh(set(B),fun(fun(C,B),fun(set(C),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(C),$o,member(C,Uuc),Uub)
        & aa(set(B),$o,member(B,aa(C,B,Uua,Uuc)),Uu) ) ) ).

% ATP.lambda_952
tff(fact_9130_ATP_Olambda__953,axiom,
    ! [B: $tType] :
      ( ( monoid_mult(B)
        & comm_ring(B) )
     => ! [Uu: B,Uua: nat,Uub: B,Uuc: nat] : aa(nat,B,aa(B,fun(nat,B),aa(nat,fun(B,fun(nat,B)),aTP_Lamp_ik(B,fun(nat,fun(B,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uu),Uuc)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_953
tff(fact_9131_ATP_Olambda__954,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_cm(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_954
tff(fact_9132_ATP_Olambda__955,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),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_oe(fun(B,C),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,C,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,C,Uu),Uuc))) ) ).

% ATP.lambda_955
tff(fact_9133_ATP_Olambda__956,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_oo(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_956
tff(fact_9134_ATP_Olambda__957,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_nu(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),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_957
tff(fact_9135_ATP_Olambda__958,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_oq(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_958
tff(fact_9136_ATP_Olambda__959,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_ns(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),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_959
tff(fact_9137_ATP_Olambda__960,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_nw(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_960
tff(fact_9138_ATP_Olambda__961,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_ny(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_961
tff(fact_9139_ATP_Olambda__962,axiom,
    ! [B: $tType,C: $tType,Uu: B,Uua: list(B),Uub: C,Uuc: list(C)] : aa(list(C),list(product_prod(B,C)),aa(C,fun(list(C),list(product_prod(B,C))),aa(list(B),fun(C,fun(list(C),list(product_prod(B,C)))),aTP_Lamp_se(B,fun(list(B),fun(C,fun(list(C),list(product_prod(B,C))))),Uu),Uua),Uub),Uuc) = 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),Uu),Uub)),zip(B,C,Uua,Uuc)) ).

% ATP.lambda_962
tff(fact_9140_ATP_Olambda__963,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: list(C),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(B,C)),aa(B,fun(list(B),list(product_prod(B,C))),aa(list(C),fun(B,fun(list(B),list(product_prod(B,C)))),aTP_Lamp_sf(C,fun(list(C),fun(B,fun(list(B),list(product_prod(B,C))))),Uu),Uua),Uub),Uuc) = 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),Uub),Uu)),zip(B,C,Uuc,Uua)) ).

% ATP.lambda_963
tff(fact_9141_ATP_Olambda__964,axiom,
    ! [B: $tType,C: $tType,Uu: filter(B),Uua: filter(C),Uub: fun(B,$o),Uuc: fun(C,$o)] :
      ( aa(fun(C,$o),$o,aa(fun(B,$o),fun(fun(C,$o),$o),aa(filter(C),fun(fun(B,$o),fun(fun(C,$o),$o)),aTP_Lamp_aox(filter(B),fun(filter(C),fun(fun(B,$o),fun(fun(C,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> ( eventually(B,Uub,Uu)
        & eventually(C,Uuc,Uua) ) ) ).

% ATP.lambda_964
tff(fact_9142_ATP_Olambda__965,axiom,
    ! [B: $tType,C: $tType,Uu: B,Uua: C,Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(C,fun(B,fun(C,$o)),aTP_Lamp_lo(B,fun(C,fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_965
tff(fact_9143_ATP_Olambda__966,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_agi(nat,fun(nat,fun(nat,fun(nat,$o))),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_966
tff(fact_9144_ATP_Olambda__967,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: set(C),Uub: fun(B,$o),Uuc: C] :
      ( aa(C,$o,aa(fun(B,$o),fun(C,$o),aa(set(C),fun(fun(B,$o),fun(C,$o)),aTP_Lamp_ir(fun(C,B),fun(set(C),fun(fun(B,$o),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(C),$o,member(C,Uuc),Uua)
        & aa(B,$o,Uub,aa(C,B,Uu,Uuc)) ) ) ).

% ATP.lambda_967
tff(fact_9145_ATP_Olambda__968,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(D,set(product_prod(B,C))),Uua: D,Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(D,fun(B,fun(C,$o)),aTP_Lamp_xs(fun(D,set(product_prod(B,C))),fun(D,fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),Uuc)),aa(D,set(product_prod(B,C)),Uu,Uua)) ) ).

% ATP.lambda_968
tff(fact_9146_ATP_Olambda__969,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,C),Uuc: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(B,C),fun(fun(B,C),fun(B,$o)),aTP_Lamp_fq(set(B),fun(fun(B,C),fun(fun(B,C),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(B),$o,member(B,Uuc),Uu)
            & ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,Uua,Uuc)),aa(B,C,Uub,Uuc)) != one_one(C) ) ) ) ) ).

% ATP.lambda_969
tff(fact_9147_ATP_Olambda__970,axiom,
    ! [B: $tType,C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,C),Uuc: B] :
          ( aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(B,C),fun(fun(B,C),fun(B,$o)),aTP_Lamp_fo(set(B),fun(fun(B,C),fun(fun(B,C),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(B),$o,member(B,Uuc),Uu)
            & ( aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,Uua,Uuc)),aa(B,C,Uub,Uuc)) != zero_zero(C) ) ) ) ) ).

% ATP.lambda_970
tff(fact_9148_ATP_Olambda__971,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: list(C),Uuc: nat] : aa(nat,B,aa(list(C),fun(nat,B),aa(B,fun(list(C),fun(nat,B)),aTP_Lamp_eq(fun(C,B),fun(B,fun(list(C),fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(C,B,Uu,aa(nat,C,nth(C,Uub),Uuc))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uuc)) ) ).

% ATP.lambda_971
tff(fact_9149_ATP_Olambda__972,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: fun(C,$o),Uub: fun(B,C),Uuc: B] :
      ( aa(B,$o,aa(fun(B,C),fun(B,$o),aa(fun(C,$o),fun(fun(B,C),fun(B,$o)),aTP_Lamp_amm(set(B),fun(fun(C,$o),fun(fun(B,C),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(C,$o,Uua,aa(B,C,Uub,Uuc))
        & aa(set(B),$o,member(B,Uuc),Uu) ) ) ).

% ATP.lambda_972
tff(fact_9150_ATP_Olambda__973,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(C,B),Uub: fun(B,$o),Uuc: C] :
      ( aa(C,$o,aa(fun(B,$o),fun(C,$o),aa(fun(C,B),fun(fun(B,$o),fun(C,$o)),aTP_Lamp_amv(set(C),fun(fun(C,B),fun(fun(B,$o),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uub,aa(C,B,Uua,Uuc))
        & aa(set(C),$o,member(C,Uuc),Uu) ) ) ).

% ATP.lambda_973
tff(fact_9151_ATP_Olambda__974,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,$o),Uua: fun(B,fun(C,$o)),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_wl(fun(B,$o),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(C,$o,aa(B,fun(C,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_974
tff(fact_9152_ATP_Olambda__975,axiom,
    ! [D: $tType,C: $tType,B: $tType,E: $tType,Uu: fun(D,set(B)),Uua: fun(E,set(C)),Uub: D,Uuc: E] : aa(E,set(product_prod(B,C)),aa(D,fun(E,set(product_prod(B,C))),aa(fun(E,set(C)),fun(D,fun(E,set(product_prod(B,C)))),aTP_Lamp_wi(fun(D,set(B)),fun(fun(E,set(C)),fun(D,fun(E,set(product_prod(B,C))))),Uu),Uua),Uub),Uuc) = product_Sigma(B,C,aa(D,set(B),Uu,Uub),aa(E,fun(B,set(C)),aTP_Lamp_wh(fun(E,set(C)),fun(E,fun(B,set(C))),Uua),Uuc)) ).

% ATP.lambda_975
tff(fact_9153_ATP_Olambda__976,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,B),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,B),fun(B,fun(B,$o)),aTP_Lamp_aoj(fun(B,$o),fun(fun(B,B),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uu,Uuc)
        & ( Uub = aa(B,B,Uua,Uuc) ) ) ) ).

% ATP.lambda_976
tff(fact_9154_ATP_Olambda__977,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_semiring_0(B)
     => ! [Uu: fun(C,B),Uua: B,Uub: C,Uuc: B] : aa(B,B,aa(C,fun(B,B),aa(B,fun(C,fun(B,B)),aTP_Lamp_bc(fun(C,B),fun(B,fun(C,fun(B,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(C,B,Uu,Uub)),aa(B,B,aa(B,fun(B,B),times_times(B),Uua),Uuc)) ) ).

% ATP.lambda_977
tff(fact_9155_ATP_Olambda__978,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( semiring_0(B)
        & comm_monoid_add(C)
        & times(C) )
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_agn(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(C,B,Uu,Uub)),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_978
tff(fact_9156_ATP_Olambda__979,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_br(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(C,B,Uu,Uub)),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_979
tff(fact_9157_ATP_Olambda__980,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,nat),Uua: fun(C,nat),Uub: B,Uuc: C] : aa(C,nat,aa(B,fun(C,nat),aa(fun(C,nat),fun(B,fun(C,nat)),aTP_Lamp_aft(fun(B,nat),fun(fun(C,nat),fun(B,fun(C,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(B,nat,Uu,Uub)),aa(C,nat,Uua,Uuc)) ).

% ATP.lambda_980
tff(fact_9158_ATP_Olambda__981,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( semiring_0(D)
     => ! [Uu: fun(B,D),Uua: fun(C,D),Uub: B,Uuc: C] : aa(C,D,aa(B,fun(C,D),aa(fun(C,D),fun(B,fun(C,D)),aTP_Lamp_afo(fun(B,D),fun(fun(C,D),fun(B,fun(C,D))),Uu),Uua),Uub),Uuc) = aa(D,D,aa(D,fun(D,D),times_times(D),aa(B,D,Uu,Uub)),aa(C,D,Uua,Uuc)) ) ).

% ATP.lambda_981
tff(fact_9159_ATP_Olambda__982,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(C,set(B)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(B),aa(C,fun(D,set(B)),aa(fun(D,set(B)),fun(C,fun(D,set(B))),aTP_Lamp_abl(fun(C,set(B)),fun(fun(D,set(B)),fun(C,fun(D,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(C,set(B),Uu,Uub)),aa(D,set(B),Uua,Uuc)) ).

% ATP.lambda_982
tff(fact_9160_ATP_Olambda__983,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_abf(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(C,B,Uu,Uub)),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_983
tff(fact_9161_ATP_Olambda__984,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(C,set(B)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(B),aa(C,fun(D,set(B)),aa(fun(D,set(B)),fun(C,fun(D,set(B))),aTP_Lamp_yi(fun(C,set(B)),fun(fun(D,set(B)),fun(C,fun(D,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(C,set(B),Uu,Uub)),aa(D,set(B),Uua,Uuc)) ).

% ATP.lambda_984
tff(fact_9162_ATP_Olambda__985,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_wn(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(C,B,Uu,Uub)),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_985
tff(fact_9163_ATP_Olambda__986,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,Uu: fun(B,filter(D)),Uua: fun(C,filter(E)),Uub: B,Uuc: C] : aa(C,filter(product_prod(D,E)),aa(B,fun(C,filter(product_prod(D,E))),aa(fun(C,filter(E)),fun(B,fun(C,filter(product_prod(D,E)))),aTP_Lamp_apb(fun(B,filter(D)),fun(fun(C,filter(E)),fun(B,fun(C,filter(product_prod(D,E))))),Uu),Uua),Uub),Uuc) = prod_filter(D,E,aa(B,filter(D),Uu,Uub),aa(C,filter(E),Uua,Uuc)) ).

% ATP.lambda_986
tff(fact_9164_ATP_Olambda__987,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,Uu: fun(D,B),Uua: fun(E,C),Uub: D,Uuc: E] : aa(E,product_prod(B,C),aa(D,fun(E,product_prod(B,C)),aa(fun(E,C),fun(D,fun(E,product_prod(B,C))),aTP_Lamp_sn(fun(D,B),fun(fun(E,C),fun(D,fun(E,product_prod(B,C)))),Uu),Uua),Uub),Uuc) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uu,Uub)),aa(E,C,Uua,Uuc)) ).

% ATP.lambda_987
tff(fact_9165_ATP_Olambda__988,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,Uu: fun(B,D),Uua: fun(C,E),Uub: B,Uuc: C] : aa(C,product_prod(D,E),aa(B,fun(C,product_prod(D,E)),aa(fun(C,E),fun(B,fun(C,product_prod(D,E))),aTP_Lamp_aaq(fun(B,D),fun(fun(C,E),fun(B,fun(C,product_prod(D,E)))),Uu),Uua),Uub),Uuc) = aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),aa(B,D,Uu,Uub)),aa(C,E,Uua,Uuc)) ).

% ATP.lambda_988
tff(fact_9166_ATP_Olambda__989,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,$o),Uua: fun(C,$o),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(C,$o),fun(B,fun(C,$o)),aTP_Lamp_vq(fun(B,$o),fun(fun(C,$o),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(C,$o,Uua,Uuc) ) ) ).

% ATP.lambda_989
tff(fact_9167_ATP_Olambda__990,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(B)
     => ! [Uu: fun(C,B),Uua: fun(list(C),B),Uub: list(C),Uuc: C] :
          ( aa(C,$o,aa(list(C),fun(C,$o),aa(fun(list(C),B),fun(list(C),fun(C,$o)),aTP_Lamp_qc(fun(C,B),fun(fun(list(C),B),fun(list(C),fun(C,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(C,B,Uu,Uuc) = aa(list(C),B,Uua,Uub) ) ) ) ).

% ATP.lambda_990
tff(fact_9168_ATP_Olambda__991,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: set(C),Uua: fun(B,fun(C,D)),Uub: fun(B,fun(C,$o)),Uuc: B] : aa(B,D,aa(fun(B,fun(C,$o)),fun(B,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D)),aTP_Lamp_fi(set(C),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D))),Uu),Uua),Uub),Uuc) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7121269368397514597t_prod(C,D),aa(B,fun(C,D),Uua,Uuc)),aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_fd(set(C),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_991
tff(fact_9169_ATP_Olambda__992,axiom,
    ! [D: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: set(C),Uua: fun(B,fun(C,D)),Uub: fun(B,fun(C,$o)),Uuc: B] : aa(B,D,aa(fun(B,fun(C,$o)),fun(B,D),aa(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D)),aTP_Lamp_fe(set(C),fun(fun(B,fun(C,D)),fun(fun(B,fun(C,$o)),fun(B,D))),Uu),Uua),Uub),Uuc) = aa(set(C),D,aa(fun(C,D),fun(set(C),D),groups7311177749621191930dd_sum(C,D),aa(B,fun(C,D),Uua,Uuc)),aa(fun(C,$o),set(C),collect(C),aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_fd(set(C),fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_992
tff(fact_9170_ATP_Olambda__993,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),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_og(fun(B,C),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_of(fun(B,C),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu)),linorder_part(B,C,Uu,aa(B,C,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,one2))))),Uua)) ) ).

% ATP.lambda_993
tff(fact_9171_ATP_Olambda__994,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: set(C),Uub: set(C),Uuc: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aa(set(C),fun(set(C),fun(B,$o)),aTP_Lamp_amk(fun(B,set(C)),fun(set(C),fun(set(C),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(C),$o,finite_finite2(C),aa(B,set(C),Uu,Uuc))
        & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),Uub),aa(B,set(C),Uu,Uuc))
        & aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(B,set(C),Uu,Uuc)),Uua) ) ) ).

% ATP.lambda_994
tff(fact_9172_ATP_Olambda__995,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( semiring_1(D)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(C,D),Uuc: C] : aa(C,D,aa(fun(C,D),fun(C,D),aa(fun(B,C),fun(fun(C,D),fun(C,D)),aTP_Lamp_ji(set(B),fun(fun(B,C),fun(fun(C,D),fun(C,D))),Uu),Uua),Uub),Uuc) = aa(D,D,aa(D,fun(D,D),times_times(D),aa(nat,D,semiring_1_of_nat(D),aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,C),fun(C,fun(B,$o)),aTP_Lamp_iy(set(B),fun(fun(B,C),fun(C,fun(B,$o))),Uu),Uua),Uuc))))),aa(C,D,Uub,Uuc)) ) ).

% ATP.lambda_995
tff(fact_9173_ATP_Olambda__996,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: fun(B,fun(B,$o)),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aTP_Lamp_ael(fun(B,fun(B,$o)),fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [A6: B] :
            ( ( Uub = A6 )
            & ( Uuc = A6 ) )
        | ? [A6: B,B9: B,C5: B] :
            ( ( Uub = A6 )
            & ( Uuc = C5 )
            & aa(B,$o,aa(B,fun(B,$o),Uua,A6),B9)
            & aa(B,$o,aa(B,fun(B,$o),Uu,B9),C5) ) ) ) ).

% ATP.lambda_996
tff(fact_9174_ATP_Olambda__997,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: fun(B,fun(B,$o)),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(B,$o)),fun(B,fun(B,$o)),aTP_Lamp_aek(fun(B,fun(B,$o)),fun(fun(B,fun(B,$o)),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [A6: B,B9: B] :
            ( ( Uub = A6 )
            & ( Uuc = B9 )
            & aa(B,$o,aa(B,fun(B,$o),Uu,A6),B9) )
        | ? [A6: B,B9: B,C5: B] :
            ( ( Uub = A6 )
            & ( Uuc = C5 )
            & aa(B,$o,aa(B,fun(B,$o),Uua,A6),B9)
            & aa(B,$o,aa(B,fun(B,$o),Uu,B9),C5) ) ) ) ).

% ATP.lambda_997
tff(fact_9175_ATP_Olambda__998,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: fun(list(B),fun(list(B),$o)),Uub: list(B),Uuc: list(B)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o)),aTP_Lamp_aen(fun(B,fun(B,$o)),fun(fun(list(B),fun(list(B),$o)),fun(list(B),fun(list(B),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [Y5: B,Ys4: list(B)] :
            ( ( Uub = nil(B) )
            & ( Uuc = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) ) )
        | ? [X3: B,Y5: B,Xs3: list(B),Ys4: list(B)] :
            ( ( Uub = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs3) )
            & ( Uuc = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
            & aa(B,$o,aa(B,fun(B,$o),Uu,X3),Y5) )
        | ? [X3: B,Y5: B,Xs3: list(B),Ys4: list(B)] :
            ( ( Uub = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),Xs3) )
            & ( Uuc = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
            & ~ aa(B,$o,aa(B,fun(B,$o),Uu,X3),Y5)
            & ~ aa(B,$o,aa(B,fun(B,$o),Uu,Y5),X3)
            & aa(list(B),$o,aa(list(B),fun(list(B),$o),Uua,Xs3),Ys4) ) ) ) ).

% ATP.lambda_998
tff(fact_9176_ATP_Olambda__999,axiom,
    ! [B: $tType,Uu: B,Uua: list(B),Uub: B,Uuc: nat] : aa(nat,list(B),aa(B,fun(nat,list(B)),aa(list(B),fun(B,fun(nat,list(B))),aTP_Lamp_hq(B,fun(list(B),fun(B,fun(nat,list(B)))),Uu),Uua),Uub),Uuc) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),list_update(B,Uua,Uuc,Uub)) ).

% ATP.lambda_999
tff(fact_9177_ATP_Olambda__1000,axiom,
    ! [B: $tType,C: $tType,Uu: $o,Uua: fun(B,fun(C,$o)),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(fun(B,fun(C,$o)),fun(B,fun(C,$o)),aTP_Lamp_lt($o,fun(fun(B,fun(C,$o)),fun(B,fun(C,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(C,$o,aa(B,fun(C,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1000
tff(fact_9178_ATP_Olambda__1001,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: fun(B,B),Uub: array(B),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(fun(B,B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_alb(nat,fun(fun(B,B),fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_Pair(array(B),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(B,Uub,Uu,aa(B,B,Uua,aa(nat,B,nth(B,array_get(B,Uuc,Uub)),Uu)),Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_1001
tff(fact_9179_ATP_Olambda__1002,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: nat,Uua: B,Uub: array(B),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),aa(B,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_alc(nat,fun(B,fun(array(B),fun(heap_ext(product_unit),product_prod(array(B),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat)),aa(array(B),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(B),product_prod(heap_ext(product_unit),nat))),product_Pair(array(B),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(B,Uub,Uu,Uua,Uuc)),one_one(nat))) ) ).

% ATP.lambda_1002
tff(fact_9180_ATP_Olambda__1003,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: fun(D,C),Uub: set(D),Uuc: B] : aa(B,set(C),aa(set(D),fun(B,set(C)),aa(fun(D,C),fun(set(D),fun(B,set(C))),aTP_Lamp_ado(fun(D,B),fun(fun(D,C),fun(set(D),fun(B,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(D),set(C),image2(D,C,Uua),aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),inf_inf(set(D)),aa(set(B),set(D),aa(fun(D,B),fun(set(B),set(D)),vimage(D,B),Uu),aa(set(B),set(B),insert(B,Uuc),bot_bot(set(B))))),Uub)) ).

% ATP.lambda_1003
tff(fact_9181_ATP_Olambda__1004,axiom,
    ! [C: $tType,B: $tType,Uu: fun(C,heap_Time_Heap(B)),Uua: C,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(C,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_na(fun(C,heap_Time_Heap(B)),fun(C,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(C,heap_Time_Heap(B),Uu,Uua)),Uub)) ).

% ATP.lambda_1004
tff(fact_9182_ATP_Olambda__1005,axiom,
    ! [B: $tType,Uu: list(B),Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),option($o),aa(B,fun(list(B),option($o)),aa(list(B),fun(B,fun(list(B),option($o))),aTP_Lamp_mw(list(B),fun(list(B),fun(B,fun(list(B),option($o)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(B,Uu,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uua),Uuc)) ).

% ATP.lambda_1005
tff(fact_9183_ATP_Olambda__1006,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,D),Uuc: D] : aa(D,C,aa(fun(B,D),fun(D,C),aa(fun(B,C),fun(fun(B,D),fun(D,C)),aTP_Lamp_ix(set(B),fun(fun(B,C),fun(fun(B,D),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(fun(B,$o),set(B),collect(B),aa(D,fun(B,$o),aa(fun(B,D),fun(D,fun(B,$o)),aTP_Lamp_iv(set(B),fun(fun(B,D),fun(D,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1006
tff(fact_9184_ATP_Olambda__1007,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(D)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,D),Uuc: C] : aa(C,D,aa(fun(B,D),fun(C,D),aa(fun(B,C),fun(fun(B,D),fun(C,D)),aTP_Lamp_ja(set(B),fun(fun(B,C),fun(fun(B,D),fun(C,D))),Uu),Uua),Uub),Uuc) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7121269368397514597t_prod(B,D),Uub),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,C),fun(C,fun(B,$o)),aTP_Lamp_iy(set(B),fun(fun(B,C),fun(C,fun(B,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1007
tff(fact_9185_ATP_Olambda__1008,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,D),Uuc: D] : aa(D,C,aa(fun(B,D),fun(D,C),aa(fun(B,C),fun(fun(B,D),fun(D,C)),aTP_Lamp_iw(set(B),fun(fun(B,C),fun(fun(B,D),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(fun(B,$o),set(B),collect(B),aa(D,fun(B,$o),aa(fun(B,D),fun(D,fun(B,$o)),aTP_Lamp_iv(set(B),fun(fun(B,D),fun(D,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1008
tff(fact_9186_ATP_Olambda__1009,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(D)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,D),Uuc: C] : aa(C,D,aa(fun(B,D),fun(C,D),aa(fun(B,C),fun(fun(B,D),fun(C,D)),aTP_Lamp_iz(set(B),fun(fun(B,C),fun(fun(B,D),fun(C,D))),Uu),Uua),Uub),Uuc) = aa(set(B),D,aa(fun(B,D),fun(set(B),D),groups7311177749621191930dd_sum(B,D),Uub),aa(fun(B,$o),set(B),collect(B),aa(C,fun(B,$o),aa(fun(B,C),fun(C,fun(B,$o)),aTP_Lamp_iy(set(B),fun(fun(B,C),fun(C,fun(B,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1009
tff(fact_9187_ATP_Olambda__1010,axiom,
    ! [B: $tType] :
      ( ord(B)
     => ! [Uu: B,Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),B,aa(B,fun(list(B),B),aa(list(B),fun(B,fun(list(B),B)),aTP_Lamp_sw(B,fun(list(B),fun(B,fun(list(B),B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),ord_min(B),Uu),min_list(B,Uua)) ) ).

% ATP.lambda_1010
tff(fact_9188_ATP_Olambda__1011,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: fun(C,B),Uua: fun(D,fun(list(D),C)),Uub: D,Uuc: list(D)] : aa(list(D),B,aa(D,fun(list(D),B),aa(fun(D,fun(list(D),C)),fun(D,fun(list(D),B)),aTP_Lamp_oj(fun(C,B),fun(fun(D,fun(list(D),C)),fun(D,fun(list(D),B))),Uu),Uua),Uub),Uuc) = aa(C,B,Uu,aa(list(D),C,aa(D,fun(list(D),C),Uua,Uub),Uuc)) ).

% ATP.lambda_1011
tff(fact_9189_ATP_Olambda__1012,axiom,
    ! [B: $tType,D: $tType,C: $tType,E: $tType,Uu: fun(C,B),Uua: fun(D,fun(E,C)),Uub: D,Uuc: E] : aa(E,B,aa(D,fun(E,B),aa(fun(D,fun(E,C)),fun(D,fun(E,B)),aTP_Lamp_lu(fun(C,B),fun(fun(D,fun(E,C)),fun(D,fun(E,B))),Uu),Uua),Uub),Uuc) = aa(C,B,Uu,aa(E,C,aa(D,fun(E,C),Uua,Uub),Uuc)) ).

% ATP.lambda_1012
tff(fact_9190_ATP_Olambda__1013,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,B,aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_dy(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(nat,B,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_1013
tff(fact_9191_ATP_Olambda__1014,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,B,aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_ei(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(nat,B,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_1014
tff(fact_9192_ATP_Olambda__1015,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,B,aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_ec(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(nat,B,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_1015
tff(fact_9193_ATP_Olambda__1016,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(nat,B),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,B,aa(nat,fun(nat,B),aa(nat,fun(nat,fun(nat,B)),aTP_Lamp_cd(fun(nat,B),fun(nat,fun(nat,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(nat,B,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_1016
tff(fact_9194_ATP_Olambda__1017,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: nat,Uub: list(B),Uuc: nat] :
      ( aa(nat,$o,aa(list(B),fun(nat,$o),aa(nat,fun(list(B),fun(nat,$o)),aTP_Lamp_qr(fun(B,$o),fun(nat,fun(list(B),fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(B,$o,Uu,aa(nat,B,nth(B,take(B,Uua,Uub)),Uuc)) ) ).

% ATP.lambda_1017
tff(fact_9195_ATP_Olambda__1018,axiom,
    ! [B: $tType,E: $tType,C: $tType,D: $tType,Uu: fun(product_prod(C,D),B),Uua: fun(E,C),Uub: E,Uuc: D] : aa(D,B,aa(E,fun(D,B),aa(fun(E,C),fun(E,fun(D,B)),aTP_Lamp_sm(fun(product_prod(C,D),B),fun(fun(E,C),fun(E,fun(D,B))),Uu),Uua),Uub),Uuc) = aa(product_prod(C,D),B,Uu,aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(E,C,Uua,Uub)),Uuc)) ).

% ATP.lambda_1018
tff(fact_9196_ATP_Olambda__1019,axiom,
    ! [B: $tType,C: $tType,D: $tType,E: $tType,Uu: fun(product_prod(C,D),B),Uua: fun(E,D),Uub: C,Uuc: E] : aa(E,B,aa(C,fun(E,B),aa(fun(E,D),fun(C,fun(E,B)),aTP_Lamp_sl(fun(product_prod(C,D),B),fun(fun(E,D),fun(C,fun(E,B))),Uu),Uua),Uub),Uuc) = aa(product_prod(C,D),B,Uu,aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),Uub),aa(E,D,Uua,Uuc))) ).

% ATP.lambda_1019
tff(fact_9197_ATP_Olambda__1020,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,Uu: fun(B,fun(D,$o)),Uua: fun(C,fun(E,$o)),Uub: B,Uuc: C] : aa(C,fun(product_prod(D,E),$o),aa(B,fun(C,fun(product_prod(D,E),$o)),aa(fun(C,fun(E,$o)),fun(B,fun(C,fun(product_prod(D,E),$o))),aTP_Lamp_arb(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(B,fun(C,fun(product_prod(D,E),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(D,fun(E,$o)),fun(product_prod(D,E),$o),product_case_prod(D,E,$o),aa(C,fun(D,fun(E,$o)),aa(B,fun(C,fun(D,fun(E,$o))),aa(fun(C,fun(E,$o)),fun(B,fun(C,fun(D,fun(E,$o)))),aTP_Lamp_ara(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(B,fun(C,fun(D,fun(E,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1020
tff(fact_9198_ATP_Olambda__1021,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,$o),Uua: fun(B,set(product_prod(C,C))),Uub: B,Uuc: C] : aa(C,fun(product_prod(B,C),$o),aa(B,fun(C,fun(product_prod(B,C),$o)),aa(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(product_prod(B,C),$o))),aTP_Lamp_acp(fun(B,$o),fun(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(product_prod(B,C),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(C,fun(B,fun(C,$o)),aa(B,fun(C,fun(B,fun(C,$o))),aa(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(B,fun(C,$o)))),aTP_Lamp_aco(fun(B,$o),fun(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(B,fun(C,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1021
tff(fact_9199_ATP_Olambda__1022,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(B,B)),Uua: set(product_prod(C,C)),Uub: B,Uuc: C] : aa(C,fun(product_prod(B,C),$o),aa(B,fun(C,fun(product_prod(B,C),$o)),aa(set(product_prod(C,C)),fun(B,fun(C,fun(product_prod(B,C),$o))),aTP_Lamp_aag(set(product_prod(B,B)),fun(set(product_prod(C,C)),fun(B,fun(C,fun(product_prod(B,C),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),aa(C,fun(B,fun(C,$o)),aa(B,fun(C,fun(B,fun(C,$o))),aa(set(product_prod(C,C)),fun(B,fun(C,fun(B,fun(C,$o)))),aTP_Lamp_aaf(set(product_prod(B,B)),fun(set(product_prod(C,C)),fun(B,fun(C,fun(B,fun(C,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1022
tff(fact_9200_ATP_Olambda__1023,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(C)
     => ! [Uu: fun(B,C),Uua: C,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(C,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_mp(fun(B,C),fun(C,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(C,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_mo(fun(B,C),fun(C,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_1023
tff(fact_9201_ATP_Olambda__1024,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( ( semiring_0(B)
        & comm_monoid_add(C)
        & times(C) )
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: multiset(D),Uuc: C] : aa(C,B,aa(multiset(D),fun(C,B),aa(fun(D,B),fun(multiset(D),fun(C,B)),aTP_Lamp_ago(fun(C,B),fun(fun(D,B),fun(multiset(D),fun(C,B))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(B,aa(multiset(D),multiset(B),image_mset(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_agn(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1024
tff(fact_9202_ATP_Olambda__1025,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(C,set(B)),Uua: fun(D,set(B)),Uub: set(D),Uuc: C] : aa(C,set(B),aa(set(D),fun(C,set(B)),aa(fun(D,set(B)),fun(set(D),fun(C,set(B))),aTP_Lamp_yj(fun(C,set(B)),fun(fun(D,set(B)),fun(set(D),fun(C,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(C,fun(D,set(B)),aa(fun(D,set(B)),fun(C,fun(D,set(B))),aTP_Lamp_yi(fun(C,set(B)),fun(fun(D,set(B)),fun(C,fun(D,set(B)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1025
tff(fact_9203_ATP_Olambda__1026,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: set(D),Uuc: C] : aa(C,B,aa(set(D),fun(C,B),aa(fun(D,B),fun(set(D),fun(C,B)),aTP_Lamp_wo(fun(C,B),fun(fun(D,B),fun(set(D),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_wn(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1026
tff(fact_9204_ATP_Olambda__1027,axiom,
    ! [B: $tType,D: $tType,E: $tType,C: $tType,Uu: set(C),Uua: fun(B,filter(D)),Uub: fun(C,filter(E)),Uuc: B] : aa(B,filter(product_prod(D,E)),aa(fun(C,filter(E)),fun(B,filter(product_prod(D,E))),aa(fun(B,filter(D)),fun(fun(C,filter(E)),fun(B,filter(product_prod(D,E)))),aTP_Lamp_apc(set(C),fun(fun(B,filter(D)),fun(fun(C,filter(E)),fun(B,filter(product_prod(D,E))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(D,E))),filter(product_prod(D,E)),complete_Inf_Inf(filter(product_prod(D,E))),aa(set(C),set(filter(product_prod(D,E))),image2(C,filter(product_prod(D,E)),aa(B,fun(C,filter(product_prod(D,E))),aa(fun(C,filter(E)),fun(B,fun(C,filter(product_prod(D,E)))),aTP_Lamp_apb(fun(B,filter(D)),fun(fun(C,filter(E)),fun(B,fun(C,filter(product_prod(D,E))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_1027
tff(fact_9205_ATP_Olambda__1028,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(C,set(B)),Uua: fun(D,set(B)),Uub: set(D),Uuc: C] : aa(C,set(B),aa(set(D),fun(C,set(B)),aa(fun(D,set(B)),fun(set(D),fun(C,set(B))),aTP_Lamp_abm(fun(C,set(B)),fun(fun(D,set(B)),fun(set(D),fun(C,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(D),set(set(B)),image2(D,set(B),aa(C,fun(D,set(B)),aa(fun(D,set(B)),fun(C,fun(D,set(B))),aTP_Lamp_abl(fun(C,set(B)),fun(fun(D,set(B)),fun(C,fun(D,set(B)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1028
tff(fact_9206_ATP_Olambda__1029,axiom,
    ! [C: $tType,B: $tType,D: $tType] :
      ( comple592849572758109894attice(B)
     => ! [Uu: fun(C,B),Uua: fun(D,B),Uub: set(D),Uuc: C] : aa(C,B,aa(set(D),fun(C,B),aa(fun(D,B),fun(set(D),fun(C,B)),aTP_Lamp_abg(fun(C,B),fun(fun(D,B),fun(set(D),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,aa(C,fun(D,B),aa(fun(D,B),fun(C,fun(D,B)),aTP_Lamp_abf(fun(C,B),fun(fun(D,B),fun(C,fun(D,B))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1029
tff(fact_9207_ATP_Olambda__1030,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_1030
tff(fact_9208_ATP_Olambda__1031,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_oc(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_1031
tff(fact_9209_ATP_Olambda__1032,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),option(product_prod(list(B),product_prod(B,list(B)))),aa(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B))))),aa(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B)))))),aTP_Lamp_rl(fun(B,$o),fun(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(B),product_prod(B,list(B))),option(product_prod(list(B),product_prod(B,list(B)))),some(product_prod(list(B),product_prod(B,list(B)))),aa(product_prod(B,list(B)),product_prod(list(B),product_prod(B,list(B))),aa(list(B),fun(product_prod(B,list(B)),product_prod(list(B),product_prod(B,list(B)))),product_Pair(list(B),product_prod(B,list(B))),takeWhile(B,aa(fun(B,$o),fun(B,$o),aa(fun($o,$o),fun(fun(B,$o),fun(B,$o)),comp($o,$o,B),fNot),Uu),Uua)),aa(list(B),product_prod(B,list(B)),aa(B,fun(list(B),product_prod(B,list(B))),product_Pair(B,list(B)),Uub),Uuc))) ).

% ATP.lambda_1032
tff(fact_9210_ATP_Olambda__1033,axiom,
    ! [B: $tType,Uu: B,Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),option(product_prod(list(B),product_prod(B,list(B)))),aa(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B))))),aa(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B)))))),aTP_Lamp_mq(B,fun(list(B),fun(B,fun(list(B),option(product_prod(list(B),product_prod(B,list(B))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(B),product_prod(B,list(B))),option(product_prod(list(B),product_prod(B,list(B)))),some(product_prod(list(B),product_prod(B,list(B)))),aa(product_prod(B,list(B)),product_prod(list(B),product_prod(B,list(B))),aa(list(B),fun(product_prod(B,list(B)),product_prod(list(B),product_prod(B,list(B)))),product_Pair(list(B),product_prod(B,list(B))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),Uua)),aa(list(B),product_prod(B,list(B)),aa(B,fun(list(B),product_prod(B,list(B))),product_Pair(B,list(B)),Uub),Uuc))) ).

% ATP.lambda_1033
tff(fact_9211_ATP_Olambda__1034,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_ou(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_1034
tff(fact_9212_ATP_Olambda__1035,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_os(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_1035
tff(fact_9213_ATP_Olambda__1036,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: set(product_prod(C,C)),Uua: fun(B,fun(C,set(D))),Uub: C,Uuc: B] : aa(B,set(D),aa(C,fun(B,set(D)),aa(fun(B,fun(C,set(D))),fun(C,fun(B,set(D))),aTP_Lamp_agp(set(product_prod(C,C)),fun(fun(B,fun(C,set(D))),fun(C,fun(B,set(D)))),Uu),Uua),Uub),Uuc) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(B,fun(C,set(D)),Uua,Uuc)),aa(set(C),set(C),image(C,C,Uu),aa(set(C),set(C),insert(C,Uub),bot_bot(set(C)))))) ).

% ATP.lambda_1036
tff(fact_9214_ATP_Olambda__1037,axiom,
    ! [D: $tType,B: $tType,C: $tType,F6: $tType,E: $tType,Uu: fun(C,fun(D,fun(E,fun(F6,set(B))))),Uua: set(product_prod(E,F6)),Uub: C,Uuc: D] : aa(D,set(B),aa(C,fun(D,set(B)),aa(set(product_prod(E,F6)),fun(C,fun(D,set(B))),aTP_Lamp_yu(fun(C,fun(D,fun(E,fun(F6,set(B))))),fun(set(product_prod(E,F6)),fun(C,fun(D,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(product_prod(E,F6)),set(set(B)),image2(product_prod(E,F6),set(B),aa(fun(E,fun(F6,set(B))),fun(product_prod(E,F6),set(B)),product_case_prod(E,F6,set(B)),aa(D,fun(E,fun(F6,set(B))),aa(C,fun(D,fun(E,fun(F6,set(B)))),Uu,Uub),Uuc))),Uua)) ).

% ATP.lambda_1037
tff(fact_9215_ATP_Olambda__1038,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: C,Uuc: fun(B,C)] :
      ( aa(fun(B,C),$o,aa(C,fun(fun(B,C),$o),aa(set(C),fun(C,fun(fun(B,C),$o)),aTP_Lamp_ail(set(B),fun(set(C),fun(C,fun(fun(B,C),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X3: B] :
          ( ( aa(set(B),$o,member(B,X3),Uu)
           => aa(set(C),$o,member(C,aa(B,C,Uuc,X3)),Uua) )
          & ( ~ aa(set(B),$o,member(B,X3),Uu)
           => ( aa(B,C,Uuc,X3) = Uub ) ) ) ) ).

% ATP.lambda_1038
tff(fact_9216_ATP_Olambda__1039,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: set(product_prod(B,D)),Uua: set(product_prod(D,C)),Uub: B,Uuc: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(set(product_prod(D,C)),fun(B,fun(C,$o)),aTP_Lamp_uy(set(product_prod(B,D)),fun(set(product_prod(D,C)),fun(B,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y5: D] :
          ( aa(set(product_prod(B,D)),$o,member(product_prod(B,D),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),Uub),Y5)),Uu)
          & aa(set(product_prod(D,C)),$o,member(product_prod(D,C),aa(C,product_prod(D,C),aa(D,fun(C,product_prod(D,C)),product_Pair(D,C),Y5),Uuc)),Uua) ) ) ).

% ATP.lambda_1039
tff(fact_9217_ATP_Olambda__1040,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: set(D),Uua: fun(D,B),Uub: fun(D,C),Uuc: product_prod(B,C)] :
      ( aa(product_prod(B,C),$o,aa(fun(D,C),fun(product_prod(B,C),$o),aa(fun(D,B),fun(fun(D,C),fun(product_prod(B,C),$o)),aTP_Lamp_vg(set(D),fun(fun(D,B),fun(fun(D,C),fun(product_prod(B,C),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A6: D] :
          ( ( Uuc = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,A6)),aa(D,C,Uub,A6)) )
          & aa(set(D),$o,member(D,A6),Uu) ) ) ).

% ATP.lambda_1040
tff(fact_9218_ATP_Olambda__1041,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,B),Uua: C,Uub: set(C),Uuc: B] :
      ( aa(B,$o,aa(set(C),fun(B,$o),aa(C,fun(set(C),fun(B,$o)),aTP_Lamp_uj(fun(C,B),fun(C,fun(set(C),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [L5: C] :
          ( ( Uuc = aa(C,B,Uu,L5) )
          & ( ( L5 = Uua )
            | aa(set(C),$o,member(C,L5),Uub) ) ) ) ).

% ATP.lambda_1041
tff(fact_9219_ATP_Olambda__1042,axiom,
    ! [C: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: B,Uub: fun(B,C),Uuc: C] :
          ( aa(C,$o,aa(fun(B,C),fun(C,$o),aa(B,fun(fun(B,C),fun(C,$o)),aTP_Lamp_abn(B,fun(B,fun(fun(B,C),fun(C,$o))),Uu),Uua),Uub),Uuc)
        <=> ? [I3: B] :
              ( ( Uuc = aa(B,C,Uub,I3) )
              & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),I3)
              & aa(B,$o,aa(B,fun(B,$o),ord_less(B),I3),Uua) ) ) ) ).

% ATP.lambda_1042
tff(fact_9220_ATP_Olambda__1043,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: set(product_prod(C,C)),Uua: fun(B,fun(C,D)),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,fun(C,D)),fun(B,fun(B,$o)),aTP_Lamp_ajy(set(product_prod(C,C)),fun(fun(B,fun(C,D)),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X3: product_prod(C,C)] :
          ( aa(set(product_prod(C,C)),$o,member(product_prod(C,C),X3),Uu)
         => aa(product_prod(C,C),$o,aa(fun(C,fun(C,$o)),fun(product_prod(C,C),$o),product_case_prod(C,C,$o),aa(B,fun(C,fun(C,$o)),aa(B,fun(B,fun(C,fun(C,$o))),aTP_Lamp_ajx(fun(B,fun(C,D)),fun(B,fun(B,fun(C,fun(C,$o)))),Uua),Uub),Uuc)),X3) ) ) ).

% ATP.lambda_1043
tff(fact_9221_ATP_Olambda__1044,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(B,$o),Uua: fun(C,$o),Uub: fun(B,fun(C,D)),Uuc: D] :
      ( aa(D,$o,aa(fun(B,fun(C,D)),fun(D,$o),aa(fun(C,$o),fun(fun(B,fun(C,D)),fun(D,$o)),aTP_Lamp_uq(fun(B,$o),fun(fun(C,$o),fun(fun(B,fun(C,D)),fun(D,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X3: B,Y5: C] :
          ( ( Uuc = aa(C,D,aa(B,fun(C,D),Uub,X3),Y5) )
          & aa(B,$o,Uu,X3)
          & aa(C,$o,Uua,Y5) ) ) ).

% ATP.lambda_1044
tff(fact_9222_ATP_Olambda__1045,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: set(product_prod(C,C)),Uub: fun(B,C),Uuc: product_prod(B,B)] :
      ( aa(product_prod(B,B),$o,aa(fun(B,C),fun(product_prod(B,B),$o),aa(set(product_prod(C,C)),fun(fun(B,C),fun(product_prod(B,B),$o)),aTP_Lamp_agz(set(B),fun(set(product_prod(C,C)),fun(fun(B,C),fun(product_prod(B,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A15: B,A26: B] :
          ( ( Uuc = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A15),A26) )
          & aa(set(B),$o,member(B,A15),Uu)
          & aa(set(B),$o,member(B,A26),Uu)
          & aa(set(product_prod(C,C)),$o,member(product_prod(C,C),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),aa(B,C,Uub,A15)),aa(B,C,Uub,A26))),Uua) ) ) ).

% ATP.lambda_1045
tff(fact_9223_ATP_Olambda__1046,axiom,
    ! [B: $tType,C: $tType,D: $tType,Uu: D,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(D,C))] :
      aa(set(product_prod(D,C)),set(product_prod(D,C)),aa(C,fun(set(product_prod(D,C)),set(product_prod(D,C))),aa(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C))))),aTP_Lamp_acf(D,fun(B,fun(B,fun(C,fun(set(product_prod(D,C)),set(product_prod(D,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(D,C)),set(product_prod(D,C)),insert(product_prod(D,C),aa(C,product_prod(D,C),aa(D,fun(C,product_prod(D,C)),product_Pair(D,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1046
tff(fact_9224_ATP_Olambda__1047,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: B,Uua: C,Uub: C,Uuc: D,Uud: set(product_prod(B,D))] :
      aa(set(product_prod(B,D)),set(product_prod(B,D)),aa(D,fun(set(product_prod(B,D)),set(product_prod(B,D))),aa(C,fun(D,fun(set(product_prod(B,D)),set(product_prod(B,D)))),aa(C,fun(C,fun(D,fun(set(product_prod(B,D)),set(product_prod(B,D))))),aTP_Lamp_acb(B,fun(C,fun(C,fun(D,fun(set(product_prod(B,D)),set(product_prod(B,D)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(B,D)),set(product_prod(B,D)),insert(product_prod(B,D),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1047
tff(fact_9225_ATP_Olambda__1048,axiom,
    ! [B: $tType,Uu: fun(B,fun(B,$o)),Uua: list(B),Uub: B,Uuc: list(B),Uud: list(B)] : aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(B,fun(list(B),fun(list(B),list(B))),aa(list(B),fun(B,fun(list(B),fun(list(B),list(B)))),aTP_Lamp_rp(fun(B,fun(B,$o)),fun(list(B),fun(B,fun(list(B),fun(list(B),list(B))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(B),list(B),quicksort_by_rel(B,Uu,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),aa(list(B),list(B),quicksort_by_rel(B,Uu,Uua),Uud))),Uuc) ).

% ATP.lambda_1048
tff(fact_9226_ATP_Olambda__1049,axiom,
    ! [B: $tType,E: $tType,D: $tType,F6: $tType,C: $tType,Uu: fun(E,fun(F6,D)),Uua: fun(B,E),Uub: fun(C,F6),Uuc: B,Uud: C] : aa(C,D,aa(B,fun(C,D),aa(fun(C,F6),fun(B,fun(C,D)),aa(fun(B,E),fun(fun(C,F6),fun(B,fun(C,D))),aTP_Lamp_aar(fun(E,fun(F6,D)),fun(fun(B,E),fun(fun(C,F6),fun(B,fun(C,D)))),Uu),Uua),Uub),Uuc),Uud) = aa(F6,D,aa(E,fun(F6,D),Uu,aa(B,E,Uua,Uuc)),aa(C,F6,Uub,Uud)) ).

% ATP.lambda_1049
tff(fact_9227_ATP_Olambda__1050,axiom,
    ! [E: $tType,C: $tType,B: $tType,D: $tType,F6: $tType,Uu: fun(C,fun(D,B)),Uua: fun(E,C),Uub: fun(F6,D),Uuc: E,Uud: F6] : aa(F6,B,aa(E,fun(F6,B),aa(fun(F6,D),fun(E,fun(F6,B)),aa(fun(E,C),fun(fun(F6,D),fun(E,fun(F6,B))),aTP_Lamp_aap(fun(C,fun(D,B)),fun(fun(E,C),fun(fun(F6,D),fun(E,fun(F6,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(D,B,aa(C,fun(D,B),Uu,aa(E,C,Uua,Uuc)),aa(F6,D,Uub,Uud)) ).

% ATP.lambda_1050
tff(fact_9228_ATP_Olambda__1051,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_apm($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uub)),Uuc),(Uud)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_apl($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_1051
tff(fact_9229_ATP_Olambda__1052,axiom,
    ! [C: $tType,B: $tType,D: $tType,Uu: fun(B,fun(D,B)),Uua: fun(B,fun(C,fun(D,C))),Uub: B,Uuc: C,Uud: D] : aa(D,product_prod(B,C),aa(C,fun(D,product_prod(B,C)),aa(B,fun(C,fun(D,product_prod(B,C))),aa(fun(B,fun(C,fun(D,C))),fun(B,fun(C,fun(D,product_prod(B,C)))),aTP_Lamp_tc(fun(B,fun(D,B)),fun(fun(B,fun(C,fun(D,C))),fun(B,fun(C,fun(D,product_prod(B,C))))),Uu),Uua),Uub),Uuc),Uud) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,aa(B,fun(D,B),Uu,Uub),Uud)),aa(D,C,aa(C,fun(D,C),aa(B,fun(C,fun(D,C)),Uua,Uub),Uuc),Uud)) ).

% ATP.lambda_1052
tff(fact_9230_ATP_Olambda__1053,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(B,fun(C,D)),Uua: B,Uub: B,Uuc: C,Uud: C] :
      ( aa(C,$o,aa(C,fun(C,$o),aa(B,fun(C,fun(C,$o)),aa(B,fun(B,fun(C,fun(C,$o))),aTP_Lamp_ajx(fun(B,fun(C,D)),fun(B,fun(B,fun(C,fun(C,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(C,D,aa(B,fun(C,D),Uu,Uua),Uuc) = aa(C,D,aa(B,fun(C,D),Uu,Uub),Uud) ) ) ).

% ATP.lambda_1053
tff(fact_9231_ATP_Olambda__1054,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: nat,Uua: B,Uub: B,Uuc: B,Uud: nat] : aa(nat,B,aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_dc(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),Uud)),Uua)),one_one(B))),Uud)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uub),Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1054
tff(fact_9232_ATP_Olambda__1055,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: nat,Uua: B,Uub: B,Uuc: B,Uud: nat] : aa(nat,B,aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_cx(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(nat,B,semiring_1_of_nat(B),Uu)),Uua)),Uud)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uub),Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1055
tff(fact_9233_ATP_Olambda__1056,axiom,
    ! [B: $tType,Uu: B,Uua: B,Uub: set(product_prod(B,B)),Uuc: B,Uud: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aa(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o))),aTP_Lamp_abr(B,fun(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uu)),transitive_trancl(B,Uub))
          | ( Uuc = Uu ) )
        & ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uua),Uud)),transitive_trancl(B,Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1056
tff(fact_9234_ATP_Olambda__1057,axiom,
    ! [B: $tType] :
      ( field_char_0(B)
     => ! [Uu: nat,Uua: B,Uub: B,Uuc: B,Uud: nat] : aa(nat,B,aa(B,fun(nat,B),aa(B,fun(B,fun(nat,B)),aa(B,fun(B,fun(B,fun(nat,B))),aTP_Lamp_cy(nat,fun(B,fun(B,fun(B,fun(nat,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,gbinomial(B,aa(B,B,uminus_uminus(B),Uua)),Uud)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,uminus_uminus(B),Uub)),Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1057
tff(fact_9235_ATP_Olambda__1058,axiom,
    ! [B: $tType,Uu: B,Uua: B,Uub: set(product_prod(B,B)),Uuc: B,Uud: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(set(product_prod(B,B)),fun(B,fun(B,$o)),aa(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o))),aTP_Lamp_adg(B,fun(B,fun(set(product_prod(B,B)),fun(B,fun(B,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uu)),transitive_rtrancl(B,Uub))
        & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uua),Uud)),transitive_rtrancl(B,Uub)) ) ) ).

% ATP.lambda_1058
tff(fact_9236_ATP_Olambda__1059,axiom,
    ! [B: $tType,Uu: set(product_prod(B,B)),Uua: B,Uub: B,Uuc: B,Uud: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(B,fun(B,fun(B,$o)),aa(B,fun(B,fun(B,fun(B,$o))),aTP_Lamp_akf(set(product_prod(B,B)),fun(B,fun(B,fun(B,fun(B,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( Uub = Uuc )
       => aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uua),Uud)),Uu) ) ) ).

% ATP.lambda_1059
tff(fact_9237_ATP_Olambda__1060,axiom,
    ! [B: $tType,Uu: heap_Time_Heap(B),Uua: heap_ext(product_unit),Uub: B,Uuc: heap_ext(product_unit),Uud: nat] :
      ( aa(nat,$o,aa(heap_ext(product_unit),fun(nat,$o),aa(B,fun(heap_ext(product_unit),fun(nat,$o)),aa(heap_ext(product_unit),fun(B,fun(heap_ext(product_unit),fun(nat,$o))),aTP_Lamp_nc(heap_Time_Heap(B),fun(heap_ext(product_unit),fun(B,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Uu),Uua) = 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)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uuc),Uud))) ) ) ).

% ATP.lambda_1060
tff(fact_9238_ATP_Olambda__1061,axiom,
    ! [D: $tType,C: $tType,B: $tType,F6: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(D,fun(B,$o)),Uub: fun(B,fun(F6,$o)),Uuc: D,Uud: F6] :
      ( aa(F6,$o,aa(D,fun(F6,$o),aa(fun(B,fun(F6,$o)),fun(D,fun(F6,$o)),aa(fun(D,fun(B,$o)),fun(fun(B,fun(F6,$o)),fun(D,fun(F6,$o))),aTP_Lamp_arr(fun(B,fun(C,$o)),fun(fun(D,fun(B,$o)),fun(fun(B,fun(F6,$o)),fun(D,fun(F6,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ? [X3: B] :
          ( aa(set(B),$o,member(B,X3),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(C,$o)),fun(B,$o),domainp(B,C),Uu)))
          & aa(B,$o,aa(D,fun(B,$o),Uua,Uuc),X3)
          & aa(F6,$o,aa(B,fun(F6,$o),Uub,X3),Uud) ) ) ).

% ATP.lambda_1061
tff(fact_9239_ATP_Olambda__1062,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( semiring_0(D)
     => ! [Uu: fun(B,D),Uua: fun(C,D),Uub: set(B),Uuc: set(C),Uud: D] :
          ( aa(D,$o,aa(set(C),fun(D,$o),aa(set(B),fun(set(C),fun(D,$o)),aa(fun(C,D),fun(set(B),fun(set(C),fun(D,$o))),aTP_Lamp_afp(fun(B,D),fun(fun(C,D),fun(set(B),fun(set(C),fun(D,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ? [A6: B,B9: C] :
              ( ( Uud = aa(D,D,aa(D,fun(D,D),times_times(D),aa(B,D,Uu,A6)),aa(C,D,Uua,B9)) )
              & aa(set(B),$o,member(B,A6),Uub)
              & aa(set(C),$o,member(C,B9),Uuc) ) ) ) ).

% ATP.lambda_1062
tff(fact_9240_ATP_Olambda__1063,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_apl($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_apk($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uue))),code_bit_cut_integer(Uud)) ).

% ATP.lambda_1063
tff(fact_9241_ATP_Olambda__1064,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,Uu: fun(B,fun(D,$o)),Uua: fun(C,fun(E,$o)),Uub: B,Uuc: C,Uud: D,Uue: E] :
      ( aa(E,$o,aa(D,fun(E,$o),aa(C,fun(D,fun(E,$o)),aa(B,fun(C,fun(D,fun(E,$o))),aa(fun(C,fun(E,$o)),fun(B,fun(C,fun(D,fun(E,$o)))),aTP_Lamp_ara(fun(B,fun(D,$o)),fun(fun(C,fun(E,$o)),fun(B,fun(C,fun(D,fun(E,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa(D,$o,aa(B,fun(D,$o),Uu,Uub),Uud)
        & aa(E,$o,aa(C,fun(E,$o),Uua,Uuc),Uue) ) ) ).

% ATP.lambda_1064
tff(fact_9242_ATP_Olambda__1065,axiom,
    ! [B: $tType,C: $tType,Uu: set(product_prod(B,B)),Uua: set(product_prod(C,C)),Uub: B,Uuc: C,Uud: B,Uue: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(C,fun(B,fun(C,$o)),aa(B,fun(C,fun(B,fun(C,$o))),aa(set(product_prod(C,C)),fun(B,fun(C,fun(B,fun(C,$o)))),aTP_Lamp_aaf(set(product_prod(B,B)),fun(set(product_prod(C,C)),fun(B,fun(C,fun(B,fun(C,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uub),Uud)),Uu)
        | ( ( Uub = Uud )
          & aa(set(product_prod(C,C)),$o,member(product_prod(C,C),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),Uuc),Uue)),Uua) ) ) ) ).

% ATP.lambda_1065
tff(fact_9243_ATP_Olambda__1066,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,set(product_prod(C,C))),Uub: B,Uuc: C,Uud: B,Uue: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aa(C,fun(B,fun(C,$o)),aa(B,fun(C,fun(B,fun(C,$o))),aa(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(B,fun(C,$o)))),aTP_Lamp_aco(fun(B,$o),fun(fun(B,set(product_prod(C,C))),fun(B,fun(C,fun(B,fun(C,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(B,$o,Uu,Uud)
        & aa(set(product_prod(C,C)),$o,member(product_prod(C,C),aa(C,product_prod(C,C),aa(C,fun(C,product_prod(C,C)),product_Pair(C,C),Uuc),Uue)),aa(B,set(product_prod(C,C)),Uua,Uud)) ) ) ).

% ATP.lambda_1066
tff(fact_9244_ATP_Olambda__1067,axiom,
    ! [Uu: fun(b,c),Uua: c,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(c,fun(b,fun(list(b),fun(list(b),fun(list(b),product_prod(list(b),product_prod(list(b),list(b))))))),aTP_Lamp_mo(fun(b,c),fun(c,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) = $let(
        x2: c,
        x2:= aa(b,c,Uu,Uub),
        $ite(
          aa(c,$o,aa(c,fun(c,$o),ord_less(c),x2),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)),
          $ite(aa(c,$o,aa(c,fun(c,$o),ord_less(c),Uua),x2),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_1067
tff(fact_9245_ATP_Olambda__1068,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_apk($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_apj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf))),code_bit_cut_integer(Uue)) ).

% ATP.lambda_1068
tff(fact_9246_ATP_Olambda__1069,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_api($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_1069
tff(fact_9247_ATP_Olambda__1070,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_apj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_api($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug))),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_1070
tff(fact_9248_ATP_Olambda__1071,axiom,
    ! [C: $tType,B: $tType,Uu: $o,Uua: B,Uub: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_vj($o,fun(B,fun(C,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1071
tff(fact_9249_ATP_Olambda__1072,axiom,
    ! [B: $tType,Uu: B,Uua: heap_ext(product_unit),Uub: nat] : aa(nat,B,aa(heap_ext(product_unit),fun(nat,B),aTP_Lamp_nh(B,fun(heap_ext(product_unit),fun(nat,B)),Uu),Uua),Uub) = Uu ).

% ATP.lambda_1072
tff(fact_9250_ATP_Olambda__1073,axiom,
    ! [B: $tType,Uu: heap_Time_Heap(B),Uua: product_unit] : aa(product_unit,heap_Time_Heap(B),aTP_Lamp_ac(heap_Time_Heap(B),fun(product_unit,heap_Time_Heap(B)),Uu),Uua) = Uu ).

% ATP.lambda_1073
tff(fact_9251_ATP_Olambda__1074,axiom,
    ! [C: $tType,B: $tType] :
      ( heap(C)
     => ! [Uu: heap_Time_Heap(B),Uua: C] : aa(C,heap_Time_Heap(B),aTP_Lamp_ak(heap_Time_Heap(B),fun(C,heap_Time_Heap(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_1074
tff(fact_9252_ATP_Olambda__1075,axiom,
    ! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,heap_ext(product_unit),aa(heap_ext(product_unit),fun(nat,heap_ext(product_unit)),aTP_Lamp_nk(heap_ext(product_unit),fun(nat,heap_ext(product_unit))),Uu),Uua) = Uu ).

% ATP.lambda_1075
tff(fact_9253_ATP_Olambda__1076,axiom,
    ! [B: $tType,C: $tType,Uu: multiset(C),Uua: B] : aa(B,multiset(C),aTP_Lamp_xl(multiset(C),fun(B,multiset(C)),Uu),Uua) = Uu ).

% ATP.lambda_1076
tff(fact_9254_ATP_Olambda__1077,axiom,
    ! [B: $tType,Uu: $o,Uua: B] :
      ( aa(B,$o,aTP_Lamp_vi($o,fun(B,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1077
tff(fact_9255_ATP_Olambda__1078,axiom,
    ! [D: $tType,E: $tType,Uu: set(E),Uua: D] : aa(D,set(E),aTP_Lamp_wj(set(E),fun(D,set(E)),Uu),Uua) = Uu ).

% ATP.lambda_1078
tff(fact_9256_ATP_Olambda__1079,axiom,
    ! [C: $tType,E: $tType,Uu: set(E),Uua: C] : aa(C,set(E),aTP_Lamp_aas(set(E),fun(C,set(E)),Uu),Uua) = Uu ).

% ATP.lambda_1079
tff(fact_9257_ATP_Olambda__1080,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_wu(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1080
tff(fact_9258_ATP_Olambda__1081,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_wv(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1081
tff(fact_9259_ATP_Olambda__1082,axiom,
    ! [C: $tType,Uu: set(C),Uua: C] : aa(C,set(C),aTP_Lamp_vz(set(C),fun(C,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1082
tff(fact_9260_ATP_Olambda__1083,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_vu(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1083
tff(fact_9261_ATP_Olambda__1084,axiom,
    ! [B: $tType,Uu: set(B),Uua: list(B)] : aa(list(B),set(B),aTP_Lamp_xi(set(B),fun(list(B),set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1084
tff(fact_9262_ATP_Olambda__1085,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_vx(set(B),fun(C,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1085
tff(fact_9263_ATP_Olambda__1086,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_zv(set(B),fun(B,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1086
tff(fact_9264_ATP_Olambda__1087,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,$o),Uua: B] : aa(B,fun(C,$o),aTP_Lamp_aoz(fun(C,$o),fun(B,fun(C,$o)),Uu),Uua) = Uu ).

% ATP.lambda_1087
tff(fact_9265_ATP_Olambda__1088,axiom,
    ! [B: $tType,D: $tType,C: $tType,Uu: fun(C,D),Uua: B] : aa(B,fun(C,D),aTP_Lamp_ti(fun(C,D),fun(B,fun(C,D)),Uu),Uua) = Uu ).

% ATP.lambda_1088
tff(fact_9266_ATP_Olambda__1089,axiom,
    ! [B: $tType,C: $tType,Uu: fun(C,C),Uua: B] : aa(B,fun(C,C),aTP_Lamp_acj(fun(C,C),fun(B,fun(C,C)),Uu),Uua) = Uu ).

% ATP.lambda_1089
tff(fact_9267_ATP_Olambda__1090,axiom,
    ! [C: $tType,D: $tType,B: $tType,Uu: fun(B,fun(D,D)),Uua: C] : aa(C,fun(B,fun(D,D)),aTP_Lamp_sv(fun(B,fun(D,D)),fun(C,fun(B,fun(D,D))),Uu),Uua) = Uu ).

% ATP.lambda_1090
tff(fact_9268_ATP_Olambda__1091,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: C,Uua: B] : aa(B,C,aTP_Lamp_vs(C,fun(B,C),Uu),Uua) = Uu ) ).

% ATP.lambda_1091
tff(fact_9269_ATP_Olambda__1092,axiom,
    ! [B: $tType,C: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [Uu: C,Uua: B] : aa(B,C,aTP_Lamp_xq(C,fun(B,C),Uu),Uua) = Uu ) ).

% ATP.lambda_1092
tff(fact_9270_ATP_Olambda__1093,axiom,
    ! [B: $tType,C: $tType] :
      ( linorder(C)
     => ! [Uu: C,Uua: B] : aa(B,C,aTP_Lamp_ok(C,fun(B,C),Uu),Uua) = Uu ) ).

% ATP.lambda_1093
tff(fact_9271_ATP_Olambda__1094,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: B] : aa(B,C,aTP_Lamp_iu(C,fun(B,C),Uu),Uua) = Uu ).

% ATP.lambda_1094
tff(fact_9272_ATP_Olambda__1095,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_yn(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1095
tff(fact_9273_ATP_Olambda__1096,axiom,
    ! [B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: B] : aa(B,B,aTP_Lamp_adz(B,fun(B,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1096
tff(fact_9274_ATP_Olambda__1097,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_gn(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1097
tff(fact_9275_ATP_Olambda__1098,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: list(B)] : aa(list(B),B,aa(B,fun(list(B),B),aTP_Lamp_pu(B,fun(list(B),B)),Uu),Uua) = Uu ) ).

% ATP.lambda_1098
tff(fact_9276_ATP_Olambda__1099,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_go(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1099
tff(fact_9277_ATP_Olambda__1100,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_au(B,fun(nat,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1100
tff(fact_9278_ATP_Olambda__1101,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),B,aa(B,fun(list(B),B),aTP_Lamp_ol(B,fun(list(B),B)),Uu),Uua) = Uu ).

% ATP.lambda_1101
tff(fact_9279_ATP_Olambda__1102,axiom,
    ! [B: $tType,Uu: B,Uua: nat] : aa(nat,B,aTP_Lamp_qq(B,fun(nat,B),Uu),Uua) = Uu ).

% ATP.lambda_1102
tff(fact_9280_ATP_Olambda__1103,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_it(B,fun(C,B)),Uu),Uua) = Uu ).

% ATP.lambda_1103
tff(fact_9281_ATP_Olambda__1104,axiom,
    ! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aa(heap_ext(product_unit),fun(nat,nat),aTP_Lamp_my(heap_ext(product_unit),fun(nat,nat)),Uu),Uua) = Uua ).

% ATP.lambda_1104
tff(fact_9282_ATP_Olambda__1105,axiom,
    ! [C: $tType,B: $tType,Uu: C,Uua: B] : aa(B,B,aa(C,fun(B,B),aTP_Lamp_tg(C,fun(B,B)),Uu),Uua) = Uua ).

% ATP.lambda_1105
tff(fact_9283_ATP_Olambda__1106,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),aTP_Lamp_qz(B,fun(list(B),list(B))),Uu),Uua) = Uua ).

% ATP.lambda_1106
tff(fact_9284_ATP_Olambda__1107,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] :
      ( aa(list(B),$o,aa(B,fun(list(B),$o),aTP_Lamp_pe(B,fun(list(B),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_1107
tff(fact_9285_ATP_Olambda__1108,axiom,
    ! [B: $tType,Uu: B,Uua: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aqr(B,fun(B,$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_1108
tff(fact_9286_ATP_Olambda__1109,axiom,
    ! [B: $tType,Uu: B,Uua: list(B)] :
      ( aa(list(B),$o,aa(B,fun(list(B),$o),aTP_Lamp_pd(B,fun(list(B),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1109
tff(fact_9287_ATP_Olambda__1110,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] :
      ( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_lv(B,fun(C,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1110
tff(fact_9288_ATP_Olambda__1111,axiom,
    ! [B: $tType,Uu: B,Uua: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_ay(B,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1111
tff(fact_9289_ATP_Olambda__1112,axiom,
    ! [Uu: product_prod(nat,nat)] : aa(product_prod(nat,nat),product_prod(nat,nat),aTP_Lamp_aqv(product_prod(nat,nat),product_prod(nat,nat)),Uu) = Uu ).

% ATP.lambda_1112
tff(fact_9290_ATP_Olambda__1113,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_da(nat,nat),Uu) = Uu ).

% ATP.lambda_1113
tff(fact_9291_ATP_Olambda__1114,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_dk(int,int),Uu) = Uu ).

% ATP.lambda_1114
tff(fact_9292_ATP_Olambda__1115,axiom,
    ! [B: $tType,Uu: fun(B,nat)] : aa(fun(B,nat),fun(B,nat),aTP_Lamp_aje(fun(B,nat),fun(B,nat)),Uu) = Uu ).

% ATP.lambda_1115
tff(fact_9293_ATP_Olambda__1116,axiom,
    ! [C: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_afb(C,C),Uu) = Uu ) ).

% ATP.lambda_1116
tff(fact_9294_ATP_Olambda__1117,axiom,
    ! [C: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_afc(C,C),Uu) = Uu ) ).

% ATP.lambda_1117
tff(fact_9295_ATP_Olambda__1118,axiom,
    ! [C: $tType,Uu: C] : aa(C,C,aTP_Lamp_rv(C,C),Uu) = Uu ).

% ATP.lambda_1118
tff(fact_9296_ATP_Olambda__1119,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_rc(B,B),Uu) = Uu ) ).

% ATP.lambda_1119
tff(fact_9297_ATP_Olambda__1120,axiom,
    ! [B: $tType] :
      ( complete_Sup(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_xo(B,B),Uu) = Uu ) ).

% ATP.lambda_1120
tff(fact_9298_ATP_Olambda__1121,axiom,
    ! [B: $tType] :
      ( complete_Inf(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_yy(B,B),Uu) = Uu ) ).

% ATP.lambda_1121
tff(fact_9299_ATP_Olambda__1122,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_hr(B,B),Uu) = Uu ) ).

% ATP.lambda_1122
tff(fact_9300_ATP_Olambda__1123,axiom,
    ! [B: $tType] :
      ( monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_at(B,B),Uu) = Uu ) ).

% ATP.lambda_1123
tff(fact_9301_ATP_Olambda__1124,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_il(B,B),Uu) = Uu ).

% ATP.lambda_1124
tff(fact_9302_ATP_Olambda__1125,axiom,
    ! [Uu: product_unit] : aa(product_unit,heap_Time_Heap(a),aTP_Lamp_a(product_unit,heap_Time_Heap(a)),Uu) = m ).

% ATP.lambda_1125
tff(fact_9303_ATP_Olambda__1126,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: C] : aa(C,B,aTP_Lamp_za(C,B),Uu) = top_top(B) ) ).

% ATP.lambda_1126
tff(fact_9304_ATP_Olambda__1127,axiom,
    ! [B: $tType,C: $tType,Uu: B] : aa(B,set(C),aTP_Lamp_vv(B,set(C)),Uu) = top_top(set(C)) ).

% ATP.lambda_1127
tff(fact_9305_ATP_Olambda__1128,axiom,
    ! [D: $tType,C: $tType,Uu: D] : aa(D,set(C),aTP_Lamp_kd(D,set(C)),Uu) = bot_bot(set(C)) ).

% ATP.lambda_1128
tff(fact_9306_ATP_Olambda__1129,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_yg(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1129
tff(fact_9307_ATP_Olambda__1130,axiom,
    ! [C: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: C] : aa(C,B,aTP_Lamp_xp(C,B),Uu) = bot_bot(B) ) ).

% ATP.lambda_1130
tff(fact_9308_ATP_Olambda__1131,axiom,
    ! [B: $tType,E: $tType,Uu: B] : aa(B,set(E),aTP_Lamp_ke(B,set(E)),Uu) = bot_bot(set(E)) ).

% ATP.lambda_1131
tff(fact_9309_ATP_Olambda__1132,axiom,
    ! [B: $tType,C: $tType,Uu: B] : aa(B,set(C),aTP_Lamp_vt(B,set(C)),Uu) = bot_bot(set(C)) ).

% ATP.lambda_1132
tff(fact_9310_ATP_Olambda__1133,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: C] : aa(C,B,aTP_Lamp_bj(C,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1133
tff(fact_9311_ATP_Olambda__1134,axiom,
    ! [C: $tType,B: $tType] :
      ( monoid_add(B)
     => ! [Uu: C] : aa(C,B,aTP_Lamp_qy(C,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1134
tff(fact_9312_ATP_Olambda__1135,axiom,
    ! [B: $tType] :
      ( mult_zero(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_as(B,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1135
tff(fact_9313_ATP_Olambda__1136,axiom,
    ! [B: $tType,Uu: B] : aa(B,nat,aTP_Lamp_am(B,nat),Uu) = zero_zero(nat) ).

% ATP.lambda_1136
tff(fact_9314_ATP_Olambda__1137,axiom,
    ! [B: $tType,C: $tType] :
      ( zero(C)
     => ! [Uu: B] : aa(B,C,aTP_Lamp_ad(B,C),Uu) = zero_zero(C) ) ).

% ATP.lambda_1137
tff(fact_9315_ATP_Olambda__1138,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: C] : aa(C,B,aTP_Lamp_dm(C,B),Uu) = one_one(B) ) ).

% ATP.lambda_1138
tff(fact_9316_ATP_Olambda__1139,axiom,
    ! [B: $tType,Uu: B] : aa(B,nat,aTP_Lamp_gr(B,nat),Uu) = one_one(nat) ).

% ATP.lambda_1139
tff(fact_9317_ATP_Olambda__1140,axiom,
    ! [B: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nj(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1140
tff(fact_9318_ATP_Olambda__1141,axiom,
    ! [D: $tType,C: $tType,Uu: D] : aa(D,option(C),aTP_Lamp_aqk(D,option(C)),Uu) = none(C) ).

% ATP.lambda_1141
tff(fact_9319_ATP_Olambda__1142,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,option(B),aTP_Lamp_jz(C,option(B)),Uu) = none(B) ).

% ATP.lambda_1142
tff(fact_9320_ATP_Olambda__1143,axiom,
    ! [B: $tType,D: $tType,Uu: B] : aa(B,option(D),aTP_Lamp_kb(B,option(D)),Uu) = none(D) ).

% ATP.lambda_1143
tff(fact_9321_ATP_Olambda__1144,axiom,
    ! [B: $tType,C: $tType,Uu: B] : aa(B,option(C),aTP_Lamp_jv(B,option(C)),Uu) = none(C) ).

% ATP.lambda_1144
tff(fact_9322_ATP_Olambda__1145,axiom,
    ! [B: $tType,C: $tType,Uu: B] : aa(B,C,aTP_Lamp_afy(B,C),Uu) = undefined(C) ).

% ATP.lambda_1145
tff(fact_9323_ATP_Olambda__1146,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_fl(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1146
tff(fact_9324_ATP_Olambda__1147,axiom,
    ! [B: $tType,Uu: B] :
      ( aa(B,$o,aTP_Lamp_hh(B,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1147
tff(fact_9325_ATP_Olambda__1148,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_fm(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1148
tff(fact_9326_ATP_Olambda__1149,axiom,
    ! [B: $tType,Uu: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aTP_Lamp_amu(fun(B,$o),$o),Uu)
    <=> $true ) ).

% ATP.lambda_1149
tff(fact_9327_ATP_Olambda__1150,axiom,
    ! [D: $tType,Uu: D] :
      ( aa(D,$o,aTP_Lamp_alk(D,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1150
tff(fact_9328_ATP_Olambda__1151,axiom,
    ! [C: $tType,Uu: C] :
      ( aa(C,$o,aTP_Lamp_alj(C,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1151
tff(fact_9329_ATP_Olambda__1152,axiom,
    ! [B: $tType,Uu: B] :
      ( aa(B,$o,aTP_Lamp_jq(B,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1152
tff(fact_9330_ATP_Olambda__1153,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_ar(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1153

% Type constructors (677)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_1,axiom,
    ! [A16: $tType] :
      ( bounded_lattice_top(A16)
     => bounded_lattice_top(option(A16)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_2,axiom,
    ! [A16: $tType] : bounded_lattice_top(filter(A16)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_3,axiom,
    bounded_lattice_top($o) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_4,axiom,
    ! [A16: $tType] : bounded_lattice_top(set(A16)) ).

tff(tcon_fun___Lattices_Obounded__lattice__top_5,axiom,
    ! [A16: $tType,A17: $tType] :
      ( bounded_lattice(A17)
     => bounded_lattice_top(fun(A16,A17)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A16: $tType,A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => condit1219197933456340205attice(fun(A16,A17)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A16: $tType,A17: $tType] :
      ( comple592849572758109894attice(A17)
     => comple592849572758109894attice(fun(A16,A17)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A16: $tType,A17: $tType] :
      ( comple489889107523837845lgebra(A17)
     => comple489889107523837845lgebra(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A16: $tType,A17: $tType] :
      ( bounded_lattice(A17)
     => bounde4967611905675639751up_bot(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A16: $tType,A17: $tType] :
      ( bounded_lattice(A17)
     => bounde4346867609351753570nf_top(fun(A16,A17)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A16: $tType,A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => comple6319245703460814977attice(fun(A16,A17)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A16: $tType,A17: $tType] :
      ( boolea8198339166811842893lgebra(A17)
     => boolea8198339166811842893lgebra(fun(A16,A17)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A16: $tType,A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => comple9053668089753744459l_ccpo(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A16: $tType,A17: $tType] :
      ( semilattice_sup(A17)
     => semilattice_sup(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A16: $tType,A17: $tType] :
      ( semilattice_inf(A17)
     => semilattice_inf(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A16: $tType,A17: $tType] :
      ( distrib_lattice(A17)
     => distrib_lattice(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A16: $tType,A17: $tType] :
      ( bounded_lattice(A17)
     => bounded_lattice(fun(A16,A17)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A16: $tType,A17: $tType] :
      ( complete_Sup(A17)
     => complete_Sup(fun(A16,A17)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A16: $tType,A17: $tType] :
      ( complete_Inf(A17)
     => complete_Inf(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A16: $tType,A17: $tType] :
      ( order_top(A17)
     => order_top(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A16: $tType,A17: $tType] :
      ( order_bot(A17)
     => order_bot(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A16: $tType,A17: $tType] :
      ( preorder(A17)
     => preorder(fun(A16,A17)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A16: $tType,A17: $tType] :
      ( ( finite_finite(A16)
        & finite_finite(A17) )
     => finite_finite(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A16: $tType,A17: $tType] :
      ( lattice(A17)
     => lattice(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A16: $tType,A17: $tType] :
      ( order(A17)
     => order(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A16: $tType,A17: $tType] :
      ( top(A17)
     => top(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A16: $tType,A17: $tType] :
      ( ord(A17)
     => ord(fun(A16,A17)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A16: $tType,A17: $tType] :
      ( bot(A17)
     => bot(fun(A16,A17)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A16: $tType,A17: $tType] :
      ( uminus(A17)
     => uminus(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Osup,axiom,
    ! [A16: $tType,A17: $tType] :
      ( semilattice_sup(A17)
     => sup(fun(A16,A17)) ) ).

tff(tcon_fun___Lattices_Oinf,axiom,
    ! [A16: $tType,A17: $tType] :
      ( semilattice_inf(A17)
     => inf(fun(A16,A17)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,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___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___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___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___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_7,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_8,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_9,axiom,
    distrib_lattice(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OSup_10,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_11,axiom,
    complete_Inf(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_12,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_13,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_14,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_15,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_16,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_17,axiom,
    sup(int) ).

tff(tcon_Int_Oint___Lattices_Oinf_18,axiom,
    inf(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___Heap_Oheap,axiom,
    heap(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_21,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_22,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_23,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_24,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_25,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_26,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_27,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_28,axiom,
    semiri6575147826004484403cancel(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___Rings_Osemiring__no__zero__divisors_41,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_42,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_43,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_44,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_45,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_46,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_47,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_48,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,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_61,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_69,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_70,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_71,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_72,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_73,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_74,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_75,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_76,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_77,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_78,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_79,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_80,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_81,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_82,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_83,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_84,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_85,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_86,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_87,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_88,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_89,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_90,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_91,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_92,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_93,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_94,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_95,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Lattices_Osup_96,axiom,
    sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Oinf_97,axiom,
    inf(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_98,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Power_Opower_99,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_100,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_101,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_102,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_103,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_104,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_105,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_106,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_107,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_108,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_109,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_110,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_111,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_112,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_113,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_114,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_115,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_116,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_117,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_118,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_119,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_120,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_121,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_122,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_123,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_124,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_125,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_126,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_127,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_128,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_129,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_130,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_131,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_132,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_133,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_134,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_135,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_136,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_137,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_138,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_139,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_140,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_141,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_142,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_143,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_144,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_145,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_146,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_147,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_148,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_149,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_150,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_151,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_152,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_153,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_154,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_155,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_156,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_157,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_158,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_159,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_160,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_161,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_162,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_163,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_164,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_165,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_166,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_167,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_168,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_169,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_170,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_171,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_172,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_173,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_174,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_175,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_176,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_177,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_178,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_179,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_180,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_181,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_182,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_183,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_184,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_185,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_186,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_187,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_188,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_189,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_190,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_191,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Lattices_Osup_192,axiom,
    sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Oinf_193,axiom,
    inf(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_194,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_195,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_196,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_197,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_198,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_199,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_200,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_201,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_202,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_203,axiom,
    ! [A16: $tType] : condit1219197933456340205attice(set(A16)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_204,axiom,
    ! [A16: $tType] : comple592849572758109894attice(set(A16)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_205,axiom,
    ! [A16: $tType] : comple489889107523837845lgebra(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_206,axiom,
    ! [A16: $tType] : bounde4967611905675639751up_bot(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_207,axiom,
    ! [A16: $tType] : bounde4346867609351753570nf_top(set(A16)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_208,axiom,
    ! [A16: $tType] : comple6319245703460814977attice(set(A16)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_209,axiom,
    ! [A16: $tType] : boolea8198339166811842893lgebra(set(A16)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_210,axiom,
    ! [A16: $tType] : comple9053668089753744459l_ccpo(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_211,axiom,
    ! [A16: $tType] : semilattice_sup(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_212,axiom,
    ! [A16: $tType] : semilattice_inf(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_213,axiom,
    ! [A16: $tType] : distrib_lattice(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_214,axiom,
    ! [A16: $tType] : bounded_lattice(set(A16)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_215,axiom,
    ! [A16: $tType] : complete_Sup(set(A16)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_216,axiom,
    ! [A16: $tType] : complete_Inf(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_217,axiom,
    ! [A16: $tType] : order_top(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_218,axiom,
    ! [A16: $tType] : order_bot(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_219,axiom,
    ! [A16: $tType] : preorder(set(A16)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_220,axiom,
    ! [A16: $tType] :
      ( finite_finite(A16)
     => finite_finite(set(A16)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_221,axiom,
    ! [A16: $tType] : lattice(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Oorder_222,axiom,
    ! [A16: $tType] : order(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Otop_223,axiom,
    ! [A16: $tType] : top(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Oord_224,axiom,
    ! [A16: $tType] : ord(set(A16)) ).

tff(tcon_Set_Oset___Orderings_Obot_225,axiom,
    ! [A16: $tType] : bot(set(A16)) ).

tff(tcon_Set_Oset___Groups_Ouminus_226,axiom,
    ! [A16: $tType] : uminus(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Osup_227,axiom,
    ! [A16: $tType] : sup(set(A16)) ).

tff(tcon_Set_Oset___Lattices_Oinf_228,axiom,
    ! [A16: $tType] : inf(set(A16)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_229,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_230,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_231,axiom,
    comple489889107523837845lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_232,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_233,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_234,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_235,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_236,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_237,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_238,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_239,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_240,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_241,axiom,
    complete_Sup($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_242,axiom,
    complete_Inf($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_243,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_244,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_245,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_246,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_247,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_248,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_249,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_250,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_251,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_252,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_253,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Lattices_Osup_254,axiom,
    sup($o) ).

tff(tcon_HOL_Obool___Lattices_Oinf_255,axiom,
    inf($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_256,axiom,
    heap($o) ).

tff(tcon_List_Olist___Heap_Oheap_257,axiom,
    ! [A16: $tType] :
      ( heap(A16)
     => heap(list(A16)) ) ).

tff(tcon_List_Olist___Nat_Osize_258,axiom,
    ! [A16: $tType] : size(list(A16)) ).

tff(tcon_Heap_Oarray___Heap_Oheap_259,axiom,
    ! [A16: $tType] : heap(array(A16)) ).

tff(tcon_Heap_Oarray___Nat_Osize_260,axiom,
    ! [A16: $tType] : size(array(A16)) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_261,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Heap_Oheap_262,axiom,
    heap(char) ).

tff(tcon_String_Ochar___Nat_Osize_263,axiom,
    size(char) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_264,axiom,
    ! [A16: $tType] : condit1219197933456340205attice(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_265,axiom,
    ! [A16: $tType] : bounde4967611905675639751up_bot(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_266,axiom,
    ! [A16: $tType] : bounde4346867609351753570nf_top(filter(A16)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_267,axiom,
    ! [A16: $tType] : comple6319245703460814977attice(filter(A16)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_268,axiom,
    ! [A16: $tType] : comple9053668089753744459l_ccpo(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_269,axiom,
    ! [A16: $tType] : semilattice_sup(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_270,axiom,
    ! [A16: $tType] : semilattice_inf(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_271,axiom,
    ! [A16: $tType] : distrib_lattice(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_272,axiom,
    ! [A16: $tType] : bounded_lattice(filter(A16)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_273,axiom,
    ! [A16: $tType] : complete_Sup(filter(A16)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_274,axiom,
    ! [A16: $tType] : complete_Inf(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_275,axiom,
    ! [A16: $tType] : order_top(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_276,axiom,
    ! [A16: $tType] : order_bot(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_277,axiom,
    ! [A16: $tType] : preorder(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_278,axiom,
    ! [A16: $tType] : lattice(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_279,axiom,
    ! [A16: $tType] : order(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_280,axiom,
    ! [A16: $tType] : top(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_281,axiom,
    ! [A16: $tType] : ord(filter(A16)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_282,axiom,
    ! [A16: $tType] : bot(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Osup_283,axiom,
    ! [A16: $tType] : sup(filter(A16)) ).

tff(tcon_Filter_Ofilter___Lattices_Oinf_284,axiom,
    ! [A16: $tType] : inf(filter(A16)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_285,axiom,
    ! [A16: $tType] :
      ( comple5582772986160207858norder(A16)
     => condit6923001295902523014norder(option(A16)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_286,axiom,
    ! [A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => condit1219197933456340205attice(option(A16)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_287,axiom,
    ! [A16: $tType] :
      ( comple592849572758109894attice(A16)
     => comple592849572758109894attice(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_288,axiom,
    ! [A16: $tType] :
      ( lattice(A16)
     => bounde4967611905675639751up_bot(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_289,axiom,
    ! [A16: $tType] :
      ( bounded_lattice_top(A16)
     => bounde4346867609351753570nf_top(option(A16)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A16: $tType] :
      ( comple5582772986160207858norder(A16)
     => comple5582772986160207858norder(option(A16)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_290,axiom,
    ! [A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple6319245703460814977attice(option(A16)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_291,axiom,
    ! [A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple9053668089753744459l_ccpo(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_292,axiom,
    ! [A16: $tType] :
      ( semilattice_sup(A16)
     => semilattice_sup(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_293,axiom,
    ! [A16: $tType] :
      ( semilattice_inf(A16)
     => semilattice_inf(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_294,axiom,
    ! [A16: $tType] :
      ( distrib_lattice(A16)
     => distrib_lattice(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_295,axiom,
    ! [A16: $tType] :
      ( bounded_lattice_top(A16)
     => bounded_lattice(option(A16)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OSup_296,axiom,
    ! [A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => complete_Sup(option(A16)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OInf_297,axiom,
    ! [A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => complete_Inf(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_298,axiom,
    ! [A16: $tType] :
      ( wellorder(A16)
     => wellorder(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_299,axiom,
    ! [A16: $tType] :
      ( order_top(A16)
     => order_top(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_300,axiom,
    ! [A16: $tType] :
      ( order(A16)
     => order_bot(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_301,axiom,
    ! [A16: $tType] :
      ( preorder(A16)
     => preorder(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_302,axiom,
    ! [A16: $tType] :
      ( linorder(A16)
     => linorder(option(A16)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_303,axiom,
    ! [A16: $tType] :
      ( finite_finite(A16)
     => finite_finite(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_304,axiom,
    ! [A16: $tType] :
      ( lattice(A16)
     => lattice(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_305,axiom,
    ! [A16: $tType] :
      ( order(A16)
     => order(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_306,axiom,
    ! [A16: $tType] :
      ( order_top(A16)
     => top(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_307,axiom,
    ! [A16: $tType] :
      ( preorder(A16)
     => ord(option(A16)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_308,axiom,
    ! [A16: $tType] :
      ( order(A16)
     => bot(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Osup_309,axiom,
    ! [A16: $tType] :
      ( sup(A16)
     => sup(option(A16)) ) ).

tff(tcon_Option_Ooption___Lattices_Oinf_310,axiom,
    ! [A16: $tType] :
      ( inf(A16)
     => inf(option(A16)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_311,axiom,
    ! [A16: $tType] :
      ( heap(A16)
     => heap(option(A16)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_312,axiom,
    ! [A16: $tType] : size(option(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_313,axiom,
    ! [A16: $tType] :
      ( preorder(A16)
     => ordere6658533253407199908up_add(multiset(A16)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_314,axiom,
    ! [A16: $tType] : cancel2418104881723323429up_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_315,axiom,
    ! [A16: $tType] : cancel1802427076303600483id_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_316,axiom,
    ! [A16: $tType] : cancel_semigroup_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_317,axiom,
    ! [A16: $tType] : comm_monoid_diff(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_318,axiom,
    ! [A16: $tType] : ab_semigroup_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_319,axiom,
    ! [A16: $tType] : comm_monoid_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OSup_320,axiom,
    ! [A16: $tType] : complete_Sup(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OInf_321,axiom,
    ! [A16: $tType] : complete_Inf(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_322,axiom,
    ! [A16: $tType] : semigroup_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_323,axiom,
    ! [A16: $tType] :
      ( preorder(A16)
     => preorder(multiset(A16)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_324,axiom,
    ! [A16: $tType] : monoid_add(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_325,axiom,
    ! [A16: $tType] :
      ( preorder(A16)
     => order(multiset(A16)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_326,axiom,
    ! [A16: $tType] :
      ( preorder(A16)
     => ord(multiset(A16)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_327,axiom,
    ! [A16: $tType] : zero(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_328,axiom,
    ! [A16: $tType] : plus(multiset(A16)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_329,axiom,
    ! [A16: $tType] : size(multiset(A16)) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_330,axiom,
    ! [A16: $tType,A17: $tType] :
      ( ( finite_finite(A16)
        & finite_finite(A17) )
     => finite_finite(product_prod(A16,A17)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_331,axiom,
    ! [A16: $tType,A17: $tType] :
      ( ( heap(A16)
        & heap(A17) )
     => heap(product_prod(A16,A17)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_332,axiom,
    ! [A16: $tType,A17: $tType] : size(product_prod(A16,A17)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_333,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_334,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_335,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_336,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_337,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_338,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_339,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_340,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_341,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_342,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_343,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_344,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_345,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_346,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_347,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_348,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_349,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_350,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_351,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_352,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_353,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_354,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_355,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_356,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_357,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_358,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_359,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_360,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osup_361,axiom,
    sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Oinf_362,axiom,
    inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_363,axiom,
    heap(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_364,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_365,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_366,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_367,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_368,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_369,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_370,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_371,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_372,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_373,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_374,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_375,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_376,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_377,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_378,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_379,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_380,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_381,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_382,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_383,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_384,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_385,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_386,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_387,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_388,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_389,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_390,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_391,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_392,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_393,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_394,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_395,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_396,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_397,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_398,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_399,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_400,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_401,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_402,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_403,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_404,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_405,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_406,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_407,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_408,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_409,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_410,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_411,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_412,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_413,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_414,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_415,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_416,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_417,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_418,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_419,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_420,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_421,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_422,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_423,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_424,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_425,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_426,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_427,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_428,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_429,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_430,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_431,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_432,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_433,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_434,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_435,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_436,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_437,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_438,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_439,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_440,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_441,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_442,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_443,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_444,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_445,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_446,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_447,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_448,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_449,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_450,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_451,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_452,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_453,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_454,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_455,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_456,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_457,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_458,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_459,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_460,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_461,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_462,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_463,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_464,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_465,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_466,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_467,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_468,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_469,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_470,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_471,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_472,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_473,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_474,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_475,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_476,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_477,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_478,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_479,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_480,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_481,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_482,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_483,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_484,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_485,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_486,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_487,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_488,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_489,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_490,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_491,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_492,axiom,
    ab_semigroup_mult(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___Groups_Osemigroup__mult_504,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_505,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_506,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_507,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_508,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_509,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_510,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_511,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_512,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_513,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_514,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_515,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_516,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_517,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_518,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_519,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_520,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_521,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom_522,axiom,
    semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_523,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Otimes_524,axiom,
    times(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_525,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_526,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_527,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_528,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_529,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_530,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_531,axiom,
    size(code_natural) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_532,axiom,
    ! [A16: $tType] : size(heap_Time_Heap(A16)) ).

% Helper facts (9)
tff(help_If_3_1_T,axiom,
    ! [P8: $o] :
      ( ( (P8) = $true )
      | ( (P8) = $false ) ) ).

tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : aa(A,A,aa(A,fun(A,A),aa($o,fun(A,fun(A,A)),if(A),$false),X),Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : aa(A,A,aa(A,fun(A,A),aa($o,fun(A,fun(A,A)),if(A),$true),X),Y) = X ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P8: fun(A,$o),X: A] :
      ( ~ aa(fun(A,$o),$o,fAll(A),P8)
      | aa(A,$o,P8,X) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P8: $o] :
      ( (P8)
      | aa($o,$o,fNot,(P8)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P8: $o] :
      ( ~ aa($o,$o,fNot,(P8))
      | ~ (P8) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y)
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P8: fun(A,$o)] :
      ( aa(A,$o,P8,fChoice(A,P8))
      = ( ? [X7: A] : aa(A,$o,P8,X7) ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    heap(b) ).

tff(tfree_1,hypothesis,
    linorder(c) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(a,heap_Time_bind(product_unit,a,refine_Imp_assert(p),aTP_Lamp_a(product_unit,heap_Time_Heap(a))),h)),t) ).

%------------------------------------------------------------------------------